Resumen de Bifurcations of limit cycles in planar differential and piecewise differential systems
Leonardo Pereira Costa da Cruz
The thesis deals with the study of isolated periodic orbits, the so called limit cycles, in some differential and piecewise differential systems in the plane. This is one of the most important topic in the Qualitative Theory of Differential Equations. The work is structured in an introduction as a first chapter and then four chapters where the results and proofs are developed.
The introduction starts with the most important historical problems studied in qualitative theory of differential equations centered in the main object studied in this work, the limit cycles. It finishes with the summary of the obtained results.
In Chapter 2 we study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The unperturbed cubic planar system simultaneously has a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. When the polynomial perturbation has degree $n$, the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of simple zeros. The maximum number of limit cycles, up to first order cubic perturbation, from the inner and outer annuli is 9 and 8, respectively. When the simultaneous cubic polynomial bifurcation problem is considered, 12 limit cycles exist. They appear in three configuration types: (9,3), (6,6), and (4,8). In the non-piecewise scenario, only 5 limit cycles were found.
The number of limit cycles bifurcating from a piecewise quadratic system is studied in Chapter 3. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. As fas as we are concerned, this is the best lower bound for the quadratic class. All the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers. We do a first and second averaging analysis. This is done perturbing all the isochronous quadratic systems having a birational linearization.
The Bendixson--Dulac Theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems in Chapter 4. This is done in the class of piecewise differential systems where the Green Theorem applies. We apply it to three different Li\'enard piecewise differential systems. The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one. This extension has no a limit cycle in the full parameters space. It presents a heteroclinic connection where the limit cycle disappears.
In Chapter 5 we study the family of quartic linear-like time reversible polynomial systems having a nondegenerate center at the origin. This family has degree one with respect to one of the variables. We are interested in systems in this class having also two extra nondegenerate centers outside the straight line of symmetry. The geometrical configuration of these centers is aligned or triangular. We solve the center problem in both situations. When the centers are in a triangular position we study the number of limit cycles appearing by a simultaneous degenerated Hopf bifurcation. Up to a first order analysis we obtain 13 limit cycles in two configuration types: (4,5,4) and (3,7,3).
