Stability Analysis Of The Non-Linear Mathieu Equation

Mond, M.; Cederbaum, G.; Khan, P.B.; Zarmi, Yair

doi:10.1006/jsvi.1993.1322

Cited by 46 publications

(18 citation statements)

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“…Although the stability analysis, even for the linear Mathieu equation, is rather cumbersome, different methods are available. Among them are the method of integral of energy and numerical integration [31], the method of perturbation (the most widely applied are the methods of multiple scales and averaging) [32][33][34][35], the method of strained parameters based on Floquet theory [32], the method of infinite determinants [35], and other methods such as the method of normal forms [36]. These analyses could give a prediction of the regions of stability and instability, i.e., the bounded and unbounded solutions of the Mathieu-Duffing equation.…”

Section: Nonlinear Differential Equations For Time Functionmentioning

confidence: 99%

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Wang

Song

et al. 2009

Arch Appl Mech

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The chaotic vibrations of a bimetallic shallow shell of revolution under time-varying temperature excitation are investigated in the present study. The governing equations are established in forms similar to those of classical single-layered shell theory by re-determination of reference surface. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to determine if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, and both the bifurcation diagrams and corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, height parameter of the shell, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaotic motion, chaos with interior crisis and period window, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converting to period orbit have been observed from these figures.

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Section: Nonlinear Differential Equations For Time Functionmentioning

confidence: 99%

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Wang

Song

et al. 2009

Arch Appl Mech

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“…Applications References u + (ω 2 + a cos(θ t))u + b cos(θt)u 3 + cu 3 = 0 Parametrically excited vibration of a column made of non-linear elastic material M. Mond and G. Cedebaum [3] u + (a + b cos(t))u + cu 3 = 0 Beams with harmonically variable length; Stability of floating offshore structures; Vibrations of beams under harmonic support motion;…”

Section: Mathieu Equation Formsmentioning

confidence: 99%

“…The generalized form of the Mathieu differential equation has been one of the main interests of researchers due to its numerous applications. Mond and Cederbaum analyzed the non-linear form of the Mathieu differential equation within the framework of the method of normal forms [3]. They obtained the analytical conditions for the explosive instability and derived expressions for the period of vibration and the vibration amplitude of the stable response.…”

Section: Introductionmentioning

confidence: 99%

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

2005

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The generalized form of the well-known Mathieu differential equation, which consists of two driving force terms, including the quadratic and cubic nonlinearities, has been analyzed in this paper. The two-dimensional Lindstedt-Poincaré's perturbation technique has been considered in order to obtain the analytical solutions. The transition curves in some special cases have been presented. It is shown that the periodic solution does indeed exist and in general they are dependent on the initial conditions. Results of this analytical approach were compared with those obtained from the numerical methods and it is found that they are in a good agreement.

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“…In 1965, Tso and Caughey [14] investigated the parametric vibration of a non-linear system through the slowly varying parameter technique, and in 1993, Mond et al [15] gave the stability analysis of non-linear Mathieu equation through the normal form technique. However, the normal forms are obtained through the simpli"cation of analytical expression of the vector "eld on the center manifold.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Motion in the Generic Separatrix Band of a Mathieu–duffing Oscillator With a Twin-Well Potential

Luo¹

2001

Journal of Sound and Vibration

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Stability Analysis Of The Non-Linear Mathieu Equation

Cited by 46 publications

References 0 publications

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

Chaotic Motion in the Generic Separatrix Band of a Mathieu–duffing Oscillator With a Twin-Well Potential

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