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Melnikov Method for a Class of Generalized Ziegler Pendulums

Author

Listed:
  • Stefano Disca

    (Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy)

  • Vincenzo Coscia

    (Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy)

Abstract

The Melnikov method is applied to a class of generalized Ziegler pendulums. We find an analytical form for the separatrix of the system in terms of Jacobian elliptic integrals, holding for a large class of initial conditions and parameters. By working in Duffing approximation, we apply the Melnikov method to the original Ziegler system, showing that the first non-vanishing Melnikov integral appears in the second order. An explicit expression for the Melnikov integral is derived in the presence of a time-periodic external force and for a suitable choice of the parameters, as well as in the presence of a dissipative term acting on the lower rod of the pendulum. These results allow us to define fundamental relationships between the Melnikov integral and a proper control parameter that distinguishes between regular and chaotic orbits for the original dynamical system. Finally, in the appendix, we present proof of a conjecture concerning the non-validity of Devaney’s chaoticity definition for a discrete map associated with the system.

Suggested Citation

  • Stefano Disca & Vincenzo Coscia, 2025. "Melnikov Method for a Class of Generalized Ziegler Pendulums," Mathematics, MDPI, vol. 13(8), pages 1-22, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1267-:d:1633131
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