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Alvarez-Socorro AJ, Clerc MG, Ferré MA, Knobloch E. Front depinning by deterministic and stochastic fluctuations: A comparison. Phys Rev E 2019; 99:062226. [PMID: 31330663 DOI: 10.1103/physreve.99.062226] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Indexed: 11/07/2022]
Abstract
Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.
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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
| | - 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
| | - 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
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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Xiong L, Cooper R, Tsimring LS. Coexistence and Pattern Formation in Bacterial Mixtures with Contact-Dependent Killing. Biophys J 2019; 114:1741-1750. [PMID: 29642042 DOI: 10.1016/j.bpj.2018.02.012] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2017] [Revised: 01/06/2018] [Accepted: 02/02/2018] [Indexed: 01/01/2023] Open
Abstract
Multistrain microbial communities often exhibit complex spatial organization that emerges because of the interplay of various cooperative and competitive interaction mechanisms. One strong competitive mechanism is contact-dependent neighbor killing enabled by the type VI secretion system. It has been previously shown that contact-dependent killing can result in bistability of bacterial mixtures so that only one strain survives and displaces the other. However, it remains unclear whether stable coexistence is possible in such mixtures. Using a population dynamics model for two interacting bacterial strains, we found that coexistence can be made possible by the interplay of contact-dependent killing and long-range growth inhibition, leading to the formation of various cellular patterns. These patterns emerge in a much broader parameter range than that required for the linear Turing-like instability, suggesting this may be a robust mechanism for pattern formation.
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Affiliation(s)
- Liyang Xiong
- Department of Physics, University of California, San Diego, La Jolla, California; BioCircuits Institute, University of California, San Diego, La Jolla, California
| | - Robert Cooper
- BioCircuits Institute, University of California, San Diego, La Jolla, California
| | - Lev S Tsimring
- BioCircuits Institute, University of California, San Diego, La Jolla, California; The San Diego Center for Systems Biology, University of California, San Diego, La Jolla, California.
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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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Ma YP, Knobloch E. Depinning, front motion, and phase slips. CHAOS (WOODBURY, N.Y.) 2012; 22:033101. [PMID: 23020440 DOI: 10.1063/1.4731268] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Pinning and depinning of fronts bounding spatially localized structures in the forced complex Ginzburg-Landau equation describing the 1:1 resonance is studied in one spatial dimension, focusing on regimes in which the structure grows via roll insertion instead of roll nucleation at either edge. The motion of the fronts is nonlocal but can be analyzed quantitatively near the depinning transition.
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Affiliation(s)
- Y-P Ma
- Department of Physics, University of California, Berkeley, California 94720, USA
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Zhang L, Song A, He J. Effect of colored noise on logical stochastic resonance in bistable dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:051106. [PMID: 21230436 DOI: 10.1103/physreve.82.051106] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2010] [Revised: 08/09/2010] [Indexed: 05/30/2023]
Abstract
Recently, Murali [Phys. Rev. Lett. 102, 104101 (2009)] showed that some noisy nonlinear dynamics could be exploited to design nonlinear logic gate based on stochastic resonance (SR). In the paper, inspired by the Murali's articles, we consider the logical stochastic resonance (LSR) phenomenon in a bistable system by cycling various combinations of two logic inputs, in the presence of exponentially correlated noise. The effect of the colored noise on the LSR is evaluated by the success probability of obtained desired logic output. The two major results are presented. First, it shows that the LSR effect can be obtained by changing noise intensity. The logic output is obtained most reliably at an intermediate noise band. There is still a plateau on performance curve of the success probability versus noise intensity. As correlation time increases, the optimal noise band shifts to higher level and peak performance is degraded from 100% accuracy. Due to the influence of the colored noise, the plateau is a bit low and not completely flat. On the other hand, the performance measure can evolve nonmonotonically as correlation time increases, and a SR-like effect is obtained as a result of correlation time. It shows that actually some finite correlation time is better than delta correlation time at certain noise intensities. The study might provide an example where LSR phenomenon can exist robustly in the presence of colored noise.
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Affiliation(s)
- Lei Zhang
- School of Instrument Science and Engineering, Southeast University, Sipailou 2, Nanjing 210096, People's Republic of China
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Burke J, Houghton SM, Knobloch E. Swift-Hohenberg equation with broken reflection symmetry. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:036202. [PMID: 19905195 DOI: 10.1103/physreve.80.036202] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2009] [Indexed: 05/28/2023]
Abstract
The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localized solutions organized in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility of the equation. We examine here the consequences of breaking spatial reversibility on the snakes-and-ladders structure. We find that the localized states now drift, and show that the snakes-and-ladders structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques.
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Affiliation(s)
- J Burke
- Center for BioDynamics, Boston University, Boston, Massachusetts 02215, USA.
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Galenko P, Danilov D, Lebedev V. Phase-field-crystal and Swift-Hohenberg equations with fast dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051110. [PMID: 19518419 DOI: 10.1103/physreve.79.051110] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2009] [Revised: 03/18/2009] [Indexed: 05/11/2023]
Abstract
A phenomenological description of transition from an unstable to a (meta)stable phase state, including microscopic and mesoscopic scales, is presented. It is based on the introduction of specific memory functions which take into account contributions to the driving force of transformation from the past. A region of applicability for phase-field crystals and Swift-Hohenberg-type models is extended by inclusion of inertia effects into the equations of motion through a memory function of an exponential form. The inertia allows us to predict fast degrees of freedom in the form of damping perturbations with finite relaxation time in the instability of homogeneous and periodic model solutions.
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Affiliation(s)
- Peter Galenko
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany.
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Burke J, Knobloch E. Homoclinic snaking: structure and stability. CHAOS (WOODBURY, N.Y.) 2007; 17:037102. [PMID: 17903009 DOI: 10.1063/1.2746816] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.
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Affiliation(s)
- John Burke
- Department of Physics, University of California, Berkeley, California 94720, USA.
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Kozyreff G, Tlidi M. Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems. CHAOS (WOODBURY, N.Y.) 2007; 17:037103. [PMID: 17903010 DOI: 10.1063/1.2759436] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
We derive asymptotically an order parameter equation in the limit where nascent bistability and long-wavelength modulation instabilities coalesce. This equation is a variant of the Swift-Hohenberg equation that generally contains nonvariational terms of the form psinabla(2)psi and /nablapsi/(2). We briefly review some of the properties already derived for this equation and derive it on three examples taken from chemical, biological, and optical contexts. Finally, we derive the equation on a general class of partial differential systems.
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Affiliation(s)
- G Kozyreff
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), Campus de la Plaine C.P. 231, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
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Burke J, Knobloch E. Localized states in the generalized Swift-Hohenberg equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056211. [PMID: 16803030 DOI: 10.1103/physreve.73.056211] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2005] [Indexed: 05/10/2023]
Abstract
The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time.
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Affiliation(s)
- John Burke
- Department of Physics, University of California, Berkeley, 94720, USA.
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