Analysis of chaotic saddles in high-dimensional dynamical systems: The Kuramoto–Sivashinsky equation

Abstract

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1759297͔In the past decades chaos theory has been revealed as a powerful way to explain the physics of low-dimensional dynamical systems, that is, dynamical systems with a small number of state variables. However, most problems of practical interest in physics and engineering are described by partial differential equations "PDEs…, and it is still unclear in which situations the physical mechanisms responsible for the onset of chaos in low-dimensional systems are applicable to systems described by PDEs. Several authors have tried to develop a dynamical systems theory for high-dimensional dynamical systems, frequently described by sets of coupled ordinary differential equations obtained as approximations to the original PDEs. 1-7 The simplest approach consists in first ''borrowing'' the tools developed for low-dimensional dynamical systems and applying them to spatially extended systems in regimes that display characteristics of lowdimensional chaos. Then, it is possible to develop the tools to be used in more complex regimes. In this paper we try to improve the understanding of the connection between low-dimensional chaotic systems and a certain class of spatiotemporal systems: the one that exhibits temporal chaos and spatial coherence. We focus on the development of a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems.

Characterization of intermittency at the onset of turbulence in the forced and damped nonlinear Schrödinger equation

“…Finally ∈ [0.1, 0.45] [25]. The equation was solved numerically using the pseudo-spectral method [31,32] and a 12 th order predictor-corrector Adams integrator [33], with time step h = 10 −2 , and a total of N = 64 Fourier modes were simulated.…”

Lyapunov exponents of the Kuramoto--Sivashinsky PDE

“…Of particular interest for periodic domains (2) are the Galilean invariance, u(x, t) → u(x − ct, t) + c for all speeds c, and the spatial translation invariance, u(x, t) → u(x + d, t) for all d. These two symmetries do not hold for odd-periodic domains (3). The Kuramoto-Sivashinsky pde (1) with oddperiodic domains (3) is particularly well studied (e.g., Rempel et al 2004;Lan and Cvitanović 2008;Foias et al 1986;Eguíluz et al 1999) compared to that with periodic domains, as the removal of periodic symmetries simplifies somewhat the analysis of the dynamics. The relatively simpler dynamics of the odd-periodic case is observed in Section 3 when comparing the periodic and odd-periodic Lyapunov spectra.…”

Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

“…This periodic window ends with an interior crisis (IC) at a IC =0.330248. To plot the chaotic saddle, for each value of a, we plot a straddle trajectory close to the chaotic saddle using the PIM triple algorithm with a precision of 10 −6 (Nusse and York, 1989;Rempel and Chian, 2004;Rempel et al, 2004a). The blue region inside the periodic window in Fig.…”

“…Kawahara and Kida (2001) numerically found an unstable periodic orbit in a three-dimensional plane Couette turbulence described by the incompressible Navier-Stokes equation. (Chian et al, 2002) and Rempel et al (2004a) showed that unstable periodic orbits and chaotic saddles can characterize an interior crisis and the intermittency induced by an interior crisis in the Kuramoto-Sivashinsky equation. Faisst and Eckhard (2003) identified a family of unstable traveling waves originating from saddle-node bifurcations in a Published by Copernicus GmbH on behalf of the European Geosciences Union and the American Geophysical Union.…”