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Suchanek T, Kroy K, Loos SAM. Time-reversal and parity-time symmetry breaking in non-Hermitian field theories. Phys Rev E 2023; 108:064123. [PMID: 38243548 DOI: 10.1103/physreve.108.064123] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2023] [Accepted: 11/06/2023] [Indexed: 01/21/2024]
Abstract
We study time-reversal symmetry breaking in non-Hermitian fluctuating field theories with conserved dynamics, comprising the mesoscopic descriptions of a wide range of nonequilibrium phenomena. They exhibit continuous parity-time (PT) symmetry-breaking phase transitions to dynamical phases. For two concrete transition scenarios, exclusive to non-Hermitian dynamics, namely, oscillatory instabilities and critical exceptional points, a low-noise expansion exposes a pretransitional surge of the mesoscale (informatic) entropy production rate, inside the static phases. Its scaling in the susceptibility contrasts conventional critical points (such as second-order phase transitions), where the susceptibility also diverges, but the entropy production generally remains finite. The difference can be attributed to active fluctuations in the wavelengths that become unstable. For critical exceptional points, we identify the coupling of eigenmodes as the entropy-generating mechanism, causing a drastic noise amplification in the Goldstone mode.
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Affiliation(s)
- Thomas Suchanek
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Klaus Kroy
- Institut für Theoretische Physik, Universität Leipzig, Postfach 100 920, D-04009 Leipzig, Germany
| | - Sarah A M Loos
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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2
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Abstract
Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.
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3
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Iverson E, Yang M, Zhang H, McCoy JH. Nontrivial amplification below the threshold for excitable cell signaling. Phys Rev E 2020; 102:032409. [PMID: 33076000 DOI: 10.1103/physreve.102.032409] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2020] [Accepted: 07/13/2020] [Indexed: 11/07/2022]
Abstract
In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.
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Affiliation(s)
- Emma Iverson
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901
| | - Minjing Yang
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901
| | - Hongyong Zhang
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901
| | - Jonathan H McCoy
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901
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4
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Biancalani T, Jafarpour F, Goldenfeld N. Giant Amplification of Noise in Fluctuation-Induced Pattern Formation. PHYSICAL REVIEW LETTERS 2017; 118:018101. [PMID: 28106453 DOI: 10.1103/physrevlett.118.018101] [Citation(s) in RCA: 32] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2016] [Indexed: 05/21/2023]
Abstract
The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature. Here, we show that a large class of spatially extended dynamical systems driven by intrinsic noise can exhibit giant amplification, yielding patterns whose amplitude is comparable to that of deterministic Turing instabilities. The giant amplification results from the interplay between noise and nonorthogonal eigenvectors of the linear stability matrix, yielding transients that grow with time, and which, when driven by the ever-present intrinsic noise, lead to persistent large amplitude patterns. This mechanism shows that fluctuation-induced Turing patterns are observable, and are not strongly limited by the amplitude of demographic stochasticity nor by the value of the diffusion coefficients.
