Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Coexistence of periods in a bisecting bifurcation

The inner structure of the attractor appearing when the Varley-Gradwell-Hassell population model bifurcates from regular to chaotic behaviour is studied. By algebraic and geometric arguments the coexistence of a continuum of neutrally stable limit cycles with different periods in the attractor is explained.Comment: 13 pages, 5 figure

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

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

Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure