N-dimensional dynamical systems exploiting instabilities in full

Abstract

We report experimental and numerical results showing how certain N-dimensional dynamical systems are able to exhibit complex time evolutions based on the nonlinear combination of N-1 oscillation modes. The experiments have been done with a family of thermo-optical systems of effective dynamical dimension varying from 1 to 6. The corresponding mathematical model is an N-dimensional vector field based on a scalar-valued nonlinear function of a single variable that is a linear combination of all the dynamic variables. We show how the complex evolutions appear associated with the occurrence of successive Hopf bifurcations in a saddle-node pair of fixed points up to exhaust their instability capabilities in N dimensions. For this reason the observed phenomenon is denoted as the full instability behavior of the dynamical system. The process through which the attractor responsible for the observed time evolution is formed may be rather complex and difficult to characterize. Nevertheless, the well-organized structure of the time signals suggests some generic mechanism of nonlinear mode mixing that we associate with the cluster of invariant sets emerging from the pair of fixed points and with the influence of the neighboring saddle sets on the flow nearby the attractor. The generation of invariant tori is likely during the full instability development and the global process may be considered as a generalized Landau scenario for the emergence of irregular and complex behavior through the nonlinear superposition of oscillatory motions. © 2000 American Institute of Physics. ͓S1054-1500͑00͒01004-1͔Oscillatory phenomena are ubiquitous in natural and social systems and their investigation is currently done within the context of nonlinear dynamics. The oscillations in a given system may appear associated either with intrinsically sustained mechanisms or with externally modulated inputs. Processes with different time scales often coexist and in certain cases the interrelation of oscillations produces complex evolutions in which, however, cyclic repetitions are usually apparent. For instance, in biology, a direct example of this behavior is found in the bursting response of small neural networks in the stomatogastric nervous system of crustacea, 1 while a more involved example could be the wake-sleep cycle of a brain. Leaving apart the case of external modulations, we find it interesting to understand how a nonlinear system can produce different characteristic frequencies and how it can mix the corresponding oscillations to yield complex time evolutions. Although the problem looks basic and simple its answer is pending. The paradigm of chaos is not useful here and among the known mechanisms of nonlinear dynamics only the Landau scenario can be invoked. The present work is a contribution to the enlightenment of this problem. We report experimental and numerical results showing the emergence of complexity through the nonlinear superposition of oscillatory motions in a dynamical system. The phenomenon develops in a generalized Landa...

A Maximally Chaotic 3d Autonomous System

“…The unstable manifold traces almost exactly the lower portion of the limit cycle; we conjecture that this branch of the unstable manifold might be an image of an inclination-flip homoclinic orbit. Notice also the back-bending of its branch towards the attractor [Sandstede, 1997;Oldeman et al, 2001;Rius et al, 2000a;Rius et al, 2000b] provide an in-depth treatment of those subjects.…”

Nonlinear oscillatory mixing in the generalized Landau scenario

“…Our previous publications [10][11][12] deal with a limited view of the oscillatory scenario, essentially based on the observation of the attractor W1, and our interpretative trials were mainly devoted to explain the influence of the rest of modes on this attractor. The tentative explanation was based on the combination of two basic mechanisms of mode mixing between pairs of periodic orbits (see, e.g., Figs 1 and 2 of Ref.…”

“…of which the first is periodic and describes the interferometric Airy function of the family of physical devices through which the oscillatory scenario was discovered [11], while the second is simply an inverted Gaussian. Figure 10 shows a graphical representation of each nonlinear function together with the corresponding steady-state solutionψ(μ C ) and distribution of p(ψ) values.…”

“…The system (1) was introduced and analyzed in Ref. [10], as a generalization of a model describing the experimental behaviors of a family of physical devices with successively increasing dimension (up to 6) [11], and it is succinctly presented in the Appendix, together with a consideration of what of its features are responsible for the good working of the oscillatory scenario.…”

“…However, the combination of such a kind of mechanisms to provide a generic way for the achievement of really complex oscillations with arbitrarily large numbers of oscillatory modes seems not obvious. To the best of our knowledge, the unique known way for such a purpose is the so-called generalized Landau scenario, which is based on the reiterative occurrence of the two most standard mechanisms of nonlinear dynamics: the saddle-node and Hopf bifurcations, and which, despite having been the object of several publications [10][11][12], remains unnoticed in the field. This article is another attempt in the form of a graphical exposition.…”