Mathematical model of a fractional nonlinear Mathieu oscillator

The work studies the fractional nonlinear Mathieu oscillator using numerical analysis methods in order to establish its various oscillatory modes. Mathieu's fractional nonlinear oscillator is an ordinary nonlinear differential equation with fractional derivatives in the Gerasimov-Caputo sense and local initial conditions (Cauchy problem). Gerasimov-Caputo fractional derivatives characterize the presence of the heredity effect in an oscillatory system. In such a system, its current state depends on the previous history. To study the Cauchy problem, a numerical method from the predictor-corrector family was used – the Adams-Bashforth-Moulton method, the algorithm of which was implemented in the Matlab computer mathematics system. Using a numerical algorithm, oscillograms and phase trajectories were constructed for various values of the parameters of the Mathieu fractional nonlinear oscillator. It is shown that in the absence of an external periodic influence, self-oscillations can arise in the oscillatory system under consideration, which are characterized by limit cycles on the phase trajectory. A study of limit cycles was carried out using computer simulation. It has been shown that aperiodic regimes can also arise, i.e. modes that are not oscillatory. Therefore, the orders of fractional derivatives can be influenced by the oscillatory mode of a nonlinear fractional Mathieu oscillator: from oscillations with a constant amplitude to damped ones and disappearing completely.