Investigations on the Chaos in the Generalized Double Sine-Gordon Planar System: Melnikov’s Approach and Applications to Generating Antenna Factors

Nikolay Kyurkchiev, Tsvetelin Zaevski and Anton Iliev ()
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Nikolay Kyurkchiev: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
Tsvetelin Zaevski: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
Anton Iliev: Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 24, Tzar Asen Str., 4000 Plovdiv, Bulgaria

Mathematics, 2025, vol. 13, issue 22, 1-23

Abstract: Many authors analyze the chaotic motion of the driven and damped double sine-Gordon equations and compute the Melnikov functions by numerical methods, taking an example to verify good agreement between numerical methods and analytical ones. Unfortunately, due to the lack of an explicit presentation of the Melnikov integral, the reader has difficulty navigating and touching upon Melnikov’s elegant theory and, in particular, the formulation of the Melnikov criterion for the occurrence of chaos in a dynamical system, based solely on the provided illustrations of dependencies between the main parameters of the model under consideration. In this paper we will try to shed additional light on this important problem. A new planar system corresponding to the generalized double sine-Gordon model with many free parameters is considered. We also look at the modeling of radiation diagrams and antenna factors as potential uses for the Melnikov functions. A number of simulations are created. We also show off a few specific modules for examining the model’s behavior. There is also discussion of one use for potential oscillation control.

Keywords: homoclinic and heteroclinic bifurcations; Melnikov function; Melnikov antenna factor; control over oscillations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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