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Affiliation(s)
- Tommaso Biancalani
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080, USA and Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1206 West Gregory Drive, Urbana, Illinois 61801, USA
| | - Farshid Jafarpour
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080, USA and Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1206 West Gregory Drive, Urbana, Illinois 61801, USA
| | - Nigel Goldenfeld
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080, USA and Carl R. Woese Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1206 West Gregory Drive, Urbana, Illinois 61801, USA
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Mamatsashvili G, Khujadze G, Chagelishvili G, Dong S, Jiménez J, Foysi H. Dynamics of homogeneous shear turbulence: A key role of the nonlinear transverse cascade in the bypass concept. Phys Rev E 2016; 94:023111. [PMID: 27627396 DOI: 10.1103/physreve.94.023111] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2016] [Indexed: 11/07/2022]
Abstract
To understand the mechanism of the self-sustenance of subcritical turbulence in spectrally stable (constant) shear flows, we performed direct numerical simulations of homogeneous shear turbulence for different aspect ratios of the flow domain with subsequent analysis of the dynamical processes in spectral or Fourier space. There are no exponentially growing modes in such flows and the turbulence is energetically supported only by the linear growth of Fourier harmonics of perturbations due to the shear flow non-normality. This non-normality-induced growth, also known as nonmodal growth, is anisotropic in spectral space, which, in turn, leads to anisotropy of nonlinear processes in this space. As a result, a transverse (angular) redistribution of harmonics in Fourier space is the main nonlinear process in these flows, rather than direct or inverse cascades. We refer to this type of nonlinear redistribution as the nonlinear transverse cascade. It is demonstrated that the turbulence is sustained by a subtle interplay between the linear nonmodal growth and the nonlinear transverse cascade. This course of events reliably exemplifies a well-known bypass scenario of subcritical turbulence in spectrally stable shear flows. These two basic processes mainly operate at large length scales, comparable to the domain size. Therefore, this central, small wave number area of Fourier space is crucial in the self-sustenance; we defined its size and labeled it as the vital area of turbulence. Outside the vital area, the nonmodal growth and the transverse cascade are of secondary importance: Fourier harmonics are transferred to dissipative scales by the nonlinear direct cascade. Although the cascades and the self-sustaining process of turbulence are qualitatively the same at different aspect ratios, the number of harmonics actively participating in this process (i.e., the harmonics whose energies grow more than 10% of the maximum spectral energy at least once during evolution) varies, but always remains quite large (equal to 36, 86, and 209) in the considered here three aspect ratios. This implies that the self-sustenance of subcritical turbulence cannot be described by low-order models.
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Affiliation(s)
- G Mamatsashvili
- Helmholtz-Zentrum Dresden-Rossendorf, P.O. Box 510119, D-01314 Dresden, Germany.,Department of Physics, Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi 0179, Georgia.,Abastumani Astrophysical Observatory, Ilia State University, Tbilisi 0162, Georgia
| | - G Khujadze
- Chair of Fluid Mechanics, Universität Siegen, Siegen 57068, Germany
| | - G Chagelishvili
- Abastumani Astrophysical Observatory, Ilia State University, Tbilisi 0162, Georgia.,Institute of Geophysics, Tbilisi State University, Tbilisi 0128, Georgia
| | - S Dong
- School of Aeronautics, Universidad Politécnica de Madrid, Madrid 28040, Spain
| | - J Jiménez
- School of Aeronautics, Universidad Politécnica de Madrid, Madrid 28040, Spain
| | - H Foysi
- Chair of Fluid Mechanics, Universität Siegen, Siegen 57068, Germany
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6
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Dixon J, Lindemann A, McCoy JH. Transient amplification limits noise suppression in biochemical networks. Phys Rev E 2016; 93:012415. [PMID: 26871109 DOI: 10.1103/physreve.93.012415] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2015] [Indexed: 06/05/2023]
Abstract
Cell physiology is orchestrated, on a molecular level, through complex networks of biochemical reactions. The propagation of random fluctuations through these networks can significantly impact cell behavior, raising challenging questions about how network design shapes the cell's ability to suppress or exploit these fluctuations. Here, drawing on insights from statistical physics, fluid dynamics, and systems biology, we explore how transient amplification phenomena arising from network connectivity naturally limit a biochemical system's ability to suppress small fluctuations around steady-state behaviors. We find that even a simple system consisting of two variables linked by a single interaction is capable of amplifying small fluctuations orders of magnitude beyond the levels predicted by linear stability theory. We also find that adding additional interactions can promote further amplification, even when these interactions implement classic design strategies known to suppress fluctuations. These results establish that transient amplification is an essential factor determining baseline noise levels in stable intracellular networks. Significantly, our analysis is not bound to specific systems or interaction mechanisms: we find that noise amplification is an emergent phenomenon found near steady states in any network containing sufficiently strong interactions, regardless of its form or function.
