2025/12/11 by Disca, Stefano, Coscia, Vincenzo · 1 citation
Mathematics · Physics and Astronomy · #34D10 #37C25 (Secondary) #70K44 (Primary) 70K55 #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.2512.10682
openalex created_date 2025/12/10 · openalex publication_date 2025/12/11 · openalex updated_date 2026/07/28
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.