Melnikov Method for a Class of Generalized Ziegler Pendulums

TL;DR

This paper applies the Melnikov method to generalized Ziegler pendulums, deriving analytical expressions for separatrices and chaos indicators, and explores conditions for regular and chaotic dynamics.

Contribution

It introduces an analytical approach to compute Melnikov integrals for generalized Ziegler pendulums, including explicit formulas under various conditions.

Findings

Analytical form of the separatrix using Jacobian elliptic integrals.

First non-vanishing Melnikov integral appears at second order.

Relationship between Melnikov integral and control parameters for chaos detection.

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.