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Stöhr M, Koch ER, Javaloyes J, Gurevich SV, Wolfrum M. Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer. CHAOS (WOODBURY, N.Y.) 2023; 33:113105. [PMID: 37909903 DOI: 10.1063/5.0173320] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2023] [Accepted: 10/13/2023] [Indexed: 11/03/2023]
Abstract
We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.
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Affiliation(s)
- Mina Stöhr
- Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany
| | - Elias R Koch
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - Julien Javaloyes
- Departament de Física & IAC-3, Universitat de les Illes Balears, C/ Valldemossa km 7.5, 07122 Mallorca, Spain
| | - Svetlana V Gurevich
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), University of Münster, Corrensstrasse 2, 48149 Münster, Germany
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2
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Walker BJ, Townsend AK, Chudasama AK, Krause AL. VisualPDE: Rapid Interactive Simulations of Partial Differential Equations. Bull Math Biol 2023; 85:113. [PMID: 37823924 PMCID: PMC10570185 DOI: 10.1007/s11538-023-01218-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2023] [Accepted: 09/25/2023] [Indexed: 10/13/2023]
Abstract
Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible.
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Affiliation(s)
- Benjamin J Walker
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
- Department of Mathematics, University College London, London, WC1E 6BT, UK
| | - Adam K Townsend
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham, DH1 3LE, UK
| | - Alexander K Chudasama
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham, DH1 3LE, UK
| | - Andrew L Krause
- Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham, DH1 3LE, UK.
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3
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Frohoff-Hülsmann T, Holl MP, Knobloch E, Gurevich SV, Thiele U. Stationary broken parity states in active matter models. Phys Rev E 2023; 107:064210. [PMID: 37464596 DOI: 10.1103/physreve.107.064210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/28/2022] [Accepted: 05/15/2023] [Indexed: 07/20/2023]
Abstract
We demonstrate that several nonvariational continuum models commonly used to describe active matter as well as other active systems exhibit nongeneric behavior: each model supports asymmetric but stationary localized states even in the absence of pinning at heterogeneities. Moreover, such states only begin to drift following a drift-transcritical bifurcation as the activity increases. Asymmetric stationary states should only exist in variational systems, i.e., in models with gradient structure. In other words, such states are expected in passive systems, but not in active systems where the gradient structure of the model is broken by activity. We identify a "spurious" gradient dynamics structure of these models that is responsible for this nongeneric behavior, and determine the types of additional terms that render the models generic, i.e., with asymmetric states that appear via drift-pitchfork bifurcations and are generically moving. We provide detailed illustrations of our results using numerical continuation of resting and steadily drifting states in both generic and nongeneric cases.
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Affiliation(s)
- Tobias Frohoff-Hülsmann
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Max Philipp Holl
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Svetlana V Gurevich
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstrasse 2, 48149 Münster, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstrasse 2, 48149 Münster, Germany
- Center for Multiscale Theory and Computation (CMTC), Westfälische Wilhelms-Universität, Corrensstrasse 40, 48149 Münster, Germany
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4
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Raja M, van Kan A, Foster B, Knobloch E. Collisions of localized patterns in a nonvariational Swift-Hohenberg equation. Phys Rev E 2023; 107:064214. [PMID: 37464667 DOI: 10.1103/physreve.107.064214] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2023] [Accepted: 06/04/2023] [Indexed: 07/20/2023]
Abstract
The cubic-quintic Swift-Hohenberg equation (SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations (DNSs), to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures, derived in the limit of weak symmetry breaking, is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperaturelike control parameter. These collisions are inelastic and result in stationary or traveling structures. Depending on system parameters and the types of structures colliding, the final state may be a simple bound state of the initial LSs, but it can also be longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system, where the free energy of the global pattern state equals that of the trivial state, is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multipulse bound states resulting from collisions lie on isolas in parameter space, disconnected from the trivial solution. In the gradient SH35, such isolas are always of figure-eight shape, but in the present nongradient case they are generically more complex, although the figure-eight shape is preserved in a small subset of cases. Some of these complex isolas are shown to terminate in T-point bifurcations. A reduced model is proposed to describe the interactions between the tails of the LSs. The model consists of two coupled ordinary differential equations (ODEs) capturing the oscillatory nature of SH35 profiles at the linear level. It contains three parameters: two interaction amplitudes and a phase, whose values are deduced from high-resolution DNSs using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy. When nonlinear interactions lead to the creation or deletion of wavelengths, the model performs less well. Finally, we propose an effective signature of a given interaction in terms of net attraction or repulsion relative to free propagation. It is found that interactions can be attractive or repulsive in the net, irrespective of whether the two closest interacting extrema are of the same or opposite signs. Our findings highlight the rich temporal dynamics described by this bistable nonvariational SH35, and show that the interactions in this system can be quantitatively captured, to a significant extent, by a highly reduced ODE model.
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Affiliation(s)
- Mathi Raja
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Adrian van Kan
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Benjamin Foster
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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5
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Groh RMJ, Pirrera A. Probing the stability landscape of cylindrical shells for buckling knockdown factors. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220032. [PMID: 36774959 PMCID: PMC9922549 DOI: 10.1098/rsta.2022.0032] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/31/2022] [Accepted: 11/11/2022] [Indexed: 06/18/2023]
Abstract
The buckling response of axially compressed cylindrical shells is well known for its imperfection sensitivity. Mapping out a stability landscape by localized probing has recently been proposed as a rational means for establishing shell buckling knockdown factors. Probing using a point force directed radially inwards and perpendicular to the cylinder wall is based on the insight that a localized single dimple exists as an edge state in the basin boundary of the stable prebuckling equilibrium. Here, we extend the idea of probing to bi-directional inwards and outwards forces to trigger both single-dimple and double-dimple edge states. We identify key features of the ensuing probing stability landscape and generalize these to derive three design curves of varying conservatism that are a function of the non-dimensional Batdorf parameter only. Interestingly, the most conservative of the three knockdown curves bounds a large dataset of experimental buckling results from below, despite being derived from probing features of geometrically perfect cylinders. Overall, the three design curves permit a more nuanced design approach than legacy knockdown factors, as different levels of conservatism can be chosen based on expected manufacturing quality. For instance, the most and least conservative of the three design guidelines differ by a factor of 3 for the most slender cylinder geometries, and the associated reduction in safety factor has profound implications for efficient structural design. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.
