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Global dynamics of mechanical systems with cubic potentials

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We study the behavior of solutions of mechanical systems with polynomial potentials of degree 3 by using a blow up of McGehee type. We first state some general properties for positive degree homogeneous potentials. In particular, we prove a very general property of transversality of the invariant manifolds of the flow along the homothetic orbit. The paper focuses in the study of global flow in the case of homogeneous polynomial potentials of degree 3 for negative energy. The flow is fairly simple because of its gradient-like structure, although for some values of the polynomial coefficients we have diverse behaviour of the separatrices on the infinity manifold, which are essential to describe the global flow.

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Correspondence to M. Falconi.

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Submitted by J. Llibre

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Falconi, M., Lacomba, E.A. & Vidal, C. Global dynamics of mechanical systems with cubic potentials. Qual. Th. Dyn. Syst 2, 429–453 (2001). https://doi.org/10.1007/BF02969350

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Key Words

  • Hamiltonian Vector Field
  • Homogeneous Polynomial Potentials
  • Invariant Manifolds
  • Global Flow