Article

Journal of Dynamics and Differential Equations

, Volume 1, Issue 3, pp 269-298

The dynamics of coupled planar rigid bodies. II. Bifurcations, periodic solutions, and chaos

  • Y. -G. OhAffiliated withDepartment of Mathematics, University of California
  • , N. SreenathAffiliated withDepartment of Electrical Engineering and the Systems Research Center, University of Maryland
  • , P. S. KrishnaprasadAffiliated withDepartment of Electrical Engineering and the Systems Research Center, University of Maryland
  • , J. E. MarsdenAffiliated withDepartment of Mathematics, University of California

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Abstract

We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.

Key words

Geometric mechanics reduction stability chaos rigid body dynamics periodic orbits

AMS subject classification

58F