Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation (original) (raw)

2: 2: 1 Resonance in the quasiperiodic Mathieu equation

Nonlinear Dynamics, 2003

In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation d 2 x dt 2 + (δ + cos t + µ cos(1 + ∆)t)x = 0, using two successive perturbation methods. The parameters and µ are assumed to be small. The parameter serves for deriving the corresponding slow flow differential system and µ serves to implement a second perturbation analysis on the slow flow system near its proper resonance. This strategy allows us to obtain analytical expressions for the transition curves in the resonant quasiperiodic Mathieu equation. We compare the analytical results with those of direct numerical integration. This work has application to parametrically excited systems in which there are two periodic drivers, each with frequency close to twice the frequency of the unforced system.

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

Nonlinear Dynamics, 2006

We investigate the damped cubic nonlinear quasiperiodic Mathieu equation d 2 x dt 2 + (δ + ε cos t + εμ cos ωt)x + εμc dx dt + εμγ x 3 = 0 in 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 .

The Quasiperiodic Mathieu Equation

2015

We obtain power series solutions to the \abc equation" dy dx = a+ b cos y + c cos x; valid for small c, and for small b. This equation is shown to determine the stability of the quasiperiodic Mathieu equation, z + ( + A 1 cos t+ A 2 cos!t)z = 0; in the small limit. Perturbation results of the abc equation are shown to compare favorably to numerical integration of the quasiperiodic Mathieu equation.

Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation

International Journal of Non-Linear Mechanics, 2002

Quasi-periodic (QP) solutions of a weakly damped non-linear QP Mathieu equation are investigated near a double primary parametric resonance. A double multiple scales method is applied to reduce the original QP oscillator to an autonomous system performing two successive reduction. The problem for approximating QP solutions of the original system is then transformed to the study of stationary regimes of the induced autonomous system. Explicit analytical approximations to QP oscillations are obtained and comparisons to numerical integration of the original QP oscillator are provided.

A Quasiperiodic Mathieu Equation

Series on Stability, Vibration and Control of Systems, Series B, 1997

We obtain power series solutions to the \abc equation" dy dx = a + b cos y + c cos x; valid for small c, and for small b. This equation is shown t o determine the stability of the quasiperiodic Mathieu equation, z + (+ A1 cos t + A2 cos!t)z = 0 ; in the small limit. Perturbation results of the abc equation are shown to compare favorably to numerical integration of the quasiperiodic Mathieu equation.

A note on nonlinear oscillations at resonance

Acta Mathematica Sinica, 1987

Abotrac*t. Existence of 2n-periodic solutions to the equation ~ + g(x)=p(t) is proved, under sharp nonresonance conditions on the interaction of g(x)/x with two consecutive eigenvalues m 2 and (m + 1)2: touch with the eigenvalues is allowed.

Weakly nonlinear and symmetric periodic systems at resonance

CITATION 1 READS 23 2 authors: Nataliya Dilna Slovak Academy of Sciences 18 PUBLICATIONS 44 CITATIONS SEE PROFILE Michal Feckan Comenius University in Bratislava 211 PUBLICATIONS 1,535 CITATIONS SEE PROFILE All content following this page was uploaded by Michal Feckan on 23 October 2014. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately.

Transition Curves for the Quasi-Periodic Mathieu Equation

SIAM Journal on Applied Mathematics, 1998

In this work we investigate an extension of Mathieu's equation, the quasi-periodic (QP) Mathieu equation given byψ + [δ + (cos t + cos ωt)] ψ = 0 for small and irrational ω. Of interest is the generation of stability diagrams that identify the points or regions in the δ-ω parameter plane (for fixed ) for which all solutions of the QP Mathieu equation are bounded. Numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots of Lyapunov exponents. In addition, we derive approximate analytic expressions for transition curves using two distinct techniques: (1) a regular perturbation method under which transition curves δ = δ(ω; ) are each expanded in powers of , and (2) the method of harmonic balance utilizing Hill's determinants. Both analytic methods deliver results in good agreement with those generated numerically in the sense that predominant regions of instability are clearly coincident. And, both analytic techniques enable us to gain insight into the structure of the corresponding numerical plots. However, the perturbation method fails in the neighborhood of resonant values of ω due to the problem of small divisors; the results obtained by harmonic balance do not display this undesirable feature.