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Bifurcations in biparametric quadratic potentials.
1Departamento de Matematicas, Universidad de la Rioja, 26004 Logrono, SpainGrupo de Mecanica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain.
Chaos (Woodbury, N.Y.)
|June 1, 1995
Summary
This study analyzes bifurcations in dynamical systems with quadratic Hamiltonians on the S(2) sphere. It identifies pitchfork, teardrop, and oyster bifurcations governing homoclinic orbit changes, with teardrop bifurcations linked to zero-Poincare index fixed points.
Area of Science:
- Mathematics
- Physics
- Dynamical Systems Theory
Background:
- Quadratic Hamiltonians on the S(2) sphere model numerous dynamical systems.
- Understanding parameter-dependent bifurcations is crucial for analyzing system behavior.
Purpose of the Study:
- To analyze bifurcations in a class of quadratic Hamiltonians on S(2) with two parameters.
- To identify bifurcation lines in the parameter plane.
- To investigate the types of bifurcations governing the appearance-disappearance of homoclinic orbits.
Main Methods:
- Analysis of quadratic Hamiltonians with phase space on the S(2) sphere.
- Parameter space analysis to determine bifurcation lines.
- Phase portrait analysis to observe homoclinic orbit dynamics.
Main Results:
- Three types of bifurcations were identified: pitchfork, teardrop, and oyster.
- Bifurcation lines were obtained in the parameter plane.
- The teardrop bifurcation is associated with a non-elementary fixed point with a Poincare index of zero.
Conclusions:
- The study provides a comprehensive analysis of bifurcations for a specific class of dynamical systems.
- The findings elucidate the mechanisms driving changes in homoclinic orbits.
- The characterization of the teardrop bifurcation offers new insights into non-elementary fixed points.