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Affiliation(s)
- R. M. J. Groh
- Bristol Composites Institute, Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK
| | - A. Pirrera
- Bristol Composites Institute, Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, UK
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6
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Al Saadi F, Parra-Rivas P. Transitions between dissipative localized structures in the simplified Gilad-Meron model for dryland plant ecology. CHAOS (WOODBURY, N.Y.) 2023; 33:033129. [PMID: 37003806 DOI: 10.1063/5.0133576] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Accepted: 02/15/2023] [Indexed: 06/19/2023]
Abstract
Spatially extended patterns and multistability of possible different states are common in many ecosystems, and their combination has an important impact on their dynamical behaviors. One potential combination involves tristability between a patterned state and two different uniform states. Using a simplified version of the Gilad-Meron model for dryland ecosystems, we study the organization, in bifurcation terms, of the localized structures arising in tristable regimes. These states are generally related to the concept of wave front locking and appear in the form of spots and gaps of vegetation. We find that the coexistence of localized spots and gaps, within tristable configurations, yields the appearance of hybrid states. We also study the emergence of spatiotemporal localized states consisting of a portion of a periodic pattern embedded in a uniform Hopf-like oscillatory background in a subcritical Turing-Hopf dynamical regime.
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Affiliation(s)
- Fahad Al Saadi
- Department of Systems Engineering, Military Technological College, Muscat, Oman
| | - Pedro Parra-Rivas
- Dipartimento di Ingegneria dell'Informazione, Elettronica e Telecomunicazioni, Sapienza Universitá di Roma, via Eudossiana 18, 00184 Rome, Italy
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7
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Stöhr M, Wolfrum M. Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback. CHAOS (WOODBURY, N.Y.) 2023; 33:023117. [PMID: 36859191 DOI: 10.1063/5.0134815] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2022] [Accepted: 01/18/2023] [Indexed: 06/18/2023]
Abstract
We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model.
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Affiliation(s)
- M Stöhr
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
| | - M Wolfrum
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
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8
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Koch ER, Seidel TG, Gurevich SV, Javaloyes J. Square-wave generation in vertical external-cavity Kerr-Gires-Tournois interferometers. OPTICS LETTERS 2022; 47:4343-4346. [PMID: 36048649 DOI: 10.1364/ol.468236] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2022] [Accepted: 08/02/2022] [Indexed: 06/15/2023]
Abstract
We study theoretically the mechanisms of square-wave (SW) formation in vertical external-cavity Kerr-Gires-Tournois interferometers in the presence of anti-resonant injection. We provide simple analytical approximations for their plateau intensities and for the conditions of their emergence. We demonstrate that SWs may appear via a homoclinic snaking scenario, leading to the formation of complex-shaped multistable SW solutions. The resulting SWs can host localized structures and robust bound states.
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9
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Moreno-Spiegelberg P, Arinyo-I-Prats A, Ruiz-Reynés D, Matias MA, Gomila D. Bifurcation structure of traveling pulses in type-I excitable media. Phys Rev E 2022; 106:034206. [PMID: 36266808 DOI: 10.1103/physreve.106.034206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2022] [Accepted: 08/01/2022] [Indexed: 06/16/2023]
Abstract
We study the scenario in which traveling pulses emerge in a prototypical type-I one-dimensional excitable medium, which exhibits two different routes to excitable behavior, mediated by a homoclinic (saddle-loop) and a saddle-node on the invariant cycle bifurcations. We characterize the region in parameter space in which traveling pulses are stable together with the different bifurcations behind either their destruction or loss of stability. In particular, some of the bifurcations delimiting the stability region have been connected, using singular limits, with the two different scenarios that mediated type-I local excitability. Finally, the existence of traveling pulses has been linked to a drift pitchfork instability of localized steady structures.
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Affiliation(s)
- Pablo Moreno-Spiegelberg
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
| | - Andreu Arinyo-I-Prats
- Institute of Computer Science, Czech Academy of Sciences, 182 07 Prague 8, Czech Republic
| | - Daniel Ruiz-Reynés
- Laboratory of Dynamics in Biological Systems, Department of Cellular and Molecular Medicine, University of Leuven (KU Leuven), B-3000 Leuven, Belgium
| | - Manuel A Matias
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
| | - Damià Gomila
- IFISC (CSIC-UIB), Instituto de Física Interdisciplinar y Sistemas Complejos, E-07122 Palma de Mallorca, Spain
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10
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Socoliuc V, Avdeev MV, Kuncser V, Turcu R, Tombácz E, Vékás L. Ferrofluids and bio-ferrofluids: looking back and stepping forward. NANOSCALE 2022; 14:4786-4886. [PMID: 35297919 DOI: 10.1039/d1nr05841j] [Citation(s) in RCA: 19] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
Ferrofluids investigated along for about five decades are ultrastable colloidal suspensions of magnetic nanoparticles, which manifest simultaneously fluid and magnetic properties. Their magnetically controllable and tunable feature proved to be from the beginning an extremely fertile ground for a wide range of engineering applications. More recently, biocompatible ferrofluids attracted huge interest and produced a considerable increase of the applicative potential in nanomedicine, biotechnology and environmental protection. This paper offers a brief overview of the most relevant early results and a comprehensive description of recent achievements in ferrofluid synthesis, advanced characterization, as well as the governing equations of ferrohydrodynamics, the most important interfacial phenomena and the flow properties. Finally, it provides an overview of recent advances in tunable and adaptive multifunctional materials derived from ferrofluids and a detailed presentation of the recent progress of applications in the field of sensors and actuators, ferrofluid-driven assembly and manipulation, droplet technology, including droplet generation and control, mechanical actuation, liquid computing and robotics.
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Affiliation(s)
- V Socoliuc
- Romanian Academy - Timisoara Branch, Center for Fundamental and Advanced Technical Research, Laboratory of Magnetic Fluids, Mihai Viteazu Ave. 24, 300223 Timisoara, Romania.
| | - M V Avdeev
- Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Joliot-Curie Str. 6, 141980 Dubna, Moscow Reg., Russia.
| | - V Kuncser
- National Institute of Materials Physics, Bucharest-Magurele, 077125, Romania
| | - Rodica Turcu
- National Institute for Research and Development of Isotopic and Molecular Technologies (INCDTIM), Donat Str. 67-103, 400293 Cluj-Napoca, Romania
| | - Etelka Tombácz
- University of Szeged, Faculty of Engineering, Department of Food Engineering, Moszkvai krt. 5-7, H-6725 Szeged, Hungary.
- University of Pannonia - Soós Ernő Water Technology Research and Development Center, H-8800 Zrínyi M. str. 18, Nagykanizsa, Hungary
| | - L Vékás
- Romanian Academy - Timisoara Branch, Center for Fundamental and Advanced Technical Research, Laboratory of Magnetic Fluids, Mihai Viteazu Ave. 24, 300223 Timisoara, Romania.
