Antonio Algaba

The integrability problem for a class of planar systems

In this paper we consider perturbations of quasi-homogeneous planar Hamiltonian systems, where the Hamiltonian function does not contain multiple factors. It is important to note that the most interesting cases (linear saddle, linear centre, nilpotent case, etc) fall into this category. For such kinds of systems, we characterize the integrability problem, by connecting it with the normal form theory.

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.

Isochronicity Via Normal Form

We analyze the isochronicity of centres, by means of a procedure to obtain hypernormal forms (simplest normal forms) for the Hopf bifurcation, that uses the theory of transformations based on the Lie transforms. We establish the relation between the period constants and the normal form coefficients, and prove that an equilibrium point is an isochronous centre if and only if a property of commutation holds. Also, we give necessary and sufficient conditions, expressed in terms of the Lie product, to determine if an equilibrium point is a centre. Several examples are also included in order to show the usefulness of the method. In particular, the isochronicity of the origin for the Lienard equation is analyzed in some cases.

Hypernormal Form for the Hopf-Zero Bifurcation

In this paper we present hypernormal forms up to an arbitrary order for equilibria of tridimensional systems having a linear degneracy corresponding to a pair of pure imaginary eigenvalues and a third one zero. These simplest normal forms are obtained assuming some generic conditions on the quadratic terms, and using C∞-conjugacy as well as C∞-equivalence. Also, the case of ℤ2-symmetric systems is considered. In this situation, the hypernormal forms are characterized under generic conditions on the cubic terms. In all the cases, we provide recursive algorithms that compute explicitly the hypernormal form coefficients, in terms of the normal form coefficients.

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.

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.

Quasi-homogeneous normal forms

In this paper, we extend the concepts of the normal form theory for vector fields that are expanded in quasi-homogeneous components of a fixed type (these expansions have been used by some authors in the analysis of the determinacy of a given singularity). Also, the use of reparametrizations in the time are considered. Namely, beyond the use of C∞-conjugation to determine normal forms, we present a method useful to determine how much a vector field can be simplified by using C∞-equivalence. The results obtained are applied in the case of the Bogdanov-Takens singularity, firstly using C∞-conjugation and later, showing the improvements provided by the C∞-equivalence.

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.