Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1 : 2 internal resonance

The nonlinear response of a two-degree-of-freedom nonlinear oscillating sys tem to parametric excitation is examined for the care of 1 : 2 internal res onance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and p hases. The steady-state solutions of the modulated equations and their stab ility are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled made solutions. The Melnikov m ethod is used to study the global bifurcation behavior, the critical parame ter is determined at which the dynamical system possesses a Smale horseshoe type of chaos.