A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens–Bogdanov normal form

2019 ◽  
Vol 97 (2) ◽  
pp. 979-990 ◽  
Author(s):  
Antonio Algaba ◽  
Kwok-Wai Chung ◽  
Bo-Wei Qin ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Normal Form ◽  
Transformation Method ◽  
Homoclinic Bifurcation ◽  
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

scholarly journals Accurate Approximation of Homoclinic Solutions in Gray–Scott Kinetic Model

2015 ◽  
Vol 25 (09) ◽  
pp. 1550125 ◽  
Author(s):  
Yu. A. Kuznetsov ◽  
H. G. E. Meijer ◽  
B. Al-Hdaibat ◽  
W. Govaerts
Keyword(s):  
Kinetic Model ◽  
Normal Form ◽  
Homoclinic Orbits ◽  
Analytic Solutions ◽  
Homoclinic Bifurcation ◽  
Second Order ◽  
Numerical Continuation ◽  
Accurate Approximation ◽  
Bifurcation Curve ◽  
Systems Of Differential Equations

The second-order predictor for the homoclinic orbit is applied to the Gray–Scott model. The problem is used to illustrate the approximation of the homoclinic orbits near a generic Bogdanov–Takens bifurcation in n-dimensional systems of differential equations. In the process, we show that it is necessary to take (usually ignored) cubic terms in the Bogdanov–Takens normal form into account to derive a correct second-order prediction for the homoclinic bifurcation curve. The analytic solutions are compared with those obtained by numerical continuation.

GEOMETRIC LIFT OF PATHS OF HAMILTONIAN EQUILIBRIA AND HOMOCLINIC BIFURCATION

2012 ◽  
Vol 22 (12) ◽  
pp. 1250304 ◽  
Author(s):  
THOMAS J. BRIDGES
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Homoclinic Orbit ◽  
Nonlinear Coefficient ◽  
Homoclinic Bifurcation ◽  
Intrinsic Curvature ◽  
Planar System ◽  
Classical View ◽  
Four Dimensions ◽  
New Formula

A saddle-center transition of eigenvalues in the linearization about Hamiltonian equilibria, and the attendant planar homoclinic bifurcation, is one of the simplest and most well-known bifurcations in dynamical systems theory. It is therefore surprising that anything new can be said about this bifurcation. In this tutorial, the classical view of this bifurcation is reviewed and the lifting of the planar system to four dimensions gives a new view. The principal practical outcome is a new formula for the nonlinear coefficient in the normal form which generates the homoclinic orbit. The new formula is based on the intrinsic curvature of the lifted path of equilibria.

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