A dynamic study of a bead sliding on a wire in fractal space with the non-perturbative technique

Drawing on the principles of fractal properties and nonlinear vibration analysis, this paper delves into the investigation of a moving bead on a vertically rotated parabola. The dynamical nonlinear equation of motion, incorporating fractal derivatives, transforms traditional derivatives within continuous space. Consequently, the equation of motion takes the form of the Duffing-Van der Pol oscillator. Utilizing a non-perturbative approach, the nonlinear oscillator is systematically transformed into a linear one, boasting an exact solution. The analytical solution yields two valid formulas governing the frequency-amplitude relationships. Numerical solutions affirm that these proposed formulas offer highly satisfactory approximations to the analytical solution. Leveraging fractal properties through Galerkin’s method, the paper successfully determines the fractalness parameter of the medium, shedding light on the intricate dynamics of the system.