Homoclinic chaos in the parameter space

Author(s)

Rene Orlando Medrano Torricos

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2004-11-26

Examining board members

Ibere Luiz Caldas; Marcus Aloizio Martinez de Aguiar; Thomas Braun; Celso Grebogi; Jose Roberto Castilho Piqueira

• Advisor: Ibere Luiz Caldas

Abstract

In this thesis we study the dynamic behavior, in the parameter space, of two versions of the Double Scroll electronic circuit, whose flows are represented by piecewise non integrable systems. The difference between these circuits is the characteristic curves of the negative resistance, one continuous and the other discontinuous. The Double Scroll circuit is known to present chaotic behavior associated to the existence of homoclinic orbits. We develop numerical methods to identify periodic and chaotic attractors in these circuits. We present a complete study of these systems manifolds and demonstrate that the discontinuous circuit cannot form homoclinic orbits. We develop a general method to obtain homoclinic and heteroclinic orbits in piecewise linear systems. This method was used in the continuous circuit to identify homoclinic orbit families in the parameter space. We develop a theoretical study about the homoclinic orbits based on the Shilnikov theorem, determining a general scaling law that describes the accumulations of the infinity homoclinic orbits in the parameter space. Using the detecting homoclinic orbits method, we show the validity of this law for the continuous Double Scroll circuit. Moreover, combining the geometry of the homoclinic or bit families with the scaling law, we show the existence of homoclinic orbits structures of the homoclinic orbits that explain the homoclinic scenario in the parameter space. These structures are present in all systems for which we can apply the Shilnikov theorem. Finally, we suggest three experiments to verify the existence of these orbits and their relation with the system dynamics. (AU)