- Politehnica University of Timisoara, Research Center for Complex Fluids Systems Engineering, Mihai Viteazul Ave. 1, 300222 Timisoara, Romania
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11
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Dolinina D, Yulin A. Dissipative switching waves and solitons in the systems with spontaneously broken symmetry. Phys Rev E 2021; 103:052207. [PMID: 34134290 DOI: 10.1103/physreve.103.052207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2021] [Accepted: 04/15/2021] [Indexed: 11/07/2022]
Abstract
The paper addresses the bistability caused by spontaneous symmetry breaking bifurcation in a one-dimensional periodically corrugated nonlinear waveguide pumped by coherent light at normal incidence. The formation and the stability of the switching waves connecting the states of different symmetries are studied numerically. It is shown that the switching waves can form stable resting and moving bound states (dissipative solitons). The protocols of the creation of the discussed nonlinear localized waves are suggested and verified by numerical simulations.
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Affiliation(s)
- D Dolinina
- Faculty of Physics, ITMO University, Saint-Petersburg 197101, Russia
| | - A Yulin
- Faculty of Physics, ITMO University, Saint-Petersburg 197101, Russia
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12
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Ophaus L, Knobloch E, Gurevich SV, Thiele U. Two-dimensional localized states in an active phase-field-crystal model. Phys Rev E 2021; 103:032601. [PMID: 33862772 DOI: 10.1103/physreve.103.032601] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2020] [Accepted: 02/08/2021] [Indexed: 11/07/2022]
Abstract
The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.
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Affiliation(s)
- Lukas Ophaus
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany.,Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstrasse 2, 48149 Münster, Germany
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Svetlana V Gurevich
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany.,Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstrasse 2, 48149 Münster, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany.,Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstrasse 2, 48149 Münster, Germany
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13
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Gopalakrishnan SS, Panajotov K, Taki M, Tlidi M. Dissipative Light Bullets in Kerr Cavities: Multistability, Clustering, and Rogue Waves. PHYSICAL REVIEW LETTERS 2021; 126:153902. [PMID: 33929226 DOI: 10.1103/physrevlett.126.153902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Revised: 02/03/2021] [Accepted: 03/19/2021] [Indexed: 06/12/2023]
Abstract
We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D) localized structures consist of either an isolated light bullet (LB), bound together, or could occur in clusters forming well-defined 3D patterns. They can be seen as stationary states in the reference frame moving with the group velocity of light within the cavity. The number of LBs and their distribution in 3D settings are determined by the initial conditions, while their maximum peak power remains constant for a fixed value of the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation of homoclinic snaking for dissipative light bullets. However, when the strength of the injected beam is increased, LBs lose their stability and the cavity field exhibits giant, short-living 3D pulses. The statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of extreme events, often called rogue waves.
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Affiliation(s)
- S S Gopalakrishnan
- Faculté des Sciences, Université libre de Bruxelles (U.L.B), CP. 231, 1050 Brussels, Belgium
- Université Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers, Atomes et Molécules, F-59000 Lille, France
| | - K Panajotov
- Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels, Belgium
- Institute of Solid State Physics, 72 Tzarigradsko Chaussee Boulevard, 1784 Sofia, Bulgaria
| | - M Taki
- Université Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers, Atomes et Molécules, F-59000 Lille, France
| | - M Tlidi
- Faculté des Sciences, Université libre de Bruxelles (U.L.B), CP. 231, 1050 Brussels, Belgium
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14
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Holl MP, Archer AJ, Thiele U. Efficient calculation of phase coexistence and phase diagrams: application to a binary phase-field-crystal model. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:115401. [PMID: 33246322 DOI: 10.1088/1361-648x/abce6e] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We show that one can employ well-established numerical continuation methods to efficiently calculate the phase diagram for thermodynamic systems described by a suitable free energy functional. In particular, this involves the determination of lines of phase coexistence related to first order phase transitions and the continuation of triple points. To illustrate the method we apply it to a binary phase-field-crystal model for the crystallisation of a mixture of two types of particles. The resulting phase diagram is determined for one- and two-dimensional domains. In the former case it is compared to the diagram obtained from a one-mode approximation. The various observed liquid and crystalline phases and their stable and metastable coexistence are discussed as well as the temperature-dependence of the phase diagrams. This includes the (dis)appearance of critical points and triple points. We also relate bifurcation diagrams for finite-size systems to the thermodynamics of phase transitions in the infinite-size limit.
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Affiliation(s)
- Max Philipp Holl
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, 48149 Münster, Germany
| | - Andrew J Archer
- Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom
- Interdisciplinary Centre for Mathematical Modelling, Loughborough University, Loughborough LE11 3TU, United Kingdom
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, 48149 Münster, Germany
- Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstr. 2, 48149 Münster, Germany
- Center for Multiscale Theory and Computation (CMTC), Westfälische Wilhelms-Universität, Corrensstr. 40, 48149 Münster, Germany
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15
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Ivars SB, Kartashov YV, Torner L, Conejero JA, Milián C. Reversible Self-Replication of Spatiotemporal Kerr Cavity Patterns. PHYSICAL REVIEW LETTERS 2021; 126:063903. [PMID: 33635689 DOI: 10.1103/physrevlett.126.063903] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2020] [Accepted: 01/13/2021] [Indexed: 06/12/2023]
Abstract
We uncover a novel and robust phenomenon that causes the gradual self-replication of spatiotemporal Kerr cavity patterns in cylindrical microresonators. These patterns are inherently synchronized multifrequency combs. Under proper conditions, the axially localized nature of the patterns leads to a fundamental drift instability that induces transitions among patterns with a different number of rows. Self-replications, thus, result in the stepwise addition or removal of individual combs along the cylinder's axis. Transitions occur in a fully reversible and, consequently, deterministic way. The phenomenon puts forward a novel paradigm for Kerr frequency comb formation and reveals important insights into the physics of multidimensional nonlinear patterns.
