Fernando Fernández-Sánchez

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.

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.

Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System

The proof of the existence of a global connection in differential systems is generally a difficult task. Some authors use numerical techniques to show this existence, even in the case of continuous piecewise linear systems. In this paper we give an analytical proof of the existence of a reversible T-point heteroclinic cycle in a continuous piecewise linear version of the widely studied Michelson system. The principal ideas of this proof can be extended to other piecewise linear systems.

Existence of homoclinic connections in continuous piecewise linear systems

Numerical methods are often used to put in evidence the existence of global connections in differential systems. The principal reason is that the corresponding analytical proofs are usually very complicated. In this work we give an analytical proof of the existence of a pair of homoclinic connections in a continuous piecewise linear system, which can be considered to be a version of the widely studied Michelson system. Although the computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

Nontransversal curves of T-points: a source of closed curves of global bifurcations

Abstract A model is derived to explain the existence of closed bifurcation curves of homoclinic and heteroclinic connections in autonomous three-dimensional systems. This scenario is related to the failure of transversality in a curve of a certain kind of codimension-two heteroclinic loops. The predictions deduced from this model strongly agree with the numerical results obtained in a modified van der Pol–Duffing electronic oscillator.

BI-SPIRALING HOMOCLINIC CURVES AROUND A T-POINT IN CHUA'S EQUATION

In this work, the existence of curves of homoclinic connections that bi-spiral around a T-point between two saddle-focus equilibria is detected in Chuas equation. That is, the homoclinic curve emerges spiraling from a T-point in a parameter bifurcation plane and ends, by a different spiral, at the same T-point. This new phenomenon is related to the existence of more than one intersection between the two-dimensional manifolds of the involved equilibria at the T-point.

CLOSED CURVES OF GLOBAL BIFURCATIONS IN CHUA'S EQUATION: A MECHANISM FOR THEIR FORMATION

In this work, the presence of closed bifurcation curves of homoclinic and heteroclinic connections has been detected in Chua’s equation. We have numerically found and qualitatively described the mechanism of the formation/destruction of such closed curves. We relate this phenomenon to a failure of transversality in a curve of T-points in a three-dimensional parameter space.

Isolas, cusps and global bifurcations in an electronic oscillator

The aim of the present work is to describe the bifurcation behaviour of a class of asymmetric periodic orbits, in an electronic oscillator. The first time we detected them they were organized in a closed branch: that is, their bifurcation diagram showed an eight-shaped isola, with a nice structure of secondary branches emerging from period-doubling bifurcations. In a two-parameter bifurcation set, the isola structure persists. We find the regions of its existence, and describe its destruction in an isola centre with a cusp of periodic orbits. Finally, the introduction of a third parameter allows us to find the relation of our orbits to symmetric periodic orbits (via a symmetry-breaking bifurcation) and to homoclinic connections of the non-trivial equilibria. The isolas are successively created by collision of two adjacent limbs of the wiggly bifurcation curve. The Shil?nikov homoclinic and heteroclinic connections, related to the symmetric and asymmetric periodic orbits, emerge from T-points and end at Sh...

HOPF BIFURCATIONS AND THEIR DEGENERACIES IN CHUA'S EQUATION

We perform an analytical study of the Hopf bifurcations and their degeneracies in Chuas equation. In the case of the equilibrium at the origin only codimension-two Hopf bifurcations appear. However, for the nontrivial equilibria we prove the existence of codimension-three Hopf bifurcations. Numerical results are in strong agreement with the analytical ones.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.