Chaos in Dynamical Systems
1992; Elsevier BV; Linguagem: Inglês
10.1016/b978-0-12-784391-9.50010-5
Autores Tópico(s)Mathematical Dynamics and Fractals
ResumoThis chapter focuses on chaos in dynamical systems. Chaos is a phenomenon that can appear in solutions to nonlinear differential equations. Chaos is easily defined and can be easily (numerically) found in some equations. For nonlinear systems exhibiting chaos, the separation of two nearby trajectories increases exponentially with time. This is referred to as sensitive dependence on initial conditions. For dissipative systems, a stretching in one direction has to be accompanied by a more-than-compensating contraction in other directions, so that the volume of an arbitrary droplet of initial conditions will contract with time. The phase-space trajectories for a chaotic system asymptotically approach a strange attractor, an attractor with a fractional dimension. The chapter highlights that Lyapunov exponents are a measure of the rate of divergence (or convergence) of initially infinitesimally separated trajectories. The ith Lyapunov exponent λi, can be found by considering the evolution of a vanishingly small set of initial conditions which form a hyperellipsoid.
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