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Affiliation(s)
- Salim B Ivars
- Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Departament de Física, Universitat Politècnica de Catalunya, 08222 Terrassa (Barcelona), Spain
| | - Yaroslav V Kartashov
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia
| | - Lluis Torner
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
| | - J Alberto Conejero
- Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain
| | - Carles Milián
- Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
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16
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Shapira AZ, Gavish N, Uecker H, Yochelis A. Bending and pinching of three-phase stripes: From secondary instabilities to morphological deformations in organic photovoltaics. Phys Rev E 2021; 102:062213. [PMID: 33466059 DOI: 10.1103/physreve.102.062213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 11/16/2020] [Indexed: 11/07/2022]
Abstract
Optimizing the properties of the mosaic nanoscale morphology of bulk heterojunction (BHJ) organic photovoltaics (OPV) is not only challenging technologically but also intriguing from the mechanistic point of view. Among the recent breakthroughs is the identification and utilization of a three-phase (donor-mixed-acceptor) BHJ, where the (intermediate) mixed phase can inhibit mesoscale morphological changes, such as phase separation. Using a mean-field approach, we reveal and distinguish between generic mechanisms that alter, through transverse instabilities, the evolution of stripes: the bending (zigzag mode) and the pinching (cross-roll mode) of the donor-acceptor domains. The results are summarized in a parameter plane spanned by the mixing energy and illumination, and show that donor-acceptor mixtures with higher mixing energy are more likely to develop pinching under charge-flux boundary conditions. The latter is notorious as it leads to the formation of disconnected domains and hence to loss of charge flux. We believe that these results provide a qualitative road map for BHJ optimization, using mixed-phase composition and, therefore, an essential step toward long-lasting OPV. More broadly, the results are also of relevance to study the coexistence of multiple-phase domains in material science, such as in ion-intercalated rechargeable batteries.
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Affiliation(s)
- Alon Z Shapira
- Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
| | - Nir Gavish
- Department of Mathematics, Technion - IIT, Haifa 3200003, Israel
| | - Hannes Uecker
- Institute for Mathematics, Carl von Ossietzky University of Oldenburg, P.F 2503, 26111 Oldenburg, Germany
| | - Arik Yochelis
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel.,Department of Physics, Ben-Gurion University of the Negev, Beer Sheva 8410501, Israel
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17
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Ophaus L, Kirchner J, Gurevich SV, Thiele U. Phase-field-crystal description of active crystallites: Elastic and inelastic collisions. CHAOS (WOODBURY, N.Y.) 2020; 30:123149. [PMID: 33380045 DOI: 10.1063/5.0019426] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2020] [Accepted: 12/07/2020] [Indexed: 06/12/2023]
Abstract
The active Phase-Field-Crystal (aPFC) model combines elements of the Toner-Tu theory for self-propelled particles and the classical Phase-Field-Crystal (PFC) model that describes the transition between liquid and crystalline phases. In the liquid-crystal coexistence region of the PFC model, crystalline clusters exist in the form of localized states that coexist with a homogeneous background. At sufficiently strong activity (related to self-propulsion strength), they start to travel. We employ numerical path continuation and direct time simulations to first investigate the existence regions of different types of localized states in one spatial dimension. The results are summarized in morphological phase diagrams in the parameter plane spanned by activity and mean density. Then we focus on the interaction of traveling localized states, studying their collision behavior. As a result, we distinguish "elastic" and "inelastic" collisions. In the former, localized states recover their properties after a collision, while in the latter, they may completely or partially annihilate, forming resting bound states or various traveling states.
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Affiliation(s)
- Lukas Ophaus
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Johannes Kirchner
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Svetlana V Gurevich
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
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18
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Shapira AZ, Uecker H, Yochelis A. Stripes on finite domains: Why the zigzag instability is only a partial story. CHAOS (WOODBURY, N.Y.) 2020; 30:073104. [PMID: 32752648 DOI: 10.1063/5.0006126] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Accepted: 06/15/2020] [Indexed: 06/11/2023]
Abstract
Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical, and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions of an infinite domain; however, on finite domains, the situation is more subtle since the unstable modes depend also on boundary conditions. Using two prototypical models, the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation, we consider finite size domains with no flux boundary conditions transversal to the stripes and reveal a distinct mixed-mode instability that lies in between the classical zigzag and the Eckhaus lines. This explains the stability of stripes in the mildly zigzag unstable regime and, after crossing the mixed-mode line, the evolution of zigzag stripes in the bulk of the domain and the formation of defects near the boundaries. The results are of particular importance for problems with large timescale separation, such as bulk-heterojunction deformations in organic photovoltaic and vegetation in semi-arid regions, where early temporal transients may play an important role.
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Affiliation(s)
- Alon Z Shapira
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
| | - Hannes Uecker
- Institute for Mathematics, Carl von Ossietzky University of Oldenburg, PF 2503, 26111 Oldenburg, Germany
| | - Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
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19
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Alvarez-Socorro AJ, Clerc MG, Ferré M, Knobloch E. Chaotic motion of localized structures. Phys Rev E 2020; 101:042212. [PMID: 32422835 DOI: 10.1103/physreve.101.042212] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 03/23/2020] [Indexed: 11/07/2022]
Abstract
Mobility properties of spatially localized structures arising from chaotic but deterministic forcing of the bistable Swift-Hohenberg equation are studied and compared with the corresponding results when the chaotic forcing is replaced by white noise. Short structures are shown to possess greater mobility, resulting in larger root-mean-square speeds but shorter displacements than longer structures. Averaged over realizations, the displacement of the structure is ballistic at short times but diffusive at larger times. Similar results hold in two spatial dimensions. The effects of chaotic forcing on the stability of these structures is also quantified. Shorter structures are found to be more fragile than longer ones, and their stability region can be displaced outside the pinning region for constant forcing. Outside the stability region the deterministic fluctuations lead either to the destruction of the structure or to its gradual growth.
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Affiliation(s)
- A J Alvarez-Socorro
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile.,Laboratorio de Investigación, Desarrollo e Innovación, Zenta Group, Diagonal Oriente 5081, Ñuñoa, Santiago, Chile
| | - Marcel G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Michel Ferré
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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20
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Schmidt H, Avitabile D. Bumps and oscillons in networks of spiking neurons. CHAOS (WOODBURY, N.Y.) 2020; 30:033133. [PMID: 32237760 DOI: 10.1063/1.5135579] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 03/03/2020] [Indexed: 06/11/2023]
Abstract
We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations.
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Affiliation(s)
- Helmut Schmidt
- Max Planck Institute for Human Cognitive and Brain Sciences, Stephanstrasse 1a, 04103 Leipzig, Germany
| | - Daniele Avitabile
- Department of Mathematics, Faculteit der Exacte Wetenschappen, Vrije Universiteit (VU University Amsterdam), De Boelelaan 1081a, 1081 HV Amsterdam, Netherlands and Mathneuro Team, Inria Sophia Antipolis, 2004 Rue des Lucioles, Sophia Antipolis, 06902 Cedex, France
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21
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Salewski M, Gibson JF, Schneider TM. Origin of localized snakes-and-ladders solutions of plane Couette flow. Phys Rev E 2019; 100:031102. [PMID: 31640040 DOI: 10.1103/physreve.100.031102] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2019] [Indexed: 11/07/2022]
Abstract
Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.