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Affiliation(s)
- John Dixon
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901, USA
| | - Anika Lindemann
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901, USA
| | - Jonathan H McCoy
- Department of Physics and Astronomy, Colby College, Waterville, Maine 04901, USA
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7
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Squire J, Bhattacharjee A. Nonmodal growth of the magnetorotational instability. PHYSICAL REVIEW LETTERS 2014; 113:025006. [PMID: 25062200 DOI: 10.1103/physrevlett.113.025006] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2014] [Indexed: 06/03/2023]
Abstract
We analyze the linear growth of the magnetorotational instability (MRI) in the short-time limit using nonmodal methods. Our findings are quite different from standard results, illustrating that shearing wave energy can grow at the maximum MRI rate -dΩ/dlnr for any choice of azimuthal and vertical wavelengths. In addition, by comparing the growth of shearing waves with static structures, we show that over short time scales shearing waves will always be dynamically more important than static structures in the ideal limit. By demonstrating that fast linear growth is possible at all wavelengths, these results suggest that nonmodal linear physics could play a fundamental role in MRI turbulence.
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Affiliation(s)
- J Squire
- Department of Astrophysical Sciences and Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA
| | - A Bhattacharjee
- Department of Astrophysical Sciences and Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA and Max Planck/Princeton Center for Plasma Physics, Princeton University, Princeton, New Jersey 08543, USA
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8
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Ortiz de Zárate JM, Sengers JV. Nonequilibrium velocity fluctuations and energy amplification in planar Couette flow. Phys Rev E 2009; 79:046308. [PMID: 19518334 DOI: 10.1103/physreve.79.046308] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2009] [Indexed: 11/07/2022]
Abstract
In this paper we investigate intrinsic thermally excited nonequilibrium velocity fluctuations in laminar planar Couette flow. For this purpose we have complemented the solution of the stochastic Orr-Sommerfeld equation for the intensity of the fluctuations of the wall-normal velocity, presented in a previous publication, with a solution of the stochastic Squire equation for the intensity of the fluctuations of the wall-normal vorticity. We have obtained exact solutions of these equations without boundary conditions and solutions in a Galerkin approximation when appropriate boundary conditions are included. These results enable us to make a quantitative assessment of the intensity of these nonequilibrium fluctuations, as well as of the related energy amplification, which are always present, even in the absence of any externally imposed noise.
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Affiliation(s)
- José M Ortiz de Zárate
- Departamento de Física Aplicada I, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain.
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Khujadze G, Oberlack M, Chagelishvili G. Direct numerical simulation of stochastically forced laminar plane couette flow: peculiarities of hydrodynamic fluctuations. PHYSICAL REVIEW LETTERS 2006; 97:034501. [PMID: 16907504 DOI: 10.1103/physrevlett.97.034501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2005] [Indexed: 05/11/2023]
Abstract
The background of three-dimensional hydrodynamic (vortical) fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated numerically. The fluctuating field is anisotropic and has well pronounced peculiarities: (i) the hydrodynamic fluctuations exhibit nonexponential, transient growth; (ii) fluctuations with the streamwise characteristic length scale about 2 times larger than the channel width are predominant in the fluctuating spectrum instead of streamwise constant ones; (iii) nonzero cross correlations of velocity (even streamwise-spanwise) components appear; (iv) stochastic forcing destroys the spanwise reflection symmetry (inherent to the linear and full Navier-Stokes equations in a case of the Couette flow) and causes an asymmetry of the dynamical processes.
