Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

Abstract

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rossler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rossler system. Using the example of the Vallis system describing the El Nino-Southern Oscillation it is demonstrated an analytical approach for localization of self-excited and hidden attractors, which allows to obtain the exact formulas or estimates of their Lyapunov dimensions.

References in corpus (8)

• Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
• Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity
• Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system
• Differences and similarities in the analysis of Lorenz, Chen, and Lu systems
• The Lyapunov dimension and its estimation via the Leonov method
• On the mathematically reliable long-term simulation of chaos of Lorenz equation in the interval [0,10000]
• Hidden attractors in fundamental problems and engineering models
• Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems