Abstract
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.
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