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Affiliation(s)
- Matthew Salewski
- Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany
| | - John F Gibson
- Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824, USA
| | - Tobias M Schneider
- Emergent Complexity in Physical Systems Laboratory (ECPS), École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
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22
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Schelte C, Pimenov A, Vladimirov AG, Javaloyes J, Gurevich SV. Tunable Kerr frequency combs and temporal localized states in time-delayed Gires-Tournois interferometers. OPTICS LETTERS 2019; 44:4925-4928. [PMID: 31613230 DOI: 10.1364/ol.44.004925] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2019] [Accepted: 08/26/2019] [Indexed: 06/10/2023]
Abstract
In this Letter, we study theoretically a new setup allowing for the generation of temporal localized states (TLSs) and frequency combs. The setup is compact (a few centimeters) and can be implemented using established technologies, while offering tunable repetition rates and potentially high power operation. It consists of a vertically emitting micro-cavity, operated in the Gires-Tournois regime, containing a Kerr medium strong time-delayed optical feedback, and detuned optical injection. We disclose sets of multistable dark and bright TLSs coexisting on their respective bistable homogeneous backgrounds.
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23
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Knobloch E, Uecker H, Wetzel D. Defectlike structures and localized patterns in the cubic-quintic-septic Swift-Hohenberg equation. Phys Rev E 2019; 100:012204. [PMID: 31499926 DOI: 10.1103/physreve.100.012204] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 11/07/2022]
Abstract
We study numerically the cubic-quintic-septic Swift-Hohenberg (SH357) equation on bounded one-dimensional domains. Under appropriate conditions stripes with wave number k≈1 bifurcate supercritically from the zero state and form S-shaped branches resulting in bistability between small and large amplitude stripes. Within this bistability range we find stationary heteroclinic connections or fronts between small and large amplitude stripes, and demonstrate that the associated spatially localized defectlike structures either snake or fall on isolas. In other parameter regimes we also find heteroclinic connections to spatially homogeneous states and a multitude of dynamically stable steady states consisting of patches of small and large amplitude stripes with different wave numbers or of spatially homogeneous patches. The SH357 equation is thus extremely rich in the types of patterns it exhibits. Some of the features of the bifurcation diagrams obtained by numerical continuation can be understood using a conserved quantity, the spatial Hamiltonian of the system.
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Affiliation(s)
- Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Hannes Uecker
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
| | - Daniel Wetzel
- Institut für Mathematik, Universität Oldenburg, D26111 Oldenburg, Germany
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24
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Gordillo L, Knobloch E. Fluid-supported elastic sheet under compression: Multifold solutions. Phys Rev E 2019; 99:043001. [PMID: 31108605 DOI: 10.1103/physreve.99.043001] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2018] [Indexed: 11/07/2022]
Abstract
The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.
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Affiliation(s)
- Leonardo Gordillo
- Departamento de Física, Universidad de Santiago de Chile, Av. Ecuador 3493, Estación Central, Santiago, Chile
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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25
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Urra H, Marín JF, Páez-Silva M, Taki M, Coulibaly S, Gordillo L, García-Ñustes MA. Localized Faraday patterns under heterogeneous parametric excitation. Phys Rev E 2019; 99:033115. [PMID: 30999404 DOI: 10.1103/physreve.99.033115] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2019] [Indexed: 11/07/2022]
Abstract
Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schrödinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength.
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Affiliation(s)
- Héctor Urra
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile
| | - Juan F Marín
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile
| | - Milena Páez-Silva
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Chile
| | - Majid Taki
- Université de Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Saliya Coulibaly
- Université de Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Leonardo Gordillo
- Departamento de Física, Universidad de Santiago de Chile Av. Ecuador 3493, Estación Central, Santiago, Chile
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26
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Continuation for Thin Film Hydrodynamics and Related Scalar Problems. COMPUTATIONAL METHODS IN APPLIED SCIENCES 2019. [DOI: 10.1007/978-3-319-91494-7_13] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/12/2023]
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27
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Clerc MG, Coulibaly S, Ferré MA, Rojas RG. Chimera states in a Duffing oscillators chain coupled to nearest neighbors. CHAOS (WOODBURY, N.Y.) 2018; 28:083126. [PMID: 30180634 DOI: 10.1063/1.5025038] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/06/2018] [Accepted: 07/31/2018] [Indexed: 06/08/2023]
Abstract
Coupled nonlinear oscillators can present complex spatiotemporal behaviors. Here, we report the coexistence of coherent and incoherent domains, called chimera states, in an array of identical Duffing oscillators coupled to their nearest neighbors. The chimera states show a significant variation of amplitude in the desynchronized domain. These intriguing states are observed in the bistability region between a homogeneous state and a spatiotemporal chaotic one. These dynamical behaviors are characterized by their Lyapunov spectra and their global phase coherence order parameter. The local coupling between oscillators prevents one domain from invading the other one. Depending on initial conditions, a family of chimera states appear, organized in a snaking-like diagram.
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Affiliation(s)
- M G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - S Coulibaly
- Université de Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - M A Ferré
- Departamento de Física and Millennium Institute for Research in Optics, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla, 487-3 Santiago, Chile
| | - R G Rojas
- Intituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla, 4059 Valparaíso, Chile
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28
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Ophaus L, Gurevich SV, Thiele U. Resting and traveling localized states in an active phase-field-crystal model. Phys Rev E 2018; 98:022608. [PMID: 30253633 DOI: 10.1103/physreve.98.022608] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2018] [Indexed: 06/08/2023]
Abstract
The conserved Swift-Hohenberg equation (or phase-field-crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles, Menzel and Löwen [Phys. Rev. Lett. 110, 055702 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.055702] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.
