Emergence of chaotic behaviour in linearly stable systems (original) (raw)
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Chaotic synchronizations of spatially extended systems as nonequilibrium phase transitions
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008
Two replicas of spatially extended chaotic systems synchronize to a common spatio-temporal chaotic state when coupled above a critical strength. As a prototype of each single spatio-temporal chaotic system a lattice of maps interacting via power-law coupling is considered. The synchronization transition is studied as a non-equilibrium phase transition, and its critical properties are analyzed at varying the spatial interaction range as well as the nonlinearity of the dynamical units composing each system. In particular, continuous and discontinuous local maps are considered. In both cases the transitions are of the second order with critical indexes varying with the exponent characterizing the interaction range. For discontinuous maps it is numerically shown that the transition belongs to the anomalous directed percolation (ADP) family of universality classes, previously identified for Lévy-flight spreading of epidemic processes. For continuous maps, the critical exponents are different from those characterizing ADP, but apart from the nearest-neighbor case, the identification of the corresponding universality classes remains an open problem. Finally, to test the influence of deterministic correlations for the studied synchronization transitions, the chaotic dynamical evolutions are substituted by suitable stochastic models. In this framework and for the discontinuous case, it is possible to derive an effective Langevin description that corresponds to that proposed for ADP.
Dynamics of perturbations in disordered chaotic systems
Physical Review E, 2008
We study the time evolution of perturbations in spatially extended chaotic systems in the presence of quenched disorder. We find that initially random perturbations tend to exponentially localize in space around static pinning centers that are selected by the particular configuration of disorder. The spatiotemporal behavior of typical perturbations ␦u͑x , t͒ is analyzed in terms of the Hopf-Cole transform h͑x , t͒ϵln͉␦u͑x , t͉͒. Our analysis shows that the associated surface h͑x , t͒ self-organizes into a faceted structure with scale-invariant correlations. Scaling analysis of critical roughening exponents reveals that there are three different universality classes for error propagation in disordered chaotic systems that correspond to different symmetries of the underlying disorder. Our conclusions are based on numerical simulations of disordered lattices of coupled chaotic elements and equations for diffusion in random potentials. We propose a phenomenological stochastic field theory that gives some insights on the path for a generalization of these results for a broad class of disordered extended systems exhibiting space-time chaos.
Collapse of Spatiotemporal Chaos
Physical Review Letters, 2003
The transient nature of spatiotemporal chaos is examined in reaction-diffusion systems with coexisting stable states. We find the apparent asymptotic spatiotemporal chaos of the Gray-Scott system to be transient, with the average transient lifetime increasing exponentially with medium size. The collapse of spatiotemporal chaos arises when statistical spatial correlations produce a quasihomogeneous medium, and the system obeys its zero-dimensional dynamics to relax to its stable asymptotic state.
The origin of diffusion: the case of non-chaotic systems
Physica D: Nonlinear Phenomena, 2003
We investigate the origin of diffusion in non-chaotic systems. As an example, we consider 1D map models whose slope is everywhere 1 (therefore the Lyapunov exponent is zero) but with random quenched discontinuities and quasi-periodic forcing. The models are constructed as non-chaotic approximations of chaotic maps showing deterministic diffusion, and represent one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the Ehrenfest wind-tree model). In particular, a simple construction shows that these maps define non-chaotic billiards in space-time. The models exhibit, in a wide range of the parameters, the same diffusive behavior of the corresponding chaotic versions. We present evidence of two sufficient ingredients for diffusive behavior in one-dimensional, non-chaotic systems: (i) a finite size, algebraic instability mechanism; (ii) a mechanism that suppresses periodic orbits.
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Physica D: Nonlinear Phenomena, 2001
We examine some new diagnostics for the behavior of a field ρ evolving in an advective-diffusive system. One of these diagnostics is approximately the Fourier second moment (denoted as χ 2 ) and the other is the linear entropy or field intensity S, the latter being significantly easier to compute or measure. We establish that as a result of chaos the increasing structure in ρ is accompanied by χ increasing exponentially rapidly in time at a rate given by ρ-dependent Lyapunov exponents Λ i and dominated by the largest one Λ max . Noise or diffusive coarse-graining of ρ causes χ to decrease as χ 2 ≈ 1 4 Dt, where D is a measure of the diffusion. When both effects are present the competition between the processes leads to metastability for χ followed by a final diffusive tail. The initial stages may be chaotic or diffusive depending upon the value of Λ −1 max 2Dχ 2 (0) but the metastable value of χ 2 is given by χ 2 * = i Λ i /2D irrespective. SinceṠ = −2Dχ 2 , similar analysis applies to S, and in particular there exists a metastable decay rate for S given byṠ * = i Λ i . These arguments are verified for a simple case, the Arnol'd Cat Map with added diffusive noise.
Breakdown of Universality in Transitions to Spatiotemporal Chaos
In this Letter we show that the transition from laminar to active behavior in extended chaotic systems can vary from a continuous transition in the universality class of Directed Percolation with infinitely many absorbing states to what appears as a first order transition. The latter occurs when finite lifetime non-chaotic structures, called "solitons", dominate the dynamics. We illustrate this scenario in an extension of the deterministic Chaté-Manneville coupled map lattice model and in a soliton including variant of the stochastic Domany-Kinzel cellular automaton.
Onset of spatiotemporal chaos in a nonlinear system
Physical Review E, 2007
We describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state. The onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on-off intermittency.
Nonlinear Phenomena and Complex Systems, 2004
We show that the presence of undulated boundaries can induce the formation of spatially chaotic, stationary, and stable structures in models as simple as the Fisher-Kolmogorov equation, which does not display any kind of chaos under common boundaries.
Noise-induced unstable dimension variability and transition to chaos in random dynamical systems
Physical Review E - PHYS REV E, 2003
Results are reported concerning the transition to chaos in random dynamical systems. In particular, situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle, which can be expected in any periodic window of a nonlinear dynamical system. Under noise, the asymptotic attractor of the system can become chaotic, as characterized by the appearance of a positive Lyapunov exponent. Generic features of the transition include the following: (1) the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions (unstable dimension variability), and this nonhyperbolicity develops as soon as the attractor becomes chaotic; (2) for systems described by differential equations, the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero Lyapunov exponent after the noisy attractor becomes chaotic; and (3) the largest Lyapu...