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.