Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields

Author(s)

Andrade, Kamila da S. ; Jeffrey, Mike R. ; Martins, Ricardo M. ; Teixeira, Marco A.

• Document type: Journal article
• Source: INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS; v. 32, n. 04, p. 27-pg., 2022-03-30.

Abstract

In a smooth dynamical system, a homoclinic connection is an orbit connecting a saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections in nonsmooth systems are complicated by their interactions with discontinuities in their vector fields. A connection may involve a regular saddle outside a discontinuity set, or a pseudo-saddle on a discontinuity set, with segments of the connection to cross or slide along the discontinuity. Even the simplest case of connection to a regular saddle, which hits a discontinuity as a parameter is varied, is surprisingly complex. In this paper, we construct bifurcation diagrams for nonresonant saddles in the plane, unfolding the homoclinic connection to a boundary saddle in a nonsmooth dynamical system. As an application, we exhibit such diagrams for a model of a forced pendulum. (AU)

• FAPESP's process: 18/13481-0 - Geometry of control, dynamical and stochastic systems
• Grantee: Marco Antônio Teixeira
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 21/08031-9 - Piecewise smooth vector fields on compact manifolds
• Grantee: Ricardo Miranda Martins
• Support Opportunities: Regular Research Grants
• FAPESP's process: 15/06903-8 - Minimal sets in non-smooth dynamical systems
• Grantee: Ricardo Miranda Martins
• Support Opportunities: Regular Research Grants
• FAPESP's process: 12/18780-0 - Geometry of control systems, dynamical and stochastics systems
• Grantee: Marco Antônio Teixeira
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 18/03338-6 - Global dynamics of piecewise smooth dynamical systems
• Grantee: Ricardo Miranda Martins
• Support Opportunities: Regular Research Grants
• FAPESP's process: 14/21259-5 - Typical cycles in piecewise smooth dynamical systems
• Grantee: Kamila da Silva Andrade
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• FAPESP's process: 13/07523-9 - Bifurcations of Degenerated Cycles in Discontinuous Systems
• Grantee: Kamila da Silva Andrade
• Support Opportunities: Scholarships in Brazil - Doctorate