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Nonlinear RLC circuits and implicit ODEs. (English) Zbl 1340.34044
Summary: We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations exhibiting I-singularities (in the sense given in [R. Riaza, Differential-algebraic systems. Analytical aspects and circuit applications. Hackensack, NJ: World Scientific (2008; Zbl 1184.34004), p. 166]). In particular, we show the persistence of such I-singularities and orbits connecting them in finite time provided a Melnikov like condition holds. We start from a concrete example, where we prove that this Melnikov condition actually holds. Then, we extend our results to more general implicit differential equations.

34A09 Implicit ordinary differential equations, differential-algebraic equations
34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
94C05 Analytic circuit theory
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations