Existence and Uniqueness of Limit Cycles in a Class of Planar Differential Systems

This paper is an extension to the recent results presented by M. Sabatini about the existence and uniqueness of limit cycles of a certain class of planar differential systems in order to include other new classes. A concrete example exhibiting the applicability of the result is introduced

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 uniqueness of limit cycles for Li\'enard equation: the legacy of G. Sansone

We give an account of the results about limit cycle's uniqueness for Li\'enard equations, from Levinson-Smith's one to the most recent ones. We present a new uniqueness theorem in the line of Sansone-Massera's geometrical approach.Comment: 2 figure

Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

Uniqueness of limit cycles for a class of planar vector fields

In this paper we give sufficient conditions to ensure uniqueness of limit cycles for a class of planar vector fields. We also exhibit a class of examples with exactly one limit cycle.Comment: 8 pages, 2 figure

Existence and uniqueness of limit cycles in a class of second order ODE's with inseparable mixed terms

We prove a uniqueness result for limit cycles of the second order ODE $\ddot x + \dot x \phi(x,\dot x) + g(x) = 0$. Under mild additional conditions, we show that such a limit cycle attracts every non-constant solution. As a special case, we prove limit cycle's uniqueness for an ODE studied in \cite{ETA} as a model of pedestrians' walk. This paper is an extension to equations with a non-linear $g(x)$ of the results presented in \cite{S}