Lyapunov Exponents for Continuous-Time Dynamical Systems
نویسندگان
چکیده
In this article, different methods of computing Lyapunov exponents for continuous-time dynamical systems are briefly reviewed. The relative merits and demerits of these methods are pointed out. 1. Preliminaries The problem of detecting and quantifying chaos in a wide variety of systems is an ongoing and important activity. In this context, computing the spectrum of Lyapunov exponents has proven to be the most useful dynamical diagnostic for chaotic systems. The Lyapunov exponents give the average exponential rates of divergence or convergence of nearby orbits in the phase-space. In systems exhibiting exponential orbital divergence, small initial differences which we may not be able to resolve get magnified rapidly leading ∗Also at Centre for Theoretical Studies; e-mail address: [email protected]
منابع مشابه
Lyapunov Exponents and Strange Attractors in Discrete and Continuous Dynamical Systems
4 Lyapunov Exponents 5 4.1 Definition and basic properties . . . . 6 4.2 Constraints on the Lyapunov exponents 7 4.3 Calculating the largest Lyapunov exponent method 1 . . . . . . . . . . . 7 4.4 Calculating the largest Lyapunov exponent method 2 . . . . . . . . . . . 8 4.4.1 Maps . . . . . . . . . . . . . . 8 4.4.2 Continuous systems . . . . . . 8 4.5 Calculating the other Lyapunov exponents ....
متن کاملOn the Computation of Lyapunov Exponents for Continuous Dynamical Systems∗
In this paper, we consider discrete and continuous QR algorithms for computing all of the Lyapunov exponents of a regular dynamical system. We begin by reviewing theoretical results for regular systems and present general perturbation results for Lyapunov exponents. We then present the algorithms, give an error analysis of them, and describe their implementation. Finally, we give several numeri...
متن کاملAn efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems
This paper proposes a new approach for computing the Lyapunov Characteristic Exponents (LCEs) for continuous dynamical systems in an ecient and numerically stable fashion. The method is adapted to systems with small dimensions. Numerical examples illustrating the accuracy of method are presented.
متن کاملLyapunov Exponents for Continuous Transformations and Dimension Theory
We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous t...
متن کاملComputing Lyapunov Exponents for Time-Delay Systems
The hall mark property of a chaotic attractor, namely sensitive dependence on initial condition, has been associated by the Lyapunov exponents to characterize the degree of exponential divergence/convergence of trajectories arising from nearby initial conditions. At first, we will describe briefly the concept of Lyapunov exponent and the procedure for computing Lyapunov exponents of the flow of...
متن کامل