A Survey on the Melnikov Theory for Implicit Ordinary Differential Equations with Applications to RLC Circuits
Our recent results are presented on the development of the Melnikov theory in investigation of implicit ordinary differential equations with small amplitude perturbations. In particular, the persistence of orbits connecting singularities in finite time is
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Abstract Our recent results are presented on the development of the Melnikov theory in investigation of implicit ordinary differential equations with small amplitude perturbations. In particular, the persistence of orbits connecting singularities in finite time is studied provided that certain Melnikov like conditions hold. Achievements on reversible implicit ordinary differential equations are also considered. Applications are given to nonlinear systems of RLC circuits. Keywords Implicit ordinary differential equations · Impasse points · IK-singularities · RLC circuits 2010 MSC 34A09 · 37C60 · 47N70
1 Introduction Implicit ordinary differential equations (IODEs for short) find applications in a large number of physical sciences and have been studied by several authors [16, 19, 20, 23–27]. In particular, IODEs naturally arise in modelling nonlinear RLC circuits as it is demonstrated also in this paper. We survey in this paper our recent results on IODEs with small amplitude perturbations by using the Melnikov theory. In particular, the persistence of orbits connecting singularities in finite time is studied This work was supported by the Slovak Research and Development Agency (grant number APVV14-0378) and the Slovak Grant Agency VEGA (grant numbers 2/0153/16 and 1/0078/17). M. Feˇckan (B) Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia e-mail: [email protected] M. Feˇckan Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia © Springer Nature Switzerland AG 2019 F. T. Smith et al. (eds.), Mathematics Applied to Engineering, Modelling, and Social Issues, Studies in Systems, Decision and Control 200, https://doi.org/10.1007/978-3-030-12232-4_4
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provided that certain Melnikov like conditions hold. Implicit differential equations have been studied also in many other papers [11, 12, 15, 18, 22] but the results of this paper do not seem to be covered by them. On the other hand, those results deal with more general implicit differential systems using analytical, topological and numerical methods. In the terminology of [27], we study persistence of global solutions terminating in finite time either to I–singularities or to IK–singularities. Related results are given in [9, 10]. We note that the Melnikov method and its extensions are mainly used to prove existence of chaotic orbits in dynamical systems [1, 21, 31]. We apply this method to study IODEs, since it is a natural way to handle our studied problems. Moreover, our kind of problems is different to former ones on the existence of chaos. Consequently, our results are original which have not yet been studied by other researchers. The paper is organized as follows. Sections 2 and 3 deal with nonlinear RLC circuits. Section 4 studies weakly coupled nonlinear RLC circuits. IODEs in arbitrary finite dimensional spaces are considered in Sect. 5. The so called
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