Limit Cycles for a class of continuous piecewise linear differential systems with three zones

Agraïments: The first author is partially supported by CNPq grand number 200293/2010-9. Both authors are also supported by the joint project CAPES-MECD grant PHB-2009-0025-PC.In this paper we consider a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map we show that these systems admit always a unique limit cycle, which is hyperbolic

Sistemas diferenciales lineales a trozos: Ciclos límite y análisis de bifurcaciones

Tesis descargada desde TESEOThe class of piecewise-linear differential systems (PWL systems, for short) is an important class of nonlinear dynamical systems. They naturally appear in realistic nonlinear engineering models, and are used in mathematical biology as well, where they constitute approximate models. Therefore, they constitute a significant subclass of piecewise-smooth dynamical systems. From the family of planar, continuous PWL systems (CPWL2, for short) we study systems with only two zones (2CPWL2 systems), and systems with three zones with or without symmetry with respect to the origin (S3CPWL2 systems). Some discontinuous PWL systems with only two zones (2DPWL2, for short) and symmetric PWL systems in dimension 3, namely S3CPWL3, are also considered. After an introduction, in Chapter 2 we review some terminology and results related to canonical forms in the study of PWL systems along with certain techniques that are useful for the bifurcation analysis of their periodic orbits. We review general results in dimension n, but we later deal only with systems in dimension 2 and 3. Next, Chapter 3 is completely devoted to planar PWL systems. Some boundary equilibrium bifurcations (BEB, for short) are characterized, putting emphasis in the ones capable of giving rise to limit cycles. We exploit and extend some recent results, which allows us to pave the way for a shorter proof of Lum-Chua conjecture. After other general results for existence and uniqueness of limit cycles in 3CPWL2 systems, we show some applications of the theory in nonlinear electronics. In a different direction of research, it is introduced a new family of algebraically computable piecewise linear nodal oscillators and shown some real electronic devices that belong to the family. The outstanding feature of this family makes it an exceptional benchmark for testing approximate methods of analysis of oscillators. Finally, we include our only contribution in the exciting world of discontinuous PWL systems: the analysis of the focus-center-limit cycle bifurcation in planar PWL systems with two zones and without a proper sliding set, which naturally includes the continuous case. Chapter 4 represents our particular incursion in PWL systems in dimension 3, namely in S3CPWL3 ones, notwithstanding some results are also interesting for 2CPWL3 vector fields. Pursuing the aim of fill in the pending gaps in the catalog of possible bifurcations, we study some unfoldings of the analogous to Hopf-pitchfork bifurcations in PWL systems. Our theorems predict the simultaneous bifurcation of 3 limit cycles but we also formulate a natural, strongly numerically based conjecture on the simultaneous bifurcation of 5 limit cycles. Finally, in Chapter 5 some conclusions and recommendations for future work are offered for consideration of interested readers. For the sake of concision, we want to specifically mention the main mathematical contributions included in this thesis. ¿ A new approach, following Massera¿s method, to get a concise proof for the Lum-Chua Conjecture in planar PWL systems with two zones (2CPWL2). ¿ Characterization for a variety of boundary equilibrium bifurcations (BEB¿s, for short) in 2CPWL2 systems. ¿ Alternative proofs of existence and uniqueness results for limit cycles in an important family of planar PWL systems with three zones (3CPWL2). ¿ Characterization for a variety of boundary equilibrium bifurcations (BEB¿s, for short) in 3CPWL2 systems, detecting some situations with two nested limit cycles surrounding the only equilibrium point. ¿ Analysis of the focus-center-limit cycle bifurcation in discontinuous planar PWL systems without sliding set. ¿ A thorough analysis of electronic Wien bridge oscillators, characterizing qualitatively (and quantitatively in some cases) the oscillatory behaviour and determining the parameter regions for oscillations. ¿ Analysis of a new family of algebraically computable nodal oscillators, including real examples of members of the family. ¿ Analysis of some specific unfolding for the Hopf-zero or Hopf-pitchfork bifurcation and its main degenerations in symmetric PWL systems in 3D (S3CPWL3), with the detection of the simultaneous bifurcation of three limit cycles. ¿ Study of some real electronic devices where the Hopf-zero bifurcation appears

On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line

In this paper we study the maximum number $N$ of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies $2\leq N \leq 3$ if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.Comment: 24 pages, 7 figure

On the birth of limit cycles for non-smooth dynamical systems

The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail

Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold

We study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve. Our main result is that 7 is a lower bound for the Hilbert number of this family. In order to get our main result, we develop the Melnikov functions for a class of nonsmooth differential systems, which generalizes, up to order 2, some previous results in the literature. Whereas the first order Melnikov function for the nonsmooth case remains the same as for the smooth one (i.e. the first order averaged function) the second order Melnikov function for the nonsmooth case is different from the smooth one (i.e. the second order averaged function). We show that, in this case, a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function