Nonlinear time transformation method for strong nonlinear oscillation systems

1992 ◽  
Vol 8 (3) ◽  
pp. 279-288 ◽  
Author(s):  
Xu Zhao
Keyword(s):  
Nonlinear Oscillation ◽  
Transformation Method ◽  
Time Transformation ◽  
Nonlinear Time Transformation

Analytical approximation of cuspidal loops using a nonlinear time transformation method

2020 ◽  
Vol 373 ◽  
pp. 125042 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Transformation Method ◽  
Analytical Approximation ◽  
Time Transformation ◽  
Nonlinear Time Transformation

Efficiency of differential transformation method for nonlinear oscillation: Comparison with HPM and VIM

2011 ◽  
Vol 11 (4) ◽  
pp. 965-971 ◽  
Author(s):  
S. Ghafoori ◽  
M. Motevalli ◽  
M.G. Nejad ◽  
F. Shakeri ◽  
D.D. Ganji ◽  
...  
Keyword(s):  
Nonlinear Oscillation ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Differential Transformation

High-Order Analysis of Global Bifurcations in a Codimension-Three Takens–Bogdanov Singularity in Reversible Systems

2020 ◽  
Vol 30 (01) ◽  
pp. 2050017 ◽  
Author(s):  
Bo-Wei Qin ◽  
Kwok-Wai Chung ◽  
Antonio Algaba ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Transformation Method ◽  
High Order ◽  
Numerical Continuation ◽  
Reversible Systems ◽  
Efficient Manner ◽  
Codimension Two ◽  
Codimension One ◽  
Melnikov Functions ◽  
Global Connections ◽  
Nonlinear Time Transformation

A codimension-three Takens–Bogdanov bifurcation in reversible systems has been very recently analyzed in the literature. In this paper, we study with the help of the nonlinear time transformation method, the codimension-one and -two homoclinic and heteroclinic connections present in the corresponding unfolding. The algorithm developed allows to obtain high-order approximations for the global connections, in such a way that it supplies in a very efficient manner the coefficients that would be obtained with high-order Melnikov functions. As we show, all our analytical predictions have excellent agreement with the numerical results. In particular we remark that, for the two different codimension-two points, the theoretical approximation coincides in six decimal digits with the numerical continuation, even being quite far from the codimension-three point. The better approximations we provide in this work will help in the study of reversible systems that exhibit this codimension-three Takens–Bogdanov bifurcation.

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