Melnikov Method for a Class of Generalized Ziegler Pendulums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915669956624384 |
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| author | Disca, Stefano Coscia, Vincenzo |
| author_facet | Disca, Stefano Coscia, Vincenzo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Melnikov Method for a Class of Generalized Ziegler Pendulums Disca, Stefano Coscia, Vincenzo Chaotic Dynamics Mathematical Physics Dynamical Systems 70K44 (Primary) 70K55, 34D10, 37C25 (Secondary) 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. |
| title | Melnikov Method for a Class of Generalized Ziegler Pendulums |
| topic | Chaotic Dynamics Mathematical Physics Dynamical Systems 70K44 (Primary) 70K55, 34D10, 37C25 (Secondary) |
| url | https://arxiv.org/abs/2512.10682 |