The non-conservative forced Toda oscillator

The vibration theory has become a hot topic in both mathematics and engineering. This work is an attempt to reduce the complexity of the Toda model and to exploit the advantages of annihilator operators. The reducing rank method is proposed for the outcoming jerk oscillator model. The modified homotopy perturbation method (HPM) is presented to solve the resulting model. Besides, to simplify the model, the outstanding features of the resonance reaction are preserved. It is declared that the obtained results are robust as they can exist in a wide range of oscillators. It is found that the amplitude of oscillation greatly affects its periodic nature and has an exponential form. The dynamics of a non-autonomous oscillator in which the frequency of the external force depends on the dynamical variable is studied. It is also revealed that the linear damping coefficient plays a dual role in the stability behavior, while the nonlinear counterpart exerts a destabilizing influence on the resonance response.