Stability and bifurcation of Mathieu–Duffing equation

A dual principle for symmetric periodic solutions Bi-stability in noninterdigitated Comb-drive MEMS

“…Another interesting remark is that the statement in Theorem 2.2 could be expected to be an if and only if characterization, similar to that given by the Ortega's principle in [17]; nevertheless, we cannot deduce a result with sufficient and necessary conditions here because the conclusion of Lemma 1 leads to two possibilities. Besides, by following the ideas in [14], it may be possible to obtain a version of the dual principle for the existence of even periodic solutions with prescribed nodal properties in nonlinear oscillators like (1). Finally, as the treatment for the stability analysis in Section 3 reveals, the sign of I m,N could change for m = 2n and n ∈ N >1 .…”

“…Introduction. Let us consider the second order differential equation ẍ + xD(t, x) = 0, (1) where D ∈ C 0,1 (R/T Z×R) for some T > 0. Hereafter, we shall assume the following hypotheses over the function D: H1 D(−t, x) ≡ D(t, x) ≡ D(t, −x), H2 D(t, 0) < D(t, x) for all t ∈ R and x ̸ = 0.…”

“…Perhaps one of the most known types of parametrically excited ODEs is given by the Mathieu equation and its nonlinear generalizations that have been considered for different problems like the vibration of bodies (elliptical membranes, vibration of columns, systems described by the Mathieu-Duffing equation), the angular oscillations of a pendulum with mobile support point (see [1,10], [22] and the references therein), and the mechanical periodic responses of electrostatic microelectromechanical systems (MEMS) [23,19]. In fact, some of these general equations were studied in the literature (parametric resonance, the existence of periodic responses and their stability/instability analysis, among other aspects) by means of tools such as fixed point theorems, normal forms, perturbation techniques, and numerical strategies.…”

On the Computation of Floquet Multipliers for Periodic Solution in Piecewise-smooth Dynamical System

“…As one kind of the most common responses arising in nonlinear dynamical systems, periodic solution has always been one of the centers of research in controlling chaotic motions [1][2] and stabilizing/destabilizing certain system responses [3][4][5][6]. For a long period of time, the Floquet multiplier theory has been an indispensable tool in the stability and bifurcation analysis for periodic responses [7][8][9][10][11]. This is an important and fundamental issue, as the stability evaluation is not only associated with system analysis [7], but also one of the cornerstones for realizing the stabilizing [4] or destabilizing [6] of certain dynamic behaviors.…”