Integrability and global dynamics of polynomial differential systems defined in R³ with invariant algebraic surfaces of degrees 1 and 2

Author(s)

Alisson de Carvalho Reinol

• Document type: Doctoral Thesis
• Press: São José do Rio Preto. 2017-07-19.
• Institution: Universidade Estadual Paulista (Unesp). Instituto de Biociências Letras e Ciências Exatas. São José do Rio Preto
• Defense date: 2017-07-05
• Advisor: Marcelo Messias

Abstract

In this work, we consider algebraic and dynamical aspects of some problems involving invariant algebraic surfaces in polynomial differential systems defined in R³. We determine the maximum number of invariant planes that a quadratic differential system can have and we study the realization and integrability of such systems. We provide the normal form for differential systems having an invariant quadric and we study in more detail the dynamics and integrability of quadratic differential systems having an elliptic paraboloid as invariant algebraic surface. Finally, we study the dynamic consequences of perturbing differential system whose phase space is foliated by invariant algebraic surfaces. For this we consider the quadratic differential system known as Sprott A system, which depends on one real parameter a and presents chaotic behavior even without having any equilibrium point, thus having a hidden attractor for suitable values of parameter a. We prove that, for a=0, the phase space of this system is foliated by invariant concentric spheres. By using the Averaging Theory and the KAM (Kolmogorov-Arnold-Moser) Theorem, we prove that, for a>0 sufficiently small, an orbitally stable periodic orbit emerges from a zero-Hopf nonisolated equilibrium point located at the origin and that invariant tori are formed around this periodic orbit. We conclude that the occurrence of these facts has an important role in the formation of the hidden attractor. (AU)

• FAPESP's process: 13/26602-7 - Integrability and global dynamics of quadratic vector fields defined on R3 with Quadrics as invariant surfaces
• Grantee: Alisson de Carvalho Reinol
• Support Opportunities: Scholarships in Brazil - Doctorate