Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

Direct approach to detect the heteroclinic bifurcation of the planar nonlinear system

“…Mikhlin [18] and Feng et al [13] used the Padé and quasi-Padé approximation to construct these types of orbits. Other methods, such as the Homotopy perturbation [14], perturbation-incremental [6], hyperbolic perturbation [7,9], elliptic Lindstedt-Poincaré (LP) [3] and hyperbolic LP [8] methods and so on, have also been used to determine the homoclinic and heteroclinic solutions of both weakly and strongly nonlinear oscillators. Chen et al [9] used the hyperbolic perturbation method, involving a new time scale in the hyperbolic function, to derive the expressions of homoclinic orbits, where the systems were characterized by quadratic, cubic, quartic, and strong nonlinearity.…”

“…According to Chen et al [9], in a preliminary computation, ω 10 equals ω 0 √ 2 in the case of zero perturbation (ε = 0). That produces the global collocation point for heteroclinic bifurcation φ 3 = π n , n = ±3.564.…”

“…In order to improve the accuracy, we use the perturbed solution in Eq. (3) to substitute the unperturbed solution in Eq (9).…”

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

“…The Lindstedt-Poincaré method. Now we apply the hyperbolic perturbation method from [10,13,14,15] to improve the prediction of the homoclinic orbit in the phase space. This method is a generalized L-P method combined with hyperbolic functions instead of the Jacobian elliptic functions used in [12,4].…”

“…Chen et al [10,14,13,11,15] used a generalization of the Lindstedt-Poincaré (L-P) method to study the homoclinic solution to a family of nonlinear oscillators. They applied a nonlinear transformation of time instead of the linear one used in the original L-P method for periodic solutions (see, e.g., [33] for the original L-P method).…”