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Affiliation(s)
- Lukas Ophaus
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany, and Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster Corrensstrasse 2, 48149 Münster, Germany
| | - Svetlana V Gurevich
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany, and Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster Corrensstrasse 2, 48149 Münster, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany, and Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster Corrensstrasse 2, 48149 Münster, Germany
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29
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Susanto H, Kusdiantara R, Li N, Kirikchi OB, Adzkiya D, Putri ERM, Asfihani T. Snakes and ghosts in a parity-time-symmetric chain of dimers. Phys Rev E 2018; 97:062204. [PMID: 30011512 DOI: 10.1103/physreve.97.062204] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2017] [Indexed: 11/07/2022]
Abstract
We consider linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time (PT)-symmetric coupler composed by a chain of dimers. We study uniform states and site-centered and bond-centered spatially localized solutions and present that each solution has a symmetric and antisymmetric configuration between the arms. The symmetric solutions can become unstable due to bifurcations of asymmetric ones, that are called ghost states, because they exist only when an otherwise real propagation constant is taken to be complex valued. When a parameter is varied, the resulting bifurcation diagrams for the existence of standing localized solutions have a snaking behavior. The critical gain and loss coefficient above which the PT symmetry is broken corresponds to the condition when bifurcation diagrams of symmetric and antisymmetric states merge. Past the symmetry breaking, the system no longer has time-independent states. Nevertheless, equilibrium solutions can be analytically continued by defining a dual equation that leads to ghost states associated with growth or decay, that are also identified and examined here. We show that ghost localized states also exhibit snaking bifurcation diagrams. We analyze the width of the snaking region and provide asymptotic approximations in the limit of strong and weak coupling where good agreement is obtained.
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Affiliation(s)
- H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung, 40132, Indonesia
| | - N Li
- School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - O B Kirikchi
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - D Adzkiya
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - E R M Putri
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - T Asfihani
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
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30
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Wetzel D. Tristability between stripes, up-hexagons, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons. Phys Rev E 2018; 97:062221. [PMID: 30011496 DOI: 10.1103/physreve.97.062221] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2017] [Indexed: 11/07/2022]
Abstract
Third-order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up-hexagons, and down-hexagons in pattern-forming systems. These amplitude equations predict the nonexistence of bistable ranges between up- and down-hexagons and tristable ranges between stripes, up-, and down-hexagons. In the present work we use fifth-order amplitude equations for finding such bistable and tristable ranges for a generalized Swift-Hohenberg equation and discuss stationary front connections between up- and down-hexagons.
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Affiliation(s)
- D Wetzel
- Institut für Mathematik, Carl-von-Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
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31
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Yochelis A. Catalytic Membrane Reactor Model as a Laboratory for Pattern Emergence in Reaction-diffusion-advection Media. Isr J Chem 2018. [DOI: 10.1002/ijch.201700145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research (SIDEER), Blaustein Institutes for Desert Research (BIDR); Ben-Gurion University of the Negev, Sede Boqer Campus; Midreshet Ben-Gurion 8499000 Israel
- Department of Physics; Ben-Gurion University of the Negev; Be'er Sheva 8410501 Israel
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32
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Alvarez-Socorro AJ, Clerc MG, Tlidi M. Spontaneous motion of localized structures induced by parity symmetry breaking transition. CHAOS (WOODBURY, N.Y.) 2018; 28:053119. [PMID: 29857691 DOI: 10.1063/1.5019734] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We consider a paradigmatic nonvariational scalar Swift-Hohenberg equation that describes short wavenumber or large wavelength pattern forming systems. This work unveils evidence of the transition from stable stationary to moving localized structures in one spatial dimension as a result of a parity breaking instability. This behavior is attributed to the nonvariational character of the model. We show that the nature of this transition is supercritical. We characterize analytically and numerically this bifurcation scenario from which emerges asymmetric moving localized structures. A generalization for two-dimensional settings is discussed.
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Affiliation(s)
- A J Alvarez-Socorro
- Departamento de Física and Millennium Institute for Research in Optics, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - M G Clerc
- Departamento de Física and Millennium Institute for Research in Optics, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - M Tlidi
- Département de Physique, Université Libre de Bruxelles (U.L.B.), CP 231, Campus Plaine, Bruxelles 1050, Belgium
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33
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Zelnik YR, Gandhi P, Knobloch E, Meron E. Implications of tristability in pattern-forming ecosystems. CHAOS (WOODBURY, N.Y.) 2018; 28:033609. [PMID: 29604648 DOI: 10.1063/1.5018925] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts. We analyze the bifurcation structure of model solutions that describe uniform states, periodic patterns, and hybrid states between the former two. We map out the parameter space where these states exist, and note how the different states interact with each other. We further focus on two special implications with ecological significance, breakdown of the snaking range and complex fronts. We find that the organization of the hybrid states within a homoclinic snaking structure breaks down as it meets a Maxwell point where simple fronts are stationary. We also discover a new series of complex fronts between the uniform states, each with its own velocity. We conclude with a brief discussion of the significance of these findings for the dynamics of regime shifts and their potential control.
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Affiliation(s)
- Yuval R Zelnik
- Centre for Biodiversity Theory and Modelling, Theoretical and Experimental Ecology Station, CNRS and Paul Sabatier University, 09200 Moulis, France
| | - Punit Gandhi
- Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Ehud Meron
- Department of Solar Energy and Environmental Physics, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
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34
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Bai L, Wang F, Wadee MA, Yang J. Nonlinear mode interaction in equal-leg angle struts susceptible to cellular buckling. Proc Math Phys Eng Sci 2017; 473:20170583. [PMID: 29225506 DOI: 10.1098/rspa.2017.0583] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2017] [Accepted: 09/28/2017] [Indexed: 11/12/2022] Open
Abstract
A variational model that describes the interactive buckling of a thin-walled equal-leg angle strut under pure axial compression is presented. A formulation combining the Rayleigh-Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Solving the equations using numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weak-axis flexural buckling mode and the strong-axis flexural-torsional buckling mode for the first time-the resulting behaviour being highly unstable. Physical experiments conducted on 10 cold-formed steel specimens are presented and the results show good agreement with the variational model.
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Affiliation(s)
- L Bai
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China.,Collaborative Innovation Centre for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, People's Republic of China
| | - F Wang
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China.,Collaborative Innovation Centre for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, People's Republic of China
| | - M A Wadee
- Department of Civil and Environmental Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
| | - J Yang
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China.,Collaborative Innovation Centre for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240, People's Republic of China
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35
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Kusdiantara R, Susanto H. Homoclinic snaking in the discrete Swift-Hohenberg equation. Phys Rev E 2017; 96:062214. [PMID: 29347380 DOI: 10.1103/physreve.96.062214] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2016] [Indexed: 11/07/2022]
Abstract
We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.
