Mathematical model of Van der Pol-Airy fractional oscillator

The paper proposes a mathematical model of the nonlinear Van der Pol-Airy oscillator taking into account heredity. The nonlinearity of the oscillator is due to the dependence of the friction coefficient on the square of the displacement function, which is typical for the Van der Pol oscillator. Also, the natural frequency of oscillations is a function of time, which increases linearly as it increases. The latter is typical for the Airy oscillator. Heredity effects are introduced into the model equation through fractional derivatives in the Gerasimov-Caputo sense. They indicate that the oscillatory system may have memory effects that manifest themselves depending on its current state from previous ones. For the proposed mathematical model, a numerical algorithm was developed based on an explicit first-order finite-difference scheme. The numerical algorithm was implemented in a computer program in the Maple language, with the help of which the simulation results were visualized. Oscillograms and phase trajectories were constructed for various values of the model parameters. It is shown that a fractional mathematical model can have various oscillatory modes: from self-oscillatory, damped and chaotic. An interpretation of the simulation results is given