Resumen de Cicles límit i períodes crítics per algunes equacions diferencials polinomials
Iván Sánchez Sánchez
The present doctoral thesis is framed in the study of the center and cyclicity problems, as well as isochronicity and criticality, in the context of the qualitative theory of differential equations. This memory is organized in three chapters.
The first chapter deals with the center and cyclicity problems. We start by giving a deeper and more exhaustive description of the center and cyclicity problems, together with a brief introduction to the main tools about polynomial ideals and some classical techniques for classifying centers, such as symmetries or Darboux Integrability Theory. We proceed then to a more detailed analysis on Lyapunov constants, by showing methods to compute them and how these calculations can be computationally implemented, stressing the importance of parallelization in the used techniques. Later, we deal with the center and cyclicity problems for some families of differential equations in the plane. We also collect a series of advances in the computation of Lyapunov constants and the determination of their structure. In the last section of Chapter 1 we study the Hopf-bifurcation in 3-dimensional polynomial vector fields, with the objective to find the highest possible number of limit cycles for different degrees. The used techniques have enabled to find 11 limit cycles for quadratic systems, 31 for cubic systems, 54 for quartic systems, and 92 for quintic systems, which to the best of our knowledge are the highest numbers found so far.
The second chapter is devoted to the study of isochronicity and criticality. We start by defining both concepts in more detail, working on the notions of period function and critical periods. Then, the equivalence between isochronicity and linearizability is introduced, together with other tools to study isochronicity which are the Lie bracket and commuting transversal systems as well as the computation of period constants. The next section in Chapter 2 aims to find the maximum number of critical periods which unfold from low degree n planar polynomial centers when perturbing reversible holomorphic isochronous centers inside the reversible class. We prove that 6 critical periods are seen for the cubic case, 10 for the quartic case, (n^2+n-2)/2 for n between 5 and 9 (both included), and (n^2+n-4)/2 for n between 10 and 16 (both included). The final section of this chapter introduces the idea of using an equivalent of Melnikov functions for the bifurcation of critical periods instead of limit cycles. This allows to obtain 10, 22, 37, 57, 80, 106, and 136 critical periods for n = 4, 6, 8, 10, 12, 14, and 16, respectively. We also classify some isochronous centers throughout this section.
The last chapter presents a new problem that, to the best of our knowledge, has never been considered before. This problem consists on simultaneously studying the bifurcation of limit cycles and critical periods for a system of differential equations in the plane, obtaining a value (k,l) which means that k limit cycles and l critical periods can simulteneously unfold. In this line, we will define the term bi-weakness [k,l] as a concept for the degree of the first nonzero coefficients in the return map k and the period function l at the same time, being both the center and isochronicity properties not kept. We study the bi-weakness for different classical families, and we give a complete classification of the simultaneous cyclicity and criticality for the cubic Liénard system. We also show the isochronicity for some Liénard families in this part.
