Route to hyperbolic hyperchaos in a nonautonomous time-delay system.

We consider a self-oscillator whose excitation parameter is varied. The frequency of the variation is much smaller than the natural frequency of the oscillator so that oscillations in the system are periodically excited and decayed. Also, a time delay is added such that when the oscillations start to grow at a new excitation stage, they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to nonlinearity, the seeding from the past arrives with a doubled phase so that the oscillation phase changes from stage to stage according to the chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period, we found a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as an alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with frequent visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; and (d) transformation of hyperchaos to the hyperbolic form.

Hyperbolic chaos of Turing patterns.

We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation longwave and shortwave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions.

Large-deviation approach to space-time chaos.

In this Letter, we show that the analysis of Lyapunov-exponent fluctuations contributes to deepen our understanding of high-dimensional chaos. This is achieved by introducing a gaussian approximation for the large-deviation function that quantifies the fluctuation probability. More precisely, a diffusion matrix D (a dynamical invariant itself) is measured and analyzed in terms of its principal components. The application of this method to three (conservative, as well as dissipative) models allows (i) quantifying the strength of the effective interactions among the different degrees of freedom, (ii) unveiling microscopic constraints such as those associated to a symplectic structure, and (iii) checking the hyperbolicity of the dynamics.

Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor.

Departing from a system of two nonautonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a one-dimensional medium as an ensemble of such local elements introducing spatial coupling via diffusion. When length of the medium is small, all spatial cells oscillate synchronously, reproducing the local hyperbolic dynamics. This regime is characterized by a single positive Lyapunov exponent. The hyperbolicity survives when the system gets larger in length so that the second Lyapunov exponent passes zero and the oscillations become inhomogeneous in space. However, at a point where the third Lyapunov exponent becomes positive, some bifurcation occur that results in violation of the hyperbolicity due to the emergence of one-dimensional intersections of contracting and expanding tangent subspaces along trajectories on the attractor. Further growth of the length results in the two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond the point of the hyperbolicity loss, the system demonstrates an extensive spatiotemporal chaos typical for extended chaotic systems: when the length of the system increases the Kaplan-Yorke dimension, the number of positive Lyapunov exponents and the upper estimate for Kolmogorov-Sinai entropy grow linearly, while the Lyapunov spectrum tends to a limiting curve.

Numerical test for hyperbolicity of chaotic dynamics in time-delay systems.

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.

Stability of flow- and diffusion-distributed structures to inlet noise effects.

Stationary flow- and diffusion-distributed structures (FDS) patterns appear in a reaction-diffusion-advection system when a constant forcing is applied at the inlet of the reactor. We show that if the forcing is subject to noise, the FDS can be destroyed via the noise-induced Hopf instability. However, the FDS patterns are restored if the flow rate is sufficiently high. We demonstrate that the critical flow rate which is required for the stabilization of FDS has a power-law dependence on the noise amplitude.

Pattern formation in a two-dimensional reaction-diffusion channel with Poiseuille flow.

We study stationary patterns arising from a combination of flow and diffusion in a two-dimensional (2D) reaction-diffusion system in a channel with Poiseuille flow. Both transverse and longitudinal modes are investigated and compared with numerical computations.

Variety of regimes of starlike networks of Hénon maps.

In this paper we categorize dynamical regimes demonstrated by starlike networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a starlike structure is the main motif of them. We analyze starlike networks of Hénon maps. They are found to demonstrate a huge diversity of regimes. Varying the coupling strength we reveal chaos, quasiperiodicity, and periodicity. The nodes can be both fully and phase synchronized. The hub node can be either synchronized with the subordinate nodes or oscillate separately from fully synchronized subordinates. There is a range of wild multistability where the zoo of regimes is the most various. One can hardly predict here even a qualitative nature of the expected solution, since each perturbation of the coupling strength or initial conditions results in a new character of dynamics.

Predictable nonwandering localization of covariant Lyapunov vectors and cluster synchronization in scale-free networks of chaotic maps.

Covariant Lyapunov vectors for scale-free networks of Hénon maps are highly localized. We revealed two mechanisms of the localization related to full and phase cluster synchronization of network nodes. In both cases the localization nodes remain unaltered in the course of the dynamics, i.e., the localization is nonwandering. Moreover, this is predictable: The localization nodes are found to have specific dynamical and topological properties and they can be found without computing of the covariant vectors. This is an example of explicit relations between the system topology, its phase-space dynamics, and the associated tangent-space dynamics of covariant Lyapunov vectors.

Fast numerical test of hyperbolic chaos.

An effective numerical method for testing the hyperbolicity of chaotic dynamics is suggested. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicates a tangency between expanding and contracting subspaces. To perform the test one has to solve several copies of equations for infinitesimal perturbations whose number is equal to the sum of numbers of positive and zero Lyapunov exponents. Since this number is normally much less than the full phase space dimension, this method provides a fast and memory saving way for numerical hyperbolicity testing.

Strict and fussy mode splitting in the tangent space of the Ginzburg-Landau equation.

In the tangent space of some spatially extended dissipative systems one can observe "physical" modes which are highly involved in the dynamics and are decoupled from the remaining set of hyperbolically "isolated" degrees of freedom representing strongly decaying perturbations. This mode splitting is studied for the Ginzburg-Landau equation at different strength of the spatial coupling. We observe that isolated modes coincide with eigenmodes of the homogeneous steady state of the system; that there is a local basis where the number of nonzero components of the state vector coincides with the number of "physical" modes; that in a system with finite number of degrees of freedom the strict mode splitting disappears at finite value of coupling; that above this value a fussy mode splitting is observed.