scholarly journals A Generalized KBM Method for Strongly Nonlinear Oscillators with Slowly Varying Parameters

2007 ◽  
Vol 12 (1) ◽  
pp. 21-30 ◽  
Author(s):  
Jianping Cai
Keyword(s):  
Nonlinear Oscillators ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Kbm Method ◽  
Strongly Nonlinear Oscillators

scholarly journals Escape time from potential wells of strongly nonlinear oscillators with slowly varying parameters

2005 ◽  
Vol 2005 (3) ◽  
pp. 365-375 ◽  
Author(s):  
Jianping Cai ◽  
Y. P. Li ◽  
Xiaofeng Wu
Keyword(s):  
Elliptic Function ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Escape Time ◽  
Potential Well ◽  
Jacobian Elliptic Function ◽  
Potential Wells ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

The effect of negative damping to an oscillatory system is to force the amplitude to increase gradually and the motion will be out of the potential well of the oscillatory system eventually. In order to deduce the escape time from the potential well of quadratic or cubic nonlinear oscillator, the multiple scales method is firstly used to obtain the asymptotic solutions of strongly nonlinear oscillators with slowly varying parameters, and secondly the character of modulus of Jacobian elliptic function is applied to derive the equations governing the escape time. The approximate potential method, instead of Taylor series expansion, is used to approximate the potential of an oscillation system such that the asymptotic solution can be expressed in terms of Jacobian elliptic function. Numerical examples verify the efficiency of the present method.

scholarly journals Approximate Potentials with Applications to Strongly Nonlinear Oscillators with Slowly Varying Parameters

2003 ◽  
Vol 10 (5-6) ◽  
pp. 379-386 ◽  
Author(s):  
Jianping Cai ◽  
Y.P. Li
Keyword(s):  
Present Method ◽  
Nonlinear Oscillator ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Elapsed Time ◽  
Strongly Nonlinear Oscillators ◽  
Strongly Nonlinear Oscillator

A method of approximate potential is presented for the study of certain kinds of strongly nonlinear oscillators. This method is to express the potential for an oscillatory system by a polynomial of degree four such that the leading approximation may be derived in terms of elliptic functions. The advantage of present method is that it is valid for relatively large oscillations. As an application, the elapsed time of periodic motion of a strongly nonlinear oscillator with slowly varying parameters is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

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