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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. [ABSTRACT FROM AUTHOR]

The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically. [ABSTRACT FROM AUTHOR]