Exact solutions for the quadratic mixed-parity Helmholtz-Duffing oscillator by bifurcation theory of dynamical systems

The dynamical behavior and exact solutions of the quadratic mixed-parity Helmholtz-Duffing oscillator are studied by using bifurcation theory of dynamical systems. As a result, all possible phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded solutions (soliton solutions, kink and anti-kink solutions and periodic solutions) are given. When parameters are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given. (C) 2015 Elsevier Ltd. All rights reserved.