Controlling activated processes of nonadiabatically, periodically driven dynamical systems: A multiple scale perturbation approach

2012 ◽  
Vol 136 (23) ◽  
pp. 234506 ◽  
Author(s):  
Anindita Shit ◽  
Sudip Chattopadhyay ◽  
Jyotipratim Ray Chaudhuri
Keyword(s):  
Dynamical Systems ◽  
Multiple Scale ◽  
Perturbation Approach ◽  
Periodically Driven ◽  
Activated Processes ◽  
Scale Perturbation

Determination of the growth rate for the linearized Zakharov—Kuznetsov equation

1993 ◽  
Vol 50 (3) ◽  
pp. 413-424 ◽  
Author(s):  
M. A. Allen ◽  
G. Rowlands
Keyword(s):  
Growth Rate ◽  
Multiple Scale ◽  
Great Accuracy ◽  
Padé Approximant ◽  
Perturbation Approach ◽  
Pade Approximant ◽  
Higher Dimensional ◽  
Analytical Work ◽  
Scale Perturbation

Studies of the Zakharov—Kuznetsov equation governing solitons in a strongly magnetized ion-acoustic plasma indicate that a perturbed flat soliton is unstable and evolves into higher-dimensional solitons. The growth rate γ = γ(k) of a small sinusoidal perturbation of wavenumber k to a flat soliton has already been found numerically, and lengthy analytical work has given the value of We introduce a more direct analytical method in the form of an extension to the usual multiple-scale perturbation approach and use it to determine a consistent expansion of γ about k = 0 and the other zero at k2 = 5.By combining these results in the form of a two-point Padé approximant, we obtain an analytical expression for γ valid over the entire range of k for which the solution is unstable. We also present a very efficient numerical method for determining the growth rate curve to great accuracy. The Padé approximant gives excellent agreement with the numerical results.

A multiple-scale perturbation approach to mode coupling in periodic plates

2013 ◽  
Vol 60 (2) ◽  
pp. 395-401 ◽  
Author(s):  
Omar Asfar ◽  
Muhammad Hawwa ◽  
Maxime Bavencoffe ◽  
Bruno Morvan ◽  
Jean-Louis Izbicki
Keyword(s):  
Mode Coupling ◽  
Multiple Scale ◽  
Perturbation Approach ◽  
Scale Perturbation

The Synchronization of a Periodically Driven Pendulum With Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Fuhong Min
Keyword(s):  
Dynamical Systems ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Invariant Domain ◽  
Periodically Driven

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in Duffing oscillator is developed through the theory of discontinuous dynamical systems. The conditions for the synchronization invariant domain are obtained, and the partial and full synchronizations are illustrated for the analytical conditions.

Simulation of Water Wave Transformation Using Higher Order Mild-Slope Equation

2009 ◽  
Author(s):  
Tai-Wen Hsu ◽  
Ta-Yuan Lin ◽  
Kuan-Yu Hsiao ◽  
Shiao-Yin Chen
Keyword(s):  
Multiple Scale ◽  
Higher Order ◽  
Wave Transformation ◽  
Radiation Boundary Conditions ◽  
Depth Function ◽  
Mild Slope Equation ◽  
Scale Perturbation ◽  
Radiation Boundary ◽  
Wave Nonlinearity ◽  
Sloping Beach

A higher-order mild-slope equation (HOMSE) was developed using classical Galerkin method in which the depth function is expanded to the third-order. Wave nonlinearity and bottom slope parameters are involved in the depth function solved on the bases of the multiple-scale perturbation method. The equation is solved subject to the radiation boundary conditions by means of the procedure of parabolic formulation. Good agreement between numerical results and experimental data has been observed for wave propagation over a submerged obstacle and a sloping beach.

Three Dimensional Fully Localized Waves on Ice-Covered Ocean

2013 ◽  
Author(s):  
Yong Liang ◽  
M.-Reza Alam
Keyword(s):  
Numerical Scheme ◽  
Perturbation Technique ◽  
Three Dimensional ◽  
Multiple Scale ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Scale Perturbation ◽  
Speed Of Propagation ◽  
Localized Waves ◽  
Mean Current

