arXiv subjects

Pavel V. Kuptsov

Publications and source records attributed to Pavel V. Kuptsov.

24 records · Page 2Linked to original sources

arXiv2011

Fast numerical test of hyperbolic chaos

The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. 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 indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for high-dimensional system this amount is normally much less then the full phase space dimension, this method provide the fast and memory saving way for numerical hyperbolicity test of such systems.

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arXiv2011

A large-deviation approach to space-time chaos

In this Letter we show that the analysis of Lyapunov-exponents 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 $\bf D$ (a dynamical invariant itself) is measured and analysed 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; (iii) checking the hyperbolicity of the dynamics.

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arXiv2010

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 non-zero 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.

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arXiv2009

Violation of hyperbolicity in a diffusive medium with local hyperbolic attractor

Departing from a system of two non-autonomous amplitude equations, demonstrating hyperbolic chaotic dynamics, we construct a 1D medium as ensemble of such local elements introducing spatial coupling via diffusion. When the 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 occurs 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 two-dimensional intersections of expanding and contracting subspaces that we classify as a stronger type of the violation. Beyond of 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.

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arXiv2008

Hyperbolic attractor of Smale-Williams type in a system of two coupled non-autonomous amplitude equations

Recently, a system with uniformly hyperbolic attractor of Smale-Williams type has been suggested [Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005]. This system consists of two coupled non-autonomous van der Pol oscillators and admits simple physical realization. In present paper we introduce amplitude equations for this system and prove that the attractor of the system of amplitude equations is also uniformly hyperbolic. Also we represent qualitative illustrations as well as quantitative characteristics of a chaotic dynamics on this attractor.

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arXiv2007

Flow- and Diffusion Distributed Structures with noise at the inlet

Flow and Diffusion Distributed Structures (FDS) are stationary spatially periodic patterns that can be observed in reaction-diffusion-advection systems. These structures arise when the flow rate exceeds a certain bifurcation point provided that concentrations of interacting species at the inlet differ from steady state values and the concentrations are held constant. Normally, theoretical studies of these patterns are developed without concerning a noise. In this paper we consider FDS for a more realistic conditions and assume that the inlet concentrations are perturbed by a small noise. When the flow rate is small, the FDS is linearly sensitive to noise at the inlet. Even weak fluctuations destroy the stationary pattern and an oscillatory solution appears instead. For higher flow rates the instability becomes nonlinear: the pattern remains unaltered for a weak noise and undergoes the destruction when the noise amplitude passes a certain threshold. We represent a detailed description of these effects and examine two scenarios for the stabilization.

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