Michal Fečkan
Department of Mathematical Analysis and Numerical Mathematics
Faculty of Mathematics Physics and Informatics
Comenius University in Bratislava, Slovakia
e-mail address: Michal.Feckan@fmph.uniba.sk
Abstract: Implicit ordinary differential equations (IODEs for short) find applications in a large number of physical sciences and have been studied by several authors [12, 14, 15, 17, 22]. In particular, IODEs naturally arise in modelling nonlinear RLC circuits, as it is also demonstrated in this talk. We survey in this talk our recent results on IODEs with small amplitude perturbations by using the Melnikov theory [28]. In particular, the persistence of orbits connecting singularities in finite time is studied, provided that certain Melnikov-like conditions hold.
Implicit differential equations have also been studied in many other papers [911, 13], but the results of this talk do not seem to be covered by them. On the other hand, those results deal with more general implicit didifferential systems using analytical, topological, and numerical methods. In the terminology of [22], we study the persistence of global solutions terminating in finite time either to I singularities or to IK singularities. We note that the Melnikov method and its extensions are mainly used to prove the existence of chaotic orbits in dynamical systems [1, 16, 23]. We apply this method to study IODEs since it is a natural way to handle our studied problems. Moreover, our kind of problem is didifferent from former ones on the existence of chaos. The talk deals with nonlinear RLC circuits, with weakly coupled nonlinear RLC circuits, with IODEs in arbitrary finite-dimensional spaces, with so-called bly-like catastrophe for reversible IODEs, with a general theory for connecting IK singularities and impase points. Many concrete examples are given to illustrate theoretical results.
References:
[1] J. Awrejcewicz and M.M. Holicke, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods, World Scientific Publishing Co., Singapore, 2007.
[2] F. Battelli and F. Fečkan, Melnikov theory for nonlinear implicit ODEs, J. Differential
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[3] F. Battelli and F. Fečkan, Melnikov theory for weakly coupled nonlinear RLC circuits, Bound. Value Probl. 2014:101 (2014).
[4] F. Battelli and F. Fečkan, Nonlinear RLC circuits and implicit ODEs, Differential Integral Equations 27 (2014), 671-690.
[5] F. Battelli and F. Fečkan, On the existence of solutions connecting singularities in nonlinear RLC, Nonlinear Anal. 116 (2015), 26-36.
[6] F. Battelli and F. Fečkan, Blue sky-like catastrophe for reversible nonlinear implicit ODEs, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), 895-922.[7] F. Battelli and F. Feckan, On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 3043-3061.
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