Analysis and synthesis of oscillator systems described by perturbed double hump Duffing equations

Abstract

SUMMARYThis paper presents an analysis of oscillator systems described by double hump Duffing equations under polynomial perturbations of fourth degree. It has been proved that such a system can have unique hyperbolic limit cycle whose properties depend on the perturbation coefficients. The analytical condition for the arising of a limit cycle has been derived. Moreover, a method for the synthesis of oscillator systems of the considered type, having preliminarily assigned properties, is proposed. The synthesis consists of an appropriate choice of the perturbation coefficients in such a way that the oscillator equation is to have in advance assigned limit cycle. Both the analysis and the synthesis are performed with the aid of the Melnikov function.

Structural characterization of classical and memristive circuits with purely imaginary eigenvalues

“…Note that this is already enough to show in practice that the circuit cannot be topologically non-hyperbolic. Proceeding analogously with the second row of (21), which comes from the second hybrid loop, we get the additional relation L 1 C 2 =L 2 C 1 , and together with the aforementioned relation, we get L 1 =L 2 , C 1 =C 2 as necessary relations for the existence of a PIE. This is of course consistent with the results in Ref.…”

“…Qualitative theory plays a key role in the analysis of nonlinear electrical and electronic circuits. Qualitative results are related, for example, to the stability properties of equilibria and operating points [5,17,22,28,60], oscillations [11,21,25,44], bifurcations [15,42,55], or chaotic phenomena [3,19,31,34,35,38,40,41,63,65]. These references are just a sample of the huge literature addressing qualitative aspects in electrical and electronic engineering (cf.…”

Horseshoe chaos and subharmonic orbits in the nanoelectromechanical Casimir nonlinear oscillator

“…This method has been successfully applied to the analysis and design of oscillating nonlinear circuits and systems [20][21][22]. The existence of such orbits implies, via the Smale-Birkhoff theorem, the presence of Smale horseshoes.…”

“…The Melnikov method also represents an effective tool to investigate the effect of periodic perturbations on the orbits inside the homoclinic loop. This method has been successfully applied to the analysis and design of oscillating nonlinear circuits and systems [20][21][22]. Assume that q m (t) is a periodic orbit of the unperturbed system, with period T m = pT/q, where p, q are relative prime integers, and define the subharmonic Melnikov function…”