We have recently shown [1] that fully-localized three-dimensional wave envelopes (so-called dromions) can exist and propagate on the surface of ice-covered waters. Here we show that the inertia of the ice can play an important role in the size, direction and speed of propagation of these structures. We use multiple-scale perturbation technique to derive governing equations for the weakly nonlinear envelope of monochromatic waves propagating over the ice-covered seas. We show that the governing equations simplify to a coupled set of one equation for the envelope amplitude and one equation for the underlying mean current. This set of nonlinear equations can be further simplified to fall in the category of Davey-Stewartson equations [2]. We then use a numerical scheme initialized with the analytical dromion solution of DSI (i.e. shallow-water and surface-tension dominated regimes of Davey-Stewartson equation) to look for dromion solution of our equations. Dromions can travel over long distances and can transport mass, momentum and energy from the ice-edge deep into the solid ice-cover that can result in the ice cracking/breaking and also in posing dangers to icebreaker ships.

TRACKING SUSTAINED CHAOS

2000 ◽  
Vol 10 (03) ◽  
pp. 571-578 ◽  
Author(s):  
IRA B. SCHWARTZ ◽  
IOANA TRIANDAF
Keyword(s):  
Dynamical Systems ◽  
Simple Method ◽  
Periodically Driven ◽  
Unstable Solutions ◽  
Chaotic Transients ◽  
Parameter Values

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO 2 laser.

Stability of solitary-wave solutions to a modified Zakharov–Kuznetsov equation

2000 ◽  
Vol 64 (4) ◽  
pp. 411-426 ◽  
Author(s):  
S. MUNRO ◽  
E. J. PARKES
Keyword(s):  
Acoustic Waves ◽  
Multiple Scale ◽  
Travelling Wave ◽  
Magnetized Plasma ◽  
Initial Growth ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Instability Range ◽  
Scale Perturbation ◽  
Weakly Nonlinear Waves

In the context of ion-acoustic waves in a magnetized plasma comprising cold ions and non-isothermal electrons, small-amplitude, weakly nonlinear waves have been shown previously by Munro and Parkes to be governed by a modified version of the Zakharov–Kuznetsov equation. In this paper, we consider solitary travelling-wave solutions to this equation that propagate along the magnetic field. We investigate the initial growth rate γ(k) of a small transverse sinusoidal perturbation of wavenumber k. The instability range is shown to be 0 < k < 3. We use the multiple-scale perturbation method developed by Allen and Rowlands to determine a consistent expansion of γ about k = 0 and k = 3. By combining these results in the form of a Padé approximant, an analytical expression for γ is found that is valid for 0 < k < 3. γ is also determined by using the variational method developed by Bettinson and Rowlands. The two results for γ are compared with a numerical determination.

Analysis of non-linear vibrations of a microresonator under piezoelectric and electrostatic actuations

2008 ◽  
Vol 223 (2) ◽  
pp. 329-344 ◽  
Author(s):  
M Zamanian ◽  
S E Khadem ◽  
S N Mahmoodi
Keyword(s):  
Piezoelectric Layer ◽  
Multiple Scale ◽  
Electrostatic Actuation ◽  
Non Linear Equation ◽  
Elasticity Module ◽  
Non Linear ◽  
Linear Behaviour ◽  
Linear Vibrations ◽  
Scale Perturbation ◽  
Electrode Plate

In this article, the non-linear vibrations of a piezoelectrically actuated microresonator is studied. The microresonator is assumed as a clamped—clamped Bernouli—Euler microbeam. In contrast to previous researches in which the piezoelectric layer has been deposited on the integral length of the microbeam, here it is assumed that the piezoelectric layer is deposited on a part of microbeam length with equal distance from two ends. The microbeam is actuated by an AC voltage between upper and lower sides of the piezoelectric layer. Also, an electrostatic actuation is applied between the microbeam and an electrode plate, for the first time. The non-linear equation of motion has been derived by using the Hamilton principle by stretching the neutral axis assumption. The obtained equations are solved using the Galerkin, Rayleigh—Ritz, and multiple scale perturbation methods. It is shown that the sensitivity and natural frequency of the piezoelectrically actuated microresonator may be altered and controlled conveniently by applying an electrostatic actuation to the microresonator. Also, it has been shown that the system shows softening or hardening behaviour depending on the value of piezoelectric actuation, electrostatic actuation, axial load, thickness, length, and elasticity module of piezoelectric layer; thickness of microbeam; and its distance from the electrode plate. It is shown that non-linear behaviour of the piezoelectrically actuated microbeam may be changed to a linear behaviour by applying a suitable electrostatic actuation to the microbeam.

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