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

Algaba, Antonio; Chung, Kwok Wai; Qin, Bo-Wei; Rodríguez-Luis, Alejandro J.

doi:10.1007/s11071-019-05025-2

Cited by 15 publications

(12 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This removes the so-called parasitic turns near the saddle point, as observed in [40]. Although, as pointed out by [2], there were mistakes in the third-order approximation with the Lindstedt-Poincaré method, the homoclinic predictor from [40] for the smooth normal form improved significantly in the phase space.…”

Section: Introductionmentioning

confidence: 88%

“…The nonlinear transformation (59) can then be used to remove the parasitic turns. In fact, using the nonlinear transformation, one can obtain a very simple form for the solution of the homoclinic orbit in phase-space, see [12,Equation 35] and [2].…”

Section: A Polynomial Lindstedt-poincaré Methodsmentioning

confidence: 99%

“…It is well known that in system (8) three codimension one bifurcation curves emanate from (β 1 , β 2 ) = (0, 0): a saddle-node, a Hopf, and a saddle-homoclinic bifurcation curve. By using either the regular perturbation or the Lindstedt-Poincaré method, an approximation to the homoclinic bifurcation curve and the corresponding solution can, theoretically, be obtained up to any order in the singular-rescaling parameter , see [40,41,1,2].…”

Section: Parameter-dependent Normal Formmentioning

confidence: 99%

“…In [2] a different approach is used to approximate the homoclinic solution near the quadratic normal form of a generic codimension 2 Bogdanov-Takens bifurcation. The approach there is an application of the so-called Perturbation-Incremental Method described in [47].…”

Section: Comparison With the Nonlinear Periodic Time-reparametrizationmentioning

confidence: 99%

“…In Section 3.2 the generalized Lindstedt-Poincaré method for the approximation of homoclinic orbits is improved by introducing an additional transformation of time after applying the usual nonlinear time transformation. The resulting algorithm solitary relies on polynomial division and does not involve any hyperbolic or trigonometric functions as in [2,1]. We show that for the quadratic Bogdanov-Takens normal form, we can represent the homoclinic solution in phase-space with only one single parameter.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Interplay between normal forms and center manifold reduction for homoclinic predictors near Bogdanov-Takens bifurcation

Bosschaert,

Kuznetsov

2021

Preprint

View full text Add to dashboard Cite

This paper provides for the first time correct third-order homoclinic predictors in ndimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincaré) to approximate the homoclinic solution near Bogdanov-Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.

show abstract

Section: Introductionmentioning

confidence: 88%

Section: A Polynomial Lindstedt-poincaré Methodsmentioning

confidence: 99%