High-Dimensional Chaos in Dissipative and Driven Dynamical Systems

Abstract

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

The stability of the suggested planet in the ν Octantis system: a numerical and statistical study

“…The second criterion utilizes the method of Lyapunov exponents (Lyapunov 1907), which is commonly used in non‐linear dynamics for determining the onset of chaos in different dynamical systems (e.g. Hilborn 1994; Musielak & Musielak 2009). A numerical algorithm that is typically adopted for calculating the spectrum of Lyapunov exponents was originally developed by Wolf et al (1985).…”

The instability transition for the restricted 3-body problem

“…A dynamical system with n degrees of freedom is represented in 2n phase space; thus, to fully determine the stability of the system all 2n Lyapunov exponents must be calculated. The Lyapunov exponents are the most commonly used tools to determine the onset of chaos and chaotic behaviour of both dissipative (e.g., Musielak & Musielak 2009, and references therein) and non-dissipative systems of orbital mechanics (e.g., Lissauer 1999,and references therein). The positive Lyapunov exponents measure the rate of divergence of neighbouring orbits, whereas negative exponents measure the convergence rates between stable manifolds.…”

“…The positive Lyapunov exponents measure the rate of divergence of neighbouring orbits, whereas negative exponents measure the convergence rates between stable manifolds. For dissipative dynamical systems the sum of all Lyapunov exponents is less than 0 (e.g., Musielak & Musielak 2009); however, for Hamiltonian (non-dissipative) systems the sum is equal to 0 (e.g., Hilborn 1994).…”

The three-body problem

“…He also showed that solutions could leave two equilibria and then return to them asymptotically, thus forming a heteroclinic orbit. The above concepts originally introduced by Poincaré are now commonly used in modern theories of chaos [Thompson and Stewart, 1986, Hilborn, 1994, Musielak and Musielak, 2009.…”

“…The third case considers a Lyapunov exponent that is exactly equal to zero, which means that the two trajectories are parallel to each other and by induction will remain so unless another force arises to disrupt the system. The mathematical definition of the Lyapunov exponent [Hilborn, 1994, Musielak andMusielak, 2009] is…”