Mathematical modeling of oscillator hereditarity

The paper considers hereditarity oscillator which is characterized by oscillation equation with derivatives of fractional order $\beta$ and $\gamma$, which are defined in terms of Gerasimova-Caputo. Using Laplace transform were obtained analytical solutions and the Green's function, which are determined through special functions of Mittag-Leffler and Wright generalized function. It is proved that for fixed values of $\beta$ = 2 and $\gamma$ = 1, the solution found becomes the classical solution for a harmonic oscillator. According to the obtained solutions were built calculated curves and the phase trajectories hereditarity oscillatory process. It was found that in the case of an external periodic influence on hereditarity oscillator may occur effects inherent in classical nonlinear oscillators.