Analytical and numerical methods for the study of attractors in dynamical systems: bifurcations, localization and dimension characteristics

This lecture is devoted to recent results on the study of attractors in dynamical systems. Effective analytical and numerical methods for the study of transition to chaos, localization of attractors, and dimension characteristics of chaotic attractors are discussed. Recently it was suggested to classify the attractors in dynamical systems as being hidden either self-excited [1, 2]: an attractor is called self-excited if its basin of attraction intersects with any vicinity of an equilibrium, otherwise it is called a hidden attractor. This allowed for combining the notions of transition processes in engineering systems, visualization in numerical mathematics, the basin of attraction, and the stability of dynamical systems. The classification, not only demonstrated difficulties of fundamental problems (e.g., the second part of Hilbert's 16th problem on the number and mutual disposition of limit cycles, Aizerman's and Kalman's conjecture on the monostability of nonlinear systems) and applied systems analysis, but also triggered the discovery of new hidden attractors in well-known physical and engineering models [1, 3, 4]. For the study of attractors and estimating the Hausdorff dimension the concept of the Lyapunov dimension was suggested by Kaplan and Yorke. Along with widely used numerical methods for computing the Lyapunov dimension it was developed an effective analytical approach, which is based on the direct Lyapunov method with special Lyapunov-like functions [5, 6]. The advantage of the method is that it allows one, in many cases, to estimate the Lyapunov dimension of an invariant set without localization of the set in the phase space, to prove Eden's conjecture for the self-excited attractors and get exact Lyapunov dimension formula for attractors of various well-known dynamical systems (e.g., such as the Chirikov, Henon, Lorenz, Shimizu-Morioka, and Glukhovsky-Dolzhansky systems). Also approaches for reliable numerical estimation of the finite-time Lyapunov exponents and finite-time Lyapunov dimension are discussed [7, 6]. The homoclinic orbits play an important role in the bifurcation theory and in scenarios of the transition to chaos. In the case of dissipative systems, the proof of the existence of homoclinic orbits is a challenging task. Recently it was developed an effective method, called Fishing principle, which allows one to obtain necessary and sufficient conditions of the existence of homoclinic orbits in various well-known dynamical systems. [2, 8, 9, 10].
References [1] G. Leonov, N. Kuznetsov, Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits, International Journal of Bifurcation and Chaos 23 (1), art. no. 1330002. doi:10.1142/S0218127413300024.1 [2] G. Leonov, N. Kuznetsov, T. Mokaev, Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective uid motion, Eur. Phys. J. Special Topics 224 (8) (2015) 1421–1458. doi:10.1140/epjst/e2015-02470-3. [3] N. Kuznetsov, Hidden attractors in fundamental problems and engineering models. A short survey, Lecture Notes in Electrical Engineering 371 (2016) 13–25, (Plenary lecture at International Conference on Advanced Engineering Theory and Applications 2015). doi:10.1007/978-3-319-27247-4 2. [4] D. Dudkowski, S. Jafari, T. Kapitaniak, N. Kuznetsov, G. Leonov, A. Prasad, Hidden attractors in dynamical systems, Physics Reports 637 (2016) 1–50.doi:10.1016/j.physrep.2016.05.002. [5] G. Leonov, Lyapunov functions in the attractors dimension theory, Journal of Applied Mathematics and Mechanics 76 (2) (2012) 129–141. [6] N. Kuznetsov, The Lyapunov dimension and its estimation via the Leonov method, Physics Letters A 380 (25–26) (2016) 2142–2149. doi:10.1016/j.physleta.2016.04.036. [7] N. Kuznetsov, T. Alexeeva, G. Leonov, Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations, Nonlinear Dynamics 85 (1) (2016) 195–201. doi:10.1007/s11071-016-2678-4. [8] G. Leonov, Shilnikov chaos in Lorenz-like systems, International Journal of Bifurcation and Chaos 23 (03), art. num. 1350058. doi:10.1142/S0218127413500582. [9] G. Leonov, The Tricomi problem on the existence of homoclinic orbits in dissipative systems, Journal of Applied Mathematics and Mechanics 77 (3) (2013) 296– 304. [10] G. Leonov, General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu-Morioka, Lu and Chen systems, Physics Letters A 376(2012) 3045–3050.