Emilio Freire

Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

Continuation of periodic orbits in conservative and Hamiltonian systems

We introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals, and in particular in Hamiltonian systems having several independent symmetries. Our method is based on a generalization of the concept of a normal periodic orbit as introduced by Sepulchre and MacKay [Nonlinearity 10 (1997) 679]. We illustrate the continuation method on some integrable Hamiltonian systems with two degrees of freedom and briefly discuss some further applications.

On simplifying and classifying piecewise-linear systems

A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feedback structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay attention to obtain equivalent state equations containing both a minimum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canonical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of different electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology.

Canonical Discontinuous Planar Piecewise Linear Systems

The family of Filippov systems constituted by planar discontinuous piecewise linear systems with two half-plane linearity zones is considered. Under generic conditions that amount to the boundedness of the sliding set, some changes of variables and parameters are used to obtain a Lienard-like canonical form with seven parameters. This canonical form is topologically equivalent to the original system if one restricts ones attention to orbits with no points in the sliding set. Under the assumption of focus-focus dynamics, a reduced canonical form with only five parameters is obtained. For the case without equilibria in both open half-planes we describe the qualitatively different phase portraits that can occur in the parameter space and the bifurcations connecting them. In particular, we show the possible existence of two limit cycles surrounding the sliding set. Such limit cycles bifurcate at certain parameter curves, organized around different codimension-two Hopf bifurcation points. The proposed canonic...

LIMIT CYCLE BIFURCATION FROM CENTER IN SYMMETRIC PIECEWISE-LINEAR SYSTEMS

The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopfs case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given. The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmanns analysis are also included.

Periodicity and chaos in an autonomous electronic system

In this paper, a simple electronic circuit is analyzed from a qualitative viewpoint. The circuit shows a great variety of dynamical behaviors (equilibrium points, periodic oscillations, chaotic motions... ) and the analysis proceeds to catalog all of them through a bifurcation study (pitchfork and Hopf bifurcations, flip bifurcations... ). This study points out the relevance of qualitative analysis in systems of simple structure but with very complex behavior. The paper includes theoretical study, numerical simulations, and actual circuit experimentation.

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.

BIFURCATION OF INVARIANT CONES IN PIECEWISE LINEAR HOMOGENEOUS SYSTEMS

Invariant surfaces in three-dimensional continuous piecewise linear homogeneous systems with two pieces separated by a plane are detected. The Poincare map associated to this plane transforms half-straight lines passing through the origin into half-straight lines of the same type. The invariant half-straight lines under this map determine invariant cones for which the existence, stability and bifurcation are studied. This analysis lead us to consider some questions about the topological type and stability of the origin.

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.