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Affiliation(s)
- G Khujadze
- Department of Mechanical Engineering, Group of Fluid Dynamics, Petersenstr. 13, Darmstadt, Germany
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10
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Fedotov S, Bashkirtseva I, Ryashko L. Stochastic dynamo model for subcritical transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:066307. [PMID: 16906976 DOI: 10.1103/physreve.73.066307] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2005] [Revised: 04/04/2006] [Indexed: 05/11/2023]
Abstract
The effects of stochastic perturbations in a nonlinear alpha Omega-dynamo model are investigated. By using transformation of variables we identify a "slow" variable that determines the global evolution of the non-normal alpha Omega-dynamo system in the subcritical case. We apply an adiabatic elimination procedure to derive a closed stochastic differential equation for the slow variable for which the dynamics is determined along one of the eigenvectors of the full system. We derive the corresponding Fokker-Planck equation and show that the generation of a large scale magnetic field can be regarded as a first-order phase transition. We show that the an advantage of the reduced system is that we have explicit expressions for both the stochastic and deterministic potentials. We also obtain the stationary solution of the Fokker-Planck equation and show that an increase in the intensity of the multiplicative noise leads to qualitative changes in the stationary probability density function. The latter can be interpreted as a noise-induced phase transition. By a numerical simulation of the stochastic galactic dynamo model, we show that the qualitative behavior of the "empirical" stationary pdf of the slow variable is accurately predicted by the stationary pdf of the reduced system.
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Affiliation(s)
- Sergei Fedotov
- School of Mathematics, The University of Manchester, M60 1QD, United Kingdom
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11
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Kulp CW, Tracy ER. Control of integrable Hamiltonian systems and degenerate bifurcations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:016205. [PMID: 15324150 DOI: 10.1103/physreve.70.016205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2003] [Indexed: 05/24/2023]
Abstract
We discuss control of low-dimensional systems which, when uncontrolled, are integrable in the Hamiltonian sense. The controller targets an exact solution of the system in a region where the uncontrolled dynamics has invariant tori. Both dissipative and conservative controllers are considered. We show that the shear flow structure of the undriven system causes a Takens-Bogdanov bifurcation to occur when control is applied. This implies extreme noise sensitivity. We then consider an example of these results using the driven nonlinear Schrödinger equation.
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Affiliation(s)
- C W Kulp
- Physics Department, College of William & Mary, Williamsburg, Virginia 23187-8795, USA
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12
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Fedotov S. Non-normal and stochastic amplification of magnetic energy in the turbulent dynamo: subcritical case. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:067301. [PMID: 14754360 DOI: 10.1103/physreve.68.067301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2003] [Indexed: 05/24/2023]
Abstract
Our attention focuses on the stochastic dynamo equation with non-normal operator that gives an insight into the role of stochastics and non-normality in magnetic field generation. The main point of this Brief Report is a discussion of the generation of a large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. The main result is a discovery of nonlinear deterministic instability and growth of finite magnetic field fluctuations in alpha beta dynamo theory. We present a simple stochastic model for the thin-disk axisymmetric alpha Omega dynamo involving three factors: (a) non-normality generated by differential rotation, (b) nonlinearity reflecting how the magnetic field affects the turbulent dynamo coefficients, and (c) stochastic perturbations. We show that even for the subcritical case (all eigenvalues are negative), there are three possible mechanisms for the generation of magnetic field. The first mechanism is a deterministic one that describes an interplay between transient growth and nonlinear saturation of the turbulent alpha effect and diffusivity. It turns out that the trivial state is nonlinearly unstable to small but finite initial perturbations. The second and third are stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation with random additive and multiplicative fluctuations.
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Affiliation(s)
- Sergei Fedotov
- Department of Mathematics, UMIST-University of Manchester, Institute of Science and Technology, P.O. Box 88, Manchester M60 1QD, United Kingdom
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Fedotov S, Bashkirtseva I, Ryashko L. Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066310. [PMID: 12513405 DOI: 10.1103/physreve.66.066310] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2002] [Indexed: 05/24/2023]
Abstract
The effects of stochastic perturbations on a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition are investigated both analytically and numerically. It is found that a nonlinear dynamical system with non-normal transient linear growth is very sensitive to the presence of weak random perturbations. The effect of non-normality on the exit probability from the zero fixed point is analyzed numerically for small values of the noise intensity parameter. It is found that an increase in the intensity of the noise, or a decrease of the non-normality parameter leads to qualitative changes in the behavior of the trajectories that can be interpreted as noise-induced phase transitions. By using the Itô formula and the adiabatic elimination procedure a stochastic equation governing the slow evolution of the energy of the non-normal system is derived.
