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arXiv · 2512.10682

Melnikov Method for a Class of Generalized Ziegler Pendulums

Stefano Disca·Vincenzo Coscia
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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.

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Stefano Disca, Vincenzo Coscia. 2025-12-11. Melnikov Method for a Class of Generalized Ziegler Pendulums. https://doi.org/10.3390/math13081267

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