Melnikov Method for a Class of Generalized Ziegler Pendulums

Stefano Disca and Vincenzo Coscia ()
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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

Mathematics, 2025, vol. 13, issue 8, 1-22

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.

Keywords: double pendulum; follower force; Melnikov integral; homoclinic intersections; time-periodic perturbations; dissipation; Devaney chaos (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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