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Affiliation(s)
- R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung 40132, Indonesia
| | - H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
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36
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Gonpe Tafo JB, Nana L, Kofane TC. Effects of nonlinear gradient terms on the defect turbulence regime in weakly dissipative systems. Phys Rev E 2017; 96:022205. [PMID: 28950606 DOI: 10.1103/physreve.96.022205] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2017] [Indexed: 06/07/2023]
Abstract
We investigate the behavior of traveling waves in a defect turbulence regime with the periodic boundary conditions by using the lowest-order complex Ginzburg-Landau equation (CGLE), and we show the effect of the nonlinear gradient terms in the system. It is found that the nonlinear gradient terms which appear at the same order as the quintic term can change the behavior of the wave patterns. The presence of the nonlinear gradient terms can cause major changes in the behavior of the solution. They can be considered like the stabilizing terms. The system which was initially unstable or chaotic can become stable by including the nonlinear gradient terms.
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Affiliation(s)
- J B Gonpe Tafo
- Laboratoire de Physique Fondamentale, Groupe Phénomènes Non Linéaires et Systèmes Complexes, UFD de Mathématique, Informatique Appliquée et Physique Fondamentale, Université de Douala, Boîte Postale 24157, Douala, Cameroon
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, Boîte Postale 812, Yaoundé, Cameroon
| | - L Nana
- Laboratoire de Physique Fondamentale, Groupe Phénomènes Non Linéaires et Systèmes Complexes, UFD de Mathématique, Informatique Appliquée et Physique Fondamentale, Université de Douala, Boîte Postale 24157, Douala, Cameroon
| | - T C Kofane
- Centre d'Excellence Africain en Technologies de l'Information et de la Communication, Boîte Postale 812, Yaoundé, Cameroon
- Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, Boîte Postale 812, Yaoundé, Cameroon
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37
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Urzagasti D, Laroze D, Pleiner H. Two-dimensional localized chaotic patterns in parametrically driven systems. Phys Rev E 2017; 95:052216. [PMID: 28618465 DOI: 10.1103/physreve.95.052216] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/26/2017] [Indexed: 11/07/2022]
Abstract
We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.
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Affiliation(s)
| | - David Laroze
- Instituto de Alta de Investigación, CEDENNA, Universidad de Tarapacá, Casilla 7D, Arica, Chile.,Max Planck Institute for Polymer Research, D-55021 Mainz, Germany
| | - Harald Pleiner
- Max Planck Institute for Polymer Research, D-55021 Mainz, Germany
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38
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Barnett J, Gurevich DR, Grigoriev RO. Streamwise localization of traveling wave solutions in channel flow. Phys Rev E 2017; 95:033124. [PMID: 28415376 DOI: 10.1103/physreve.95.033124] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2016] [Indexed: 11/07/2022]
Abstract
Channel flow of an incompressible fluid at Reynolds numbers above 2400 possesses a number of different spatially localized solutions that approach laminar flow far upstream and downstream. We use one such relative time-periodic solution, which corresponds to a spatially localized version of a Tollmien-Schlichting wave, to illustrate how the upstream and downstream asymptotics can be computed analytically. In particular, we show that for these spanwise uniform states the asymptotics predict the exponential localization that has been observed for numerically computed solutions of several canonical shear flows but never properly understood theoretically.
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Affiliation(s)
- Joshua Barnett
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
| | - Daniel R Gurevich
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
| | - Roman O Grigoriev
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
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39
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Desertification by front propagation? J Theor Biol 2017; 418:27-35. [DOI: 10.1016/j.jtbi.2017.01.029] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/19/2016] [Revised: 11/26/2016] [Accepted: 01/19/2017] [Indexed: 11/19/2022]
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40
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Tabbert F, Schelte C, Tlidi M, Gurevich SV. Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model. Phys Rev E 2017; 95:032213. [PMID: 28415377 DOI: 10.1103/physreve.95.032213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2016] [Indexed: 06/07/2023]
Abstract
We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.
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Affiliation(s)
- Felix Tabbert
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
| | - Christian Schelte
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
| | - Mustapha Tlidi
- Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, C.P. 231, Brussels B-1050, Belgium
| | - Svetlana V Gurevich
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), University of Münster, Corrensstrasse 2, D-48149 Münster, Germany
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41
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Clerc MG, Coulibaly S, Ferré MA, García-Ñustes MA, Rojas RG. Chimera-type states induced by local coupling. Phys Rev E 2016; 93:052204. [PMID: 27300877 DOI: 10.1103/physreve.93.052204] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2015] [Indexed: 06/06/2023]
Abstract
Coupled oscillators can exhibit complex self-organization behavior such as phase turbulence, spatiotemporal intermittency, and chimera states. The latter corresponds to a coexistence of coherent and incoherent states apparently promoted by nonlocal or global coupling. Here we investigate the existence, stability properties, and bifurcation diagram of chimera-type states in a system with local coupling without different time scales. Based on a model of a chain of nonlinear oscillators coupled to adjacent neighbors, we identify the required attributes to observe these states: local coupling and bistability between a stationary and an oscillatory state close to a homoclinic bifurcation. The local coupling prevents the incoherent state from invading the coherent one, allowing concurrently the existence of a family of chimera states, which are organized by a homoclinic snaking bifurcation diagram.
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Affiliation(s)
- M G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - S Coulibaly
- Laboratoire de Physique des Lasers, Atomes et Molécules, CNRS UMR 8523, Université de Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq Cedex, France
| | - M A Ferré
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
| | - M A García-Ñustes
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
| | - R G Rojas
- Instituto de Física, Pontificia Universidad Católica de Valparaíso, casilla 4059, Valparaíso, Chile
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42
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Draelants D, Avitabile D, Vanroose W. Localized auxin peaks in concentration-based transport models of the shoot apical meristem. J R Soc Interface 2016; 12:rsif.2014.1407. [PMID: 25878130 DOI: 10.1098/rsif.2014.1407] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
Abstract
We study the formation of auxin peaks in a generic class of concentration-based auxin transport models, posed on static plant tissues. Using standard asymptotic analysis, we prove that, on bounded domains, auxin peaks are not formed via a Turing instability in the active transport parameter, but via simple corrections to the homogeneous steady state. When the active transport is small, the geometry of the tissue encodes the peaks' amplitude and location: peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We test our theory and perform numerical bifurcation analysis on two models that are known to generate auxin patterns for biologically plausible parameter values. In the same parameter regimes, we find that realistic tissues are capable of generating a multitude of stationary patterns, with a variable number of auxin peaks, that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. The competition between active transport and production rate determines whether peaks remain localized or cover the entire domain. In particular, changes in the auxin production that are fast with respect to the cellular life cycle affect the auxin peak distribution, switching from localized spots to fully patterned states. We relate the occurrence of localized patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media, but has not yet been reported in plant models.
