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In this article we prove that for a $C^{1+\alpha}$ diffeomorphism on a compact Riemannian manifold, if there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then the topological entropy of the set of irregular…
The chaotic properties of Newton-Leipnik system are discussed from the view point of strange attractors. Previously, two strange attractors of this system were illustrated which occured from two different initial conditions under the same…
We use numerical and analytical tools to demonstrate arguments in favor of the existence of a family of smooth, symplectic diffeomorphisms of the two-dimensional torus that have both a positive measure set with positive Lyapunov exponent…
We show that the threshold of complete synchronization in a lattice of coupled non-smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace. As a result, the numerically…
In this paper, a numerical study on the complete synchronization phenomenon exhibited by coupled forced negative conductance circuits is presented. The nonlinear system exhibiting two types of chaotic attractors is studied for complete…
Repulsive and attractive interactions usually lead to very different physics. Striking exceptions exist in the dynamics of driven-dissipative quantum systems. For the example of a photonic Bose-Hubbard dimer, we establish a one-to-one…
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…
We present a moduli space for all hyperbolic basic sets of diffeomorphisms on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. To do this we introduce two new invariants: the measure…
This article is devoted to the study of the multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local…
We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov…
We answer a question of Burns and Wilkinson, showing that there are open families of volume-preserving partially hyperbolic diffeomorphisms which are accessible and center bunched and neither dynamically coherent nor Anosov. We also show in…
Chaotic maps are very important for establishing chaos-based image encryption systems. This paper introduces a coupling chaotic system based on a certain unit transform, which can combine any two 1D chaotic maps to generate a new one with…
We propose a general algorithm for computer assisted verification of uniform hyperbolicity for maps which exhibit a robust attractor. The method has been successfully applied to a Poincare map for a system of coupled non-autonomous van der…
A method for the quantitative analysis of the degree and parameters of synchronization of the chaotic oscillations in two coupled oscillators is proposed, which makes it possible to reveal a change in the structure of attractors. The…
In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if…
The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and…
In four-dimensional symplectic maps complex instability of periodic orbits is possible, which cannot occur in the two-dimensional case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter…
In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian…
Dynamics of a cluster of chaotic oscillators on a network are studied using coupled maps. By introducing the association schemes, we obtain coupling strength in the adjacency matrices form, which satisfies Markov matrices property. We…
Examples of one-dimensional lattice systems are considered, in which patterns of different spatial scales arise alternately, so that the spatial phase over a full cycle undergo transformation according to expanding circle map that implies…