Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

Abstract

This paper examines the most probable route to chaos in high-dimensional dynamical systems in a very general computational setting. The most probable route to chaos in high-dimensional, discrete-time maps is observed to be a sequence of Neimark-Sacker bifurcations into chaos. A means for determining and understanding the degree to which the Landau-Hopf route to turbulence is non-generic in the space of C r mappings is outlined. The results comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. In their first edition of Fluid Mechanics [26], Landau and Lifschitz proposed a route to turbulence in fluid systems. Since then, much work, in dynamical systems, experimental fluid dynamics, and many other fields has been done concerning the routes to turbulence. In this paper, we present early results from the first large statistical study of the route to chaos in a very general class of high-dimensional, C r , dynamical systems. Our results contain both some reassurances based on a wealth of previous results and some surprises. We conclude that, for high-dimensional discrete-time maps, the most probable route to chaos (in our general construction) from a fixed point is via at least one Neimark-Sacker bifurcation, followed by persistent zero Lyapunov exponents, and finally a bifurcation into chaos. We observe both the Ruelle-Takens

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

“…Utilizing the neural network framework discussed here [4] conclude that the most likely route to chaos is a quasi-periodic one -however these results are likely subject to the measures imposed upon the weight matrices. Likewise, Doyon et.…”

“…Thus, choosing s to be small forces the dynamics to be mostly linear, yielding fixed-point behavior. Increasing s yields a route to chaos [5] [4]. Due to this, s provides a unique bifurcation parameter that sweeps from linear to highly nonlinear parameter regimes.…”

“…they can approximate C r (r ≥ 0 mappings and their derivatives on compact, metrizible sets) and also came to similar conclusions despite the significant differences in the constructions. In [4] the authors claimed that as the dimension of the dynamical system is increased, flip, fold, or any bifurcations due to real eigenvalues, and strong resonance bifurcations will be vanishingly rare and the route to chaos from a fixed point in parameter space will consist of periodic orbits with high-period (> 4) and quasi-periodic orbits. The basic idea of the above arguments follows from the matrix theory of Girko [19], Edelman [13], and Bai [8] and is the following: given a square matrix whose elements are real random variables drawn from a distribution with a finite sixth moment, in the limit of infinite dimensions, the normalized spectrum (or eigenvalues) of the matrix will converge to a uniform distribution on the unit disk in the complex plane.…”

Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions

“…Floquet multipliers replace in the stability analysis of limit cycles (Floquet theory) often the eigenvalues used to analyse fixed point stability [21]. In [4] a sequence of Neimark-Sacker bifurcations into chaos is mentioned as one possible route to chaos.…”

Structural stability and hyperbolicity violation in high-dimensional dynamical systems

“…We could consider parameter ranges considerably larger, but for s very large, the round-off error begins to play a significant role, and the networks become binary. This region has been briefly explored in [54]; further analysis is necessary for a more complete understanding [55].…”