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Affiliation(s)
- Delphine Draelants
- Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020 Antwerpen, Belgium
| | - Daniele Avitabile
- Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
| | - Wim Vanroose
- Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020 Antwerpen, Belgium
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43
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Descalzi O, Brand HR. Non-unique results of collisions of quasi-one-dimensional dissipative solitons. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2015; 373:rsta.2015.0115. [PMID: 26527813 DOI: 10.1098/rsta.2015.0115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/10/2015] [Indexed: 06/05/2023]
Abstract
We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mons. Álvaro del Portillo 12.455, Las Condes, Santiago, Chile Department of Physics, University of Bayreuth, Bayreuth 95440, Germany
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, Bayreuth 95440, Germany
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44
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Golubitsky M, Stewart I. Recent advances in symmetric and network dynamics. CHAOS (WOODBURY, N.Y.) 2015; 25:097612. [PMID: 26428565 DOI: 10.1063/1.4918595] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network.
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Affiliation(s)
- Martin Golubitsky
- Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210, USA
| | - Ian Stewart
- Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
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45
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Chantry M, Kerswell RR. Localization in a spanwise-extended model of plane Couette flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:043005. [PMID: 25974578 DOI: 10.1103/physreve.91.043005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2014] [Indexed: 06/04/2023]
Abstract
We consider a nine-partial-differential-equation (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localization in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Spatially localized, time-periodic solutions were found instead, which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localized states in very wide domains. This smooth localization behavior, which has also been seen in fully resolved duct flow (S. Okino, Ph.D. thesis, Kyoto University, Kyoto, 2011), indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains.
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Affiliation(s)
- M Chantry
- School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
| | - R R Kerswell
- School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
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46
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Gandhi P, Knobloch E, Beaume C. Localized states in periodically forced systems. PHYSICAL REVIEW LETTERS 2015; 114:034102. [PMID: 25659000 DOI: 10.1103/physrevlett.114.034102] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2014] [Indexed: 06/04/2023]
Abstract
The theory of stationary spatially localized patterns in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing, related but time-dependent structures may result. These may consist of breathing localized patterns, or states that grow for part of the cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during the remainder of the cycle. These two competing processes lead to a complex phase diagram whose structure is a consequence of a series of resonances between the nucleation time and the forcing period. The resulting diagram is computed for the periodically forced quadratic-cubic Swift-Hohenberg equation, and its details are interpreted in terms of the properties of the depinning transition for the fronts bounding the localized state on either side. The results are expected to shed light on localized states in a large variety of periodically driven systems.
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Affiliation(s)
- Punit Gandhi
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Cédric Beaume
- Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
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47
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Breña-Medina V, Champneys A. Subcritical Turing bifurcation and the morphogenesis of localized patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032923. [PMID: 25314520 DOI: 10.1103/physreve.90.032923] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 05/03/2023]
Abstract
Subcritical Turing bifurcations of reaction-diffusion systems in large domains lead to spontaneous onset of well-developed localized patterns via the homoclinic snaking mechanism. This phenomenon is shown to occur naturally when balancing source and loss effects are included in a typical reaction-diffusion system, leading to a super- to subcritical transition. Implications are discussed [corrected]for a range of physical problems, arguing that subcriticality leads to naturally robust phase transitions to localized patterns.
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Affiliation(s)
- Víctor Breña-Medina
- Departamento de Nanotecnología, Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Juriquilla No. 3001, Querétaro 76230, Mexico and Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
| | - Alan Champneys
- Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
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Li X, Xu D, Liao S. Observations of highly localized oscillons with multiple crests and troughs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:031001. [PMID: 25314386 DOI: 10.1103/physreve.90.031001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2014] [Indexed: 06/04/2023]
Abstract
Two types of stable, highly localized Faraday resonant standing waves with multiple crests and troughs are observed in an ethanol-water solution partly filled in a Hele-Shaw cell vertically oscillated with a single frequency. Systematical experiments are performed to investigate the properties of these oscillons. It is found that the wave height of these oscillons is independent of fluid depth from 1 to 5 cm. In particular, some experiments are performed to indicate the high localization of the oscillons, which suggests that these oscillons may be regarded as a combination of the two elementary oscillons discovered by Rajchenbach et al. [Phys. Rev. Lett. 107, 024502 (2011)], for instance, (2,3)=(1,1)+(1,2), where (m,n) denotes an oscillon with m crests and n troughs. So, our experiments also reveal an elegant "arithmetic" of these oscillons. These experimental phenomena are helpful to deepen and enrich our understanding about Faraday waves.
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Affiliation(s)
- Xiaochen Li
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
| | - Dali Xu
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
| | - Shijun Liao
- State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China and Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia
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Abstract
An analytical model based on variational principles for a thin-walled stiffened plate subjected to axial compression is presented. A system of nonlinear differential and integral equations is derived and solved using numerical continuation. The results show that the system is susceptible to highly unstable local–global mode interaction after an initial instability is triggered. Moreover, snap-backs in the response showing sequential destabilization and restabilization, known as cellular buckling or snaking, arise. The analytical model is compared with static finite element (FE) models for joint conditions between the stiffener and the main plate that have significant rotational restraint. However, it is known from previous studies that the behaviour, where the same joint is insignificantly restrained rotationally, is captured better by an analytical approach than by standard FE methods; the latter being unable to capture cellular buckling behaviour even though the phenomenon is clearly observed in laboratory experiments.
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Affiliation(s)
- M. Ahmer Wadee
- Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
| | - Maryam Farsi
- Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
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Descalzi O, Brand HR. Class of compound dissipative solitons as a result of collisions in one and two spatial dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:020901. [PMID: 25215679 DOI: 10.1103/physreve.90.020901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2014] [Indexed: 06/03/2023]
Abstract
We study the interaction of quasi-one-dimensional (quasi-1D) dissipative solitons (DSs). Starting with quasi-1D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as the initial condition, we find, as a function of the approach velocity and the real part of the cubic interaction of the two counterpropagating envelopes: interpenetration, one compound state made of both envelopes or two compound states. For the latter class both envelopes show DSs superposed at two different locations. The stability of this class of compound states is traced back to the quasilinear growth rate associated with the coupled system. We show that this mechanism also works for 1D coupled cubic-quintic CGL equations. For quasi-1D states that are not in their asymptotic state before the collision, a breakup along the crest can be observed, leading to nonunique results after the collision of quasi-1D states.
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Affiliation(s)
- Orazio Descalzi
- Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo 12.455, Las Condes, Santiago, Chile and Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
| | - Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany
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