The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance

Abstract

We investigate the damped cubic nonlinear quasiperiodic Mathieu equationin the vicinity of the principal 2:2:1 resonance. By using a double perturbation method which assumes that both ε and μ are small, we approximate analytical conditions for the existence and bifurcation of nonlinear quasiperiodic motions in the neighborhood of the middle of the principal instability region associated with 2:2:1 resonance. The effect of damping and nonlinearity on the resonant quasiperiodic motions of the quasiperiodic Mathieu equation is also provided. We show that the existence of quasiperiodic solutions does not depend upon the nonlinearity coefficient γ , whereas the amplitude of the associated quasiperiodic motion does depend on γ .

Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

“…For example, the nature of the stability chart near = 0:25; ! = 1, see Figure 14b, involves a 2:2:1 resonance and has been studied in [45], [46]. The diagram near = 0:25, !…”

Krylov Bogoliubov Type Analysis of Variants of the Mathieu Equation

“…To compare this result with Ref. [6] we first change variable from A to R in (9). We then take their Eqs.…”

“…A closely related work is [5] where a 2:2:1 resonance has been analysed using the scale separation method. An extension to the case of nonlinear Mathieu equation has been done in Reference [6] which has a viscous damping term and a Duffing type nonlinear term. Once again, the authors work near the 2:2:1 resonance and use a scale separation approach to obtain their primary equations, Eqs.…”

“…The Mathieu equation for dynamical systems (and its variant, the Bloch equation of quantum mechanics) have been around for about a century [1], but it is only recently, with the advent of modern computational technology, that a comprehensive analysis of extensions and variations on the basic equation has been performed. One of the primary workers in the field is RICHARD RAND; he and his collaborators have performed extensive research into quasiperiodic Mathieu equations [2][3][4][5] and equations with nonlinearity and damping [3,6]. An extension to twodimensional systems has been done by THOMAS WATERS [7].…”

Multi-level approach for parametric roll analysis

“…For the quasi-periodic Mathieu equation, mathematical and numerical approaches have been introduced by Zounes and Rand (1998) to achieve the solution and the stability with same amplitude of fluctuation in two components. Further improvements are achieved in different amplitude of fluctuation (Rand et al, 2003), in damped and nonlinear restoring models (Guennoun et al, 2001;Abouhazim et al, 2005). Based on such mathematical background, stability diagram of roll motion in bichromatic waves is developed.…”