Center-Type Dynamics and Nonintegrability in Rocard’s Oscillator

• Martha Alvarez-Ramírez ^[1] ; Johanna D. García-Saldaña ^[2] ; Mario Medina ^[1]
  1. [1] Universidad Autónoma Metropolitana

     Universidad Autónoma Metropolitana

     México

     
  2. [2] Universidad Católica de la Santísima Concepción

     Universidad Católica de la Santísima Concepción

     Comuna de Concepción, Chile

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
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• Resumen

  • Rocard’s oscillator is a third-order nonlinear equation given by ...

    z + εω¨z + ω2 ˙z + ω3 ε + η 1 − z2 − ˙z2 ω2 z = 0, depending on the parameters ε, η and ω = 0, where ω sets a characteristic time scale and is treated as fixed throughout the analysis.We prove that, for η = 0, the equilibrium at the origin exhibits center-type behavior on a two-dimensional center manifold and admits a local analytic first integral. This explains the vanishing of the first Lyapunov coefficient and rules out aHopf bifurcation at this parameter value.We then analyze the conservative limit ε = 0, showing that the system is reversible and volume-preserving, in a setting compatible with the nonintegrability results of Hu and Tang (2025). Finally, we discuss these nonintegrability results within a geometric framework connecting the center-type dynamics at η = 0 with the conservative reversible structure arising in the dissipation-free regime.

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