BIMSA Integrable Systems Seminar | On robust chaos

Abstract

One of the most fundamental problems in multidimensional chaos theory is the study of strange attractors which are robustly chaotic (i.e., they remain chaotic after small perturbations of the system). It was hypothesized in [1] that the robustness of chaoticity is equivalent to the pseudohyperbolicity of the attractor. Pseudohyperbolicity is a generalization of hyperbolicity. The main characteristic property of a pseudohyperbolic attractor is that each of its orbits has a positive maximal Lyapunov exponent. In addition, this property must be preserved under small perturbations. The foundations of the theory of pseudohyperbolic attractors were laid by Turaev and Shilnikov [2,3], who showed that the class of pseudohyperbolic attractors, besides the classical Lorenz and hyperbolic attractors, also includes wild attractors which contain orbits with a homoclinic tangency. In this talk we give a review on the theory of pseudohyperbolic attractors arising in both systems with continuous and discrete time. At first, we explain what is meant under pseudohyperbolic attractors. Then, we describe our methods for the pseudohyperbolicity verification. We demonstrate the applicability of these methods for several well-known systems (with both pseudohyperbolic and non-pseudohyperbolic attractors). Finally, we present new examples of pseudohyperbolic attractors. [1] Gonchenko, S., Kazakov, A., & Turaev, D. (2021). Wild pseudohyperbolic attractor in a four-dimensional Lorenz system. Nonlinearity, 34(4), 2018. [2] Turaev, D. V., & Shilnikov, L. P. (1998). An example of a wild strange attractor. Sbornik: Mathematics, 189(2), 291. [3] Turaev, D. V., & Shilnikov, L. P. (2008, February). Pseudohyperbolicity and the problem on periodic perturbations of Lorenz-type attractors. In Doklady Mathematics (Vol. 77, pp. 17-21).