Dynamic analysis of the fractal nonlinear oscillators with coordinate-dependent mass

The dynamic behavior of nonlinear oscillators can be researched more accurately in the micro-scale. In this paper, a modified nonlinear oscillator with coordinate-dependent mass by He’s fractal derivative is first given. Then, the energy balance method and modified harmonic balance method are utilized to constructed the first-order and second-order approximate solutions of the fractal model. Next, two sets of parameters are choosen, the obtained numerical solution are compared with the Runge-kuta (RK) solution, and the results demonstrate that the second-order approximate solution is more accurate. In addition, by comparing the solutions under the different fractal dimensions, one can be found that the fractal dimension does not change global properties of the oscillators, but the vibration behaviors gradually accelerates with the increase of the fractal dimension, which means that we can study the oscillation behavior more clearly in the micro-scale.