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Affiliation(s)
- Sergei Fedotov
- Department of Mathematics, UMIST-University of Manchester Institute of Science and Technology, United Kingdom
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Longhi S. Pulse dynamics in actively mode-locked lasers with frequency shifting. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:056607. [PMID: 12513623 DOI: 10.1103/physreve.66.056607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2002] [Indexed: 05/24/2023]
Abstract
The dynamics of wave packet splitting in a dissipative Schrödinger-like dynamical system is theoretically studied by considering the model of an amplitude-modulated mode-locked laser with frequency shifting provided by an intracavity frequency modulation. It is shown that as the strength of the frequency modulation is increased, a bifurcation takes place which corresponds to a transition from a single-pulse steady-state oscillation to a two-pulse coherent oscillatory dynamics. An analytical model for pulse splitting bifurcation and onset of two-mode oscillatory dynamics, based on a Gaussian pulse analysis, is presented and compared with numerical simulations.
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Affiliation(s)
- S Longhi
- INFM, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy
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15
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Finn JM. The effect of Lagrangian chaos on locking bifurcations in shear flows. CHAOS (WOODBURY, N.Y.) 2002; 12:508-517. [PMID: 12779581 DOI: 10.1063/1.1468246] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.
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Affiliation(s)
- John M. Finn
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
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16
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Longhi S, Laporta P. Excess noise in intracavity laser frequency modulation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:R989-R992. [PMID: 11046520 DOI: 10.1103/physreve.61.r989] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/1999] [Indexed: 05/23/2023]
Abstract
The response to perturbations and to stochastic noise of a laser below threshold subjected to an intracavity periodic frequency modulation is theoretically studied. It is shown that, when the modulation frequency is close to the cavity axial mode separation but yet detuned from exact resonance, the laser exhibits a strongly enhanced sensitivity to external noise, which includes large transient energy amplification of perturbations in the deterministic case and enhancement of field fluctuations in presence of a continuous stochastic noise. This large excess noise is due to the nonorthogonality of Floquet laser modes which makes it possible continuous energy transfer from the forcing noise to transient growing perturbations.
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Affiliation(s)
- S Longhi
- Istituto Nazionale di Fisica per la Materia, Dipartimento di Fisica and CEQSE-CNR, Politecnico di Milano, Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy
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17
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Gog JR, Oprea I, Proctor MRE, Rucklidge AM. Destabilization by noise of transverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory. Proc Math Phys Eng Sci 1999. [DOI: 10.1098/rspa.1999.0498] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- J. R. Gog
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
| | - I. Oprea
- Faculty of Mathematics, University of Bucharest, Str. Academiei 14, Sector 1, Bucharest, Romania
| | - M. R. E. Proctor
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
| | - A. M. Rucklidge
- Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK
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18
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Eckhardt B, Mersmann A. Transition to turbulence in a shear flow. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:509-17. [PMID: 11969790 DOI: 10.1103/physreve.60.509] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/1997] [Revised: 02/16/1999] [Indexed: 04/18/2023]
Abstract
We analyze the properties of a 19-dimensional Galerkin approximation to a parallel shear flow. The laminar flow with a sinusoidal shape is stable for all Reynolds numbers Re. For sufficiently large Re additional stationary flows occur; they are all unstable. The lifetimes of finite amplitude perturbations shows a fractal dependence on amplitude and Reynolds number. These findings are in accord with observations on plane Couette flow and suggest a universality of this transition scenario in shear flows.
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Affiliation(s)
- B Eckhardt
- Fachbereich Physik, Philipps Universität Marburg, D-35032 Marburg, Germany
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19
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Gebhardt T, Grossmann S. Chaos transition despite linear stability. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:3705-3711. [PMID: 9962424 DOI: 10.1103/physreve.50.3705] [Citation(s) in RCA: 88] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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