Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in Higher Dimensions

Abstract

Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices. These transients are caused by the presence of chaotic saddles, and they are a common phenomenon in higher dimensional dynamical systems. For many physical systems, chaotic saddles have a big impact on laboratory measurements, but there has been no way to observe these chaotic saddles directly. We present the first general method to locate and visualize chaotic saddles in higher dimensions.

A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill's problem

“…A recently developed numerical method [21] may play a role for this purpose. Finally, we remark that our results are not limited to 3 DOF.…”

Amplitude-Phase Synchronization at the Onset of Permanent Spatiotemporal Chaos

“…The degree of phase synchronization due to multiscale interactions can be quantified by the Fourier phase It has been shown that the on-off states in the time series of spatiotemporal intermittency at the onset of permanent STC are linked to chaotic saddles embedded in the chaotic attractor [6]. We choose a Poincaré map defined by Refû 1 ðtÞg ¼ 0 and dRefû 1 ðtÞg=dt > 0, then apply the stagger-and-step method [12] to find chaotic saddles, before and after the transition to permanent STC. Figure 2 shows a three-dimensional projection of the Poincaré map defined by (Refû 2 g, Imfû 2 g, Refû 3 g).…”

Edge of chaos and genesis of turbulence

“…We define a new initial condition u 1 = u(t 1 ), with lifetime T (u 1 ) = T c , and generate random perturbations r such that T (u 1 + r) > T c . Sweet et al [14] found that the random search is faster when ||r|| = 10 −s , with s being a uniformly distributed random number between 3 and the machine precision, 15 in our case. The perturbation r which increases the lifetime of the initial condition u 1 is called a "stagger", and the trajectory obtained integrating u 1 + r is the "step".…”

“…The maximum Lyapunov exponent quantifies the degree of temporal chaoticity of the system. In order to characterize the spatiotemporal dynamics of the chaotic saddle STCS created at > ∼ 0.11 as a function of , first we generate arbitrarily long trajectories near the STCS by using the stagger-and-step method [14]. To quantify the degree of spatial disorder of the STCS we compute the time-average of the Fourier power spectral Shannon entropy [10], given by…”