Abstract
In this article, on the example of the known low-order dynamical models, namely Lorenz, Rössler 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 Rössler system. Using the example of the Vallis system describing the El Ninõ-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.
Highlights
History of the turbulence phenomena study is associated with the consideration of various models, which include the Navier-Stokes equations, their Galerkin approximations, and the development of the theory of chaos [1, 2, 3, 4]
Lorenz numerically found a chaotic attractor in the model
In numerical computation of a trajectory over a nite-time interval it is dicult to distinguish a sustained chaos from a transient chaos [21, 22], it is reasonable to give a similar classication for transient chaotic sets [23, 15]: a transient chaotic set is called a hidden transient chaotic set if it does not involve and attract trajectories from a small neighborhood of equilibria; otherwise, it is called self-excited
Summary
History of the turbulence phenomena study is associated with the consideration of various models, which include the Navier-Stokes equations, their Galerkin approximations, and the development of the theory of chaos [1, 2, 3, 4]. Lorenz numerically found a chaotic attractor in the model. If for a particular attractor its basin of attraction is connected with the unstable manifold of unstable equilibrium, the localization procedure is quite simple. Whether for some parameters there exists a hidden Lorenz attractor. This question is related to the chaotic generalization [11] of the second part of Hilbert's 16th problem on the number and mutual disposition of attractors and repellers in the chaotic multidimensional dynamical. We note that the Lorenz system (1) with parameters r = 28, σ = 10, b = 8/3 is dissipative in the sense of Levinson, and for any initial data (except for equilibria) the trajectorytends to the attractor. Systems and, in particular, their dependence on the degree of polynomials in the model
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
