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

Abstract

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.

Remarks on homoclinic structure in Bogdanov-Takens map

“…Such a predictor should combine correct asymptotics of the bifurcating homoclinic orbits in the normal form with the parameter-dependent reduction to the two-dimensional center manifold. The former would probably require to add some extra terms to the BT map (1), while the latter can easily be done using the homological equation technique applied in the ODE-case in [1,17].…”

“…The Bogdanov-Takens (BT) bifurcation plays an important role in the study of dynamical systems since it implies a global (homoclinic) bifurcation [2]. Improved asymptotics of the homoclinic bifurcation in a neighborhood of a BT bifurcation point for vector-fields have been obtained recently [1,17]. In the present paper we discuss the homoclinic structure in the two-parameter map G(u, ν) :…”

“…The dynamic behavior of (1) is different from that of the approximating system. In the approximating system, the parameters that correspond to the saddle homoclinic bifurcation form a curve, while a homoclinic zone bounded by two curves corresponding to primary homoclinic tangencies exists in (1), see Fig.1. If parameters (ν 1 , ν 2 ) are located inside the homoclinic zone, then the BT map possesses transversal homoclinic orbits.…”

“…The new asymptotic is derived by (a) considering all second-order terms w.r.t. coordinates and parameters in the approximating system; (b) using the accurate homoclinic asymptotic predictor from [1,17], instead of an incorrect asymptotic from [4] (which is also used in [19]) or a rough Melnikov approximation employed in [7,9,10].…”

Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

“…Matcont also uses orthogonal collocation for the discretization of periodic and homoclinic orbits. For homoclinic orbits, it also uses the Lindstedt-Poincaré method to achieve a better rendering of these orbits compared to the regular perturbation method [22,23]. Therefore, the analytical results developed here serve as the theoretical foundations, which is the mathematically solid part that gives support to what is calculated by numerical simulation software.…”