Absrrucf-This paper provides a rigorous mathematical proof that the double scroll is indeed chaotic. Our approach is to derive a linearly equitdent class of piecewise-linear differential equations which includes the double scroll as a special case. A necessary and sufficient condition for two such piecewise-linear vector fields to be linearly equivalent is that their respective eigenvalues be a scaled version of each other. In the special case where they are identical, we have exact equivalence in the sense of hear conjugacy. An explicit normalform equation in the context of global bifurcation is derived and parametrized by their eigenvalues. Analytical expressions for various Poincur ~ maps are then derived and used to characterize the birtli and the &4t/1 of the double scroll, as well as to derive an approximate one-dimensional map in analytic form which is useful for further bifurca-tion analysis. In particular, the analytical expressions characterizing vari-ous La/f-return maps associated with the Poincare map are used in a T 1073 crucial way to prove the existence of a Shilnihov-type homoclinic orbit, thereby establishing rigorously the chaotic nature of the double scroll. These analytical expressions are also fundamental in our in-depth analysis of the birth (onset of the double scroll) and &&I (extinction of chaos) of the double scroll. The unifying theme throughout this paper is to analyze the double scroll system as an unfolding of a large family of piecewise-linear vector fields in R3. Using this approach, we were able to prove that the chaotic dyrumics of the double scroll is quite common, and is robust because the associated horseshoes predicted from Shilnikov’s theorem are structurally stable. In fact, it is exhibited by a large family (in fact, infinitely many linearly-equideni circuits) of vector fields whose associated piecewise-linear dif-ferential equations bear no resemblance to each other. It is therefore remarkable that the normalized eigenvalues, which is a local concept, completely determine the system’s global qualitative behavior. Part I: Rigorous Proof of’chaos I.

Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...

In this paper, we provide an analytical insight into the observed nonlinear behaviour of the buck converter and link this with the study of a wider class of piecewise-smooth systems. After introducing the buck converter model and background, we describe the most fascinating features of its dynamical behaviour. We then introduce the so-called grazing and sliding solutions and discuss their role in determining many of the buck converter's dynamical oddities. In particular, a local map is studied which explains how the grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, we show how these orbits accumulate onto a sliding trajectory through a "spiralling" impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered stru...

Periodic orbits of a third order ODE are topological knots. Using results from the theory of branched 2-manifolds, we prove the existence of simple ODEs whose periodic orbit set contains every type of knot and collection of knots (link). The construction depends on the dynamics near certain con gurations of Shil'nikov connections, and can be applied to, among other things, a model of a nonlinear electric circuit.

st in the modelling community, and we relate our results to simulations of other models. c 1999 Society for Mathematical Biology 1. INTRODUCTION The sunlit surface waters of the world's oceans are populated by tiny plankton. Plankton is a general term used to describe freely-floating and weakly-swimming marine and freshwater organisms. It comes from a Greek word (######o# ) meaning wandering or drifting, and was introduced by the German scientist Victor Hensen in 1887 (Thurman, 1997). At around the same time, the French mathematician Henri Author to whom correspondence should be addressed: Biological Oceanography Division, Bedford Institute of Oceanography, B240, Dartmouth, Nova Scotia, Canada B2Y 4A2. 0092-8240/99/020303 + 37 $30.00/0 c 1999 Society for Mathematical Biology 304 A. M. Edwards and J. Brindley Poincare was laying down the foundati

Experimental measurements of nonlinear vibrations of a string are analyzed using new techniques of nonlinear modeling. Previous theoretical and numerical work suggested that the motions of a string can be chaotic and a Shil'nikov mechanism is responsible. We show that the experimental data is consistent with a Shil'nikov mechanism. We also reveal a period doubling cascade with a period three window which is not immediately observable because there is sufficient noise, probably of a dynamical origin, to mask the period-doubling bifurcation and the period three window. 1 Introduction Bajaj and Johnson [2] have conducted an analysis of weakly nonlinear partial differential equations describing the forced vibrations of stretched uniform strings. The equations take into account motions transverse to the plan of forcing, which are induced by a coupling with longitudinal displacements, and changes in tension that occur in large amplitude motions. The averaged equations of a resonant system c...

We study the interaction of a transcritical (or saddle-node) bifurcation with a codim0 /codim-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reactiondiffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codim-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbitflip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found. Keywords: heteroclinic cycle; T-point; orbit-flip bifurcation; inclination-flip bifurcation. Running title: Bifurcations near a twisted heteroclinic cycle. 1 Introduction The traveling wave ODE of the FitzHugh-Nagumo reaction-diffusion (RD) e...

on the electroencephalogram (EEG). The ictal discharges (seizures) often spread to involve widespread regions of the ipsilateral then the contralateral cerebral hemispheres. These diffuse discharges often persist for approximately 1 to 5 minutes and are followed by a postictal pattern of asynchronous low amplitude slow waves in the EEG. It is a widely held view that seizures arise from mesial temporal structures because of damage to hippocampal circuitry. The characteristic circuit abnormalities include drop out of neurons, simplification of the dendritic tree (reduced synaptic input), sprouting of dentate granule cell axons (increasing the number of excitatory-excitatory feedback connections), and increase in glial cell elements (sclerosis). There is a concomitant loss in neurotransmitter receptors in the hippocampus. Physiologic studies in epileptogenic hippocampi have demonstrated loss of neuronal inhibition. It is generally believed that loss of inhibition is, at least i

We describe a Sil'nikov-Saddle-Node interaction in the proximity of a HopfSaddle -Node bifurcation both from numerical and theoretical viewpoints, inspired in numerical observations of a 3D model of a laser with injected signal. We describe the interaction near the codimension-2 point using a return map controlled by the parameters of the bifurcation. The theoretical predictions compare well with numerical experiments. Finally, we discuss the reasons for the existence of the Sil'nikov-Saddle-Node bifurcation in the laser with injected signal, and correspondingly, the characteristics of other systems that might display this bifurcation. PACS: 05.45.+b, 42.50.Ne, 47.20.Ky e-mail: martin.zimmermann@kvac.uu.se 2 1 Introduction. In a numerical study [30] of a 3 dimensional model of a laser with injected signal (LIS) derived in [25], a strange attractor and periodic orbit organization of the Sil'nikov type are reported (see fig. 1); i.e., there is a homoclinic orbit to a hyperbolic fi...

We study a model of the interaction of a saddle-node bifurcation with a codim-0/codim-2 heteroclinic cycle. The manifold organization is such that the two-parameter unfolding of this global bifurcation, creates two branches of homoclinic orbits to each fixed point. It is shown that the homoclinic orbits become degenerate in an orientation-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found. Keywords: heteroclinic cycle; T-point; orientation-flip bifurcation; inclination-flip bifurcation. Running title: Bifurcations near a twisted heteroclinic cycle. 1 Introduction The traveling wave ODE of the FitzHugh-Nagumo reaction-diffusion equation was analyzed by several authors (Rinzel and Terman, 1982). The existence of a heteroclinic cycle between 2 equilibrium points has been shown. This cycle is composed o...