Stability of a Duffing oscillator with a position-dependent mass

This paper reports high-resolution stability diagrams classifying the different solutions of a driven Duffing oscillator with a position-dependent mass. The Duffing oscillator is a prototypical model to produce reference charts for experimentalists and to study stability phases normally present in nonlinear systems. The diagrams obtained reveal the size and organization of the oscillation phases present in the control plane defined by a mass index and the amplitude of the external drive. The range of values of the mass index and the force amplitude which were investigated display a variety of dynamical behaviors, as sequences of periodic orbits with number of spikes increasing by spike-adding and by spike-doubling routes, and spike-doubling routes ending in regions of chaotic dynamics. Chaotic situations reported in the literature are seen as particular cases of a complex scenario, which includes the occurrence of quint points, where five different stability phases meet. The phase organization is also investigated as a function of the angular frequency of the external force. The results show that the system is free of chaos for sufficiently small frequency of the driving force, and that chaotic regions increase in size and occur for higher values of the force amplitude, with the increase of the driving frequency.