Alejandro J. Rodríguez-Luis

Chen's attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system.

In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by x=a(y-x), y=(c-a)x+cy-xz, ż=-bz+xy, is, generically (c≠0), a special case of the Lorenz system. First, we infer that it is enough to consider two parameters to study its dynamics. Furthermore, we prove that there exists a homothetic transformation between the Chen and the Lorenz systems and, accordingly, all the dynamical behavior exhibited by the Chen system is present in the Lorenz system (since the former is a special case of the second). We illustrate our results relating Hopf bifurcations, periodic orbits, invariant surfaces, and chaotic attractors of both systems. Since there has been a large literature that has ignored this equivalence, the aim of this paper is to review and clarify this field. Unfortunately, a lot of the previous papers on the Chen system are unnecessary or incorrect.

A NOTE ON THE TRIPLE-ZERO LINEAR DEGENERACY: NORMAL FORMS, DYNAMICAL AND BIFURCATION BEHAVIORS OF AN UNFOLDING

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rossler equation.

SOME RESULTS ON CHUA'S EQUATION NEAR A TRIPLE-ZERO LINEAR DEGENERACY

In this work we study a wide class of symmetric control systems that has the Chua’s circuit as a prototype. Namely, we compute normal forms for Takens{Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a rst guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua’s circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three-parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.

The non-transverse Shil'nikov-Hopf bifurcation: uncoupling of homoclinic orbits and homoclinic tangencies

Abstract It is known that in a neighbourhood of a codimension-two Shil’nikov–Hopf bifurcation, primary periodic orbits lie on a single wiggly curve in period–parameter space, with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit. In contrast, it has recently been shown by Hirschberg and Laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit. This paper analyses the codimension-three bifurcation caused by a non-transverse (i.e., degenerately parametrically unfolded) Shil’nikov–Hopf bifurcation, which contains nearby dynamics akin to both degeneracies. Two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle-focus homoclinic orbits to equilibria, or the uncoupling of this locus from the locus of Hopf bifurcations. We undertake a heuristic analysis of the unfolding, showing that in both cases it contains codimension-two non-transversal homoclinic orbits to equilibria and non-transversal homoclinic tangencies to periodic orbits. Unfolding the former non-transverse orbit, is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits. Unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas. Asymptotic expressions are given for the accumulation of two types of isola-forming bifurcations. The implications of Z 2 -equivariance on the unfolding is discussed. Finally, numerical evidence is presented for both upward and downward pointing non-transverse Shil’nikov–Hopf bifurcations occuring in a model of an autonomous non-linear electronic circuit, both with respect to a Z 2 -equivariant and a Z 2 -non-equivariant equilibrium. Numerical computation of curves of periodic orbits, homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications.

A Tame Degenerate Hopf-Pitchfork Bifurcation in a Modified van der Pol–Duffing Oscillator

We consider a modified van der Pol–Duffing electronic circuit,focusing on the case where a Hopf-pitchfork bifurcation takes places.The analysis of this bifurcation is a simple way to detect andcharacterize purely three-dimensional behaviour (an oscillatory regimein three variables, quasiperiodic motion, etc.).The normal formanalysis provides the classification of different kinds ofHopf-pitchfork bifurcation, organized according to some degeneratecases. One of these degenerate cases is analyzed, by considering acodimension-three unfolding of a reflectionally symmetric planar vectorfield. Later, the implications for three-dimensional flows arepresented. Unlike another degenerate Hopf-pitchfork bifurcationsexhibited by the system, the one studied here does not involvequasiperiodic behaviour, so that the complexity related to quasiperiodicmotion is not present.

T-Points in a Z2-Symmetric Electronic Oscillator. (I) Analysis

In this work we study the presence of T-points, a kind of codimension-two heteroclinic loop, in a Z2-symmetric electronicoscillator. Our analysis proves that, in the parameter plane, whenthe equilibria involved are saddle-focus,three spiraling curves of global codimension-onebifurcations emerge from this T-point, corresponding tohomoclinic of the origin, homoclinic of the nontrivialequilibria and heteroclinic between the nontrivial equilibriaconnections. Some first-order features of these three curves are also shown.The analytical results, valid for all three-dimensionalZ2-symmetric systems, are successfully checked in themodified van der Pol–Duffing electronic oscillator considered.

A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation

A codimension-three unfolding for the 2-symmetric Hopf-pitchfork bifurcation, in the presence of an additional nonlinear degeneracy, is analysed. Up to ten distinct topological equivalence classes for the unfolding are found. A rich variety of dynamical and bifurcation behaviours is pointed out. Beyond the bifurcations present in the nondegenerate case, we show that the following bifurcations appear locally: Takens - Bogdanov of periodic orbits, degenerate pitchfork of periodic orbits, and global connections involving equilibria and/or periodic orbits. The local results achieved, extended by means of numerical continuation methods, are used to understand the dynamics of a modified van der Pol - Duffing electronic oscillator, for a certain range of the parameters.

ON THE HOPF PITCHFORK BIFURCATION IN THE CHUA'S EQUATION

We study some periodic and quasiperiodic behaviors exhibited by the Chuas equation with a cubic nonlinearity, near a Hopf–pitchfork bifurcation. We classify the types of this bifurcation in the nondegenerate cases, and point out the presence of a degenerate Hopf–pitchfork bifurcation. In this degenerate situation, analytical and numerical study shows a diversity of bifurcations of periodic orbits. We find a secondary Hopf bifurcation of periodic orbits, where invariant torus appears. This secondary Hopf bifurcation is bounded by a Takens–Bogdanov bifurcation of periodic orbits. Here, a sequence of period-doubling bifurcations of invariant tori is detected. Resonance phenomena are also analyzed. In the case of strong resonance 1:4, we show a new sequence of period-doubling bifurcations of 4T invariant tori.

Stationary instabilities in a dielectric liquid layer subjected to an arbitrary unipolar injection and an adverse thermal gradient

The effect of temperature variations of both ionic mobility and dielectric constant on the stability of a horizontal layer of dielectric liquid subjected to a DC electric field with heating and an arbitrary injection of unipolar charge from below is analysed. The marginal stability boundaries for steady convection are determined by numerical solution of the linearised thermo-electrohydrodynamic perturbation equations.

TAKENS BOGDANOV BIFURCATIONS OF PERIODIC ORBITS AND ARNOLD'S TONGUES IN A THREE-DIMENSIONAL ELECTRONIC MODEL

In this paper we study Arnolds tongues in a ℤ2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The existence of cascades of the cubic Takens–Bogdanov bifurcations is also pointed out. On the other hand, we study the dynamics inside the tongues showing different Poincare sections. Several bifurcation diagrams show the presence of cusps of periodic orbits and homoclinic bifurcations. We show the relation that exists between two codimension-two bifurcations of equilibria, Takens–Bogdanov and Hopf–pitchfork, via homoclinic connections, period-doubling and quasiperiodic motions.