Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbits in MATLAB

Witte, Virginie De; Govaerts, W.; Kuznetsov, Yu. А.; Friedman, Mark J.

doi:10.1145/2168773.2168776

Cited by 30 publications

(36 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can fix the phase using an integral condition as in [Doedel & Kernévez, 1986;Champneys et al, 1996] and [De Witte et al, 2012]. Namely, we can fix the phase of u(s) by requiring its minimal L 2 -distance to the Hamiltonian approximation u 0 (s).…”

Section: Computational Formulas For N-dimensional Systemsmentioning

confidence: 99%

“…In this system we have two equilibria ± − 1 a β 1 , 0 , which lie on the folded surface S = {(w 0 , w 1 , β 1 , β 2 ) = (s, 0, −as 2 , β 2 ) : s, β 2 ∈ R}. We want to compare the predicted homoclinic solution with that obtained via the high-accuracy numerical continuation in MatCont [De Witte et al, 2012].…”

Section: Truncated Normal Formmentioning

confidence: 99%

“…There are standard methods to continue fold (limit point) and Hopf bifurcations of equilibria of ODEs in two parameters [Govaerts, 2000], as well as techniques for the two-parameter continuation of saddle homoclinic orbits given a suitable initial point [Champneys et al, 1996;De Witte et al, 2012]. One needs, therefore, a special predictor method to initialize the continuation of homoclinic solutions from a BT point.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

Kouznetsov

Meijer

Al-Hdaibat

et al. 2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first-and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.

show abstract

Section: Computational Formulas For N-dimensional Systemsmentioning

confidence: 99%

Section: Truncated Normal Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

Kouznetsov

Meijer

Al-Hdaibat

et al. 2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

“…In general, to continue the homoclinic orbits in two free parameters, MatCont uses an extended defining system that consists of several components (see [24,18]). [17,27].…”

Section: Practical Initialization Issuesmentioning

confidence: 99%

“…The projection boundary conditions place the solution at the two end points in the unstable and stable eigenspaces of A(x 0 , α), respectively [9]. MatCont uses a specific algorithm that depends on the real Schur factorization and a Riccati equation to construct and update Q S and Q U ; for more details, see [22,19,23,24,7,18].…”

Section: Practical Initialization Issuesmentioning

confidence: 99%

Initialization of Homoclinic Solutions near Bogdanov--Takens Points: Lindstedt--Poincaré Compared with Regular Perturbation Method

Al-Hdaibat¹,

Govaerts²,

Kuznetsov³

et al. 2016

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

Abstract. To continue a branch of homoclinic solutions starting from a Bogdanov-Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt-Poincaré (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an n-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a "parasitic turn" near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the MATLAB continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Two examples with multidimensional state spaces are included.

show abstract

Bifurcation set for a disregarded Bogdanov-Takens unfolding: Application to 3D cubic memristor oscillators

2021

View full text Add to dashboard Cite

We derive the bifurcation set for a not previously considered threeparametric Bogdanov-Takens unfolding, showing that it is possible express its vector field as two different perturbed cubic Hamiltonians. By using several firstorder Melnikov functions, we obtain for the first time analytical approximations for the bifurcation curves corresponding to homoclinic and heteroclinic connections, which along with the curves associated to local bifurcations organize the parametric regions with different structures of periodic orbits.As an application of these results, we study a family of 3D memristor oscillators, for which the characteristic function of the memristor is a cubic polynomial. We show that these systems have an infinity number of invariant manifolds, and by adding one parameter that stratifies the 3D dynamics of the family, it is shown that the dynamics in each stratum is topologically equivalent to a representant of the above unfolding. Also, based upon the bifurcation set obtained, we show the existence of closed surfaces in the 3D state space which are foliated by periodic orbits. Finally, we clarify some misconceptions that arise from the numerical simulations of these systems, emphasizing the important role played by the existence of invariant manifolds.

show abstract

Interactive Initialization and Continuation of Homoclinic and Heteroclinic Orbits in MATLAB

Cited by 30 publications

References 20 publications

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

Improved Homoclinic Predictor for Bogdanov–Takens Bifurcation

Initialization of Homoclinic Solutions near Bogdanov--Takens Points: Lindstedt--Poincaré Compared with Regular Perturbation Method

Bifurcation set for a disregarded Bogdanov-Takens unfolding: Application to 3D cubic memristor oscillators

Contact Info

Product

Resources

About