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In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol’s equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol’s coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol’ oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.
Content available remote Time-delay two-dimension mathieu equation in synchrotron dynamics
Two dimensions Mathieu equation containing periodic terms as well as the delayed parameters has been investigated in the present work. The present system represents to a generalized form of the one-dimension delay Mathieu equation. The mathematical difficulty for delay the coupled Mathieu equation has been overcome by using the matrices method. Properties of inverse complex matrices enable us to transform the vector form of the solvability conditions to the scalar form. Small oscillation about a marginal state is introduced by using the method of multiple scales. Stability criteria for the complex matrices have been established and lead to obtain resonance curves. The analysis has been extended so that the delay 2-dimensions Mathieu equation containing weak complex damping part. Stability conditions and the transition curves that included the influence of both the delayed as well the complex damping terms has been obtained. The transition curves are analyzed using the method of harmonic balance. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators.
In the present work the stability criterion for two coupled nonlinear Schrödinger equations having parametric terms is derived. In this investigation, two different types of coupled nonlinear Schrödinger equations are discussed. Two coupled parametric nonlinear Schrödinger equations govern the wave behavior at the self-secondary resonance interaction and other two coupled parametric equations describe the wave-wave interaction at self-cubic resonance case. Stability criterion governing resonance mechanisms is performed in view of temporal periodic perturbations. Moreover, stability criterion at the perfect resonance case is achieved. Further, some numerical calculations are made to screen the stability pictures at the self-second resonance case.
Content available remote Instability of Darcian flow in an alternating magnetic field
This paper treats the stability of an interface between two different fluids moving through two different porous media. There is an alternating magnetic field parallel to the interface and to the flow direction, and there is a concentrated sheet of electric current at the interface which produces jump in the magnetic field strength. The evolution of the amplitude of propagation surface waves is governed by a complex Mathieu equation which have damping terms. In the limiting case of non-streaming fluids a simplified damped Mathieu equation has been imposed. At a critical value of the stratified magnetic field, the ordinary Mathieu equation without the damping terms is derived. The contribution of viscosity to the existence of free electric surface currents on the fluid interface is discussed. It is found that at the critical stratified magnetic field, the surface currents density has disappeared from the interface whence the stratified viscosity has a unite value. The stability criteria are discussed theoretically and numerically in which stability diagrams are obtained. Regions of stability and instability are identified for the wave number versus the coefficient of free surface currents. It is found that the increase of the fluid velocity plays a destabilizing influence in the stability criteria. Porous permeability and viscosity ratio play a stabilizing or a destabilizing role in certain cases. It is found that the viscosity ratio plays a dual role in the stability behaviour at the resonance case. The field frequency plays a stabilizing influence in the case of weak viscosity analysis and at a special value of the magnetic field ratio. The destabilizing influence for the field frequency is observed for the case of the Rayleigh- Taylor model and at the resonance case.
Content available remote Nonlinear interfacial instability of two electrified miscible fluids
In a previews paper [1], a simplified formulation was presented to deal with the intcrfacial linear stability problem with mass and heat transfer, considering the presence of a periodic electric field. The present paper treats the same problem with a nonlinear approach. This approach is achieved by considering the multiple time scale method. The analysis reveals the existence of both resonant and non - resonant cases. Three types of nonlinear Schődinger equations are derived . The necessary and sufficient stability of conditions in obtained and the results are confirmed numerically. Graphs are drawn to illustrate the stability regions.
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