Chaotic modes of a non-linear fractional oscillator

A non-linear fractional oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractional oscillator can be mathematically described using integro-differential equations with difference kernels or fractional order derivatives. In this paper, a fractional non-linear oscillator has been investigated to identify chaotic oscillatory modes. The quantitative measure of chaotic regimes is the largest (maximal) Lyapunov exponents. For calculating the maximal Lyapunov exponents, the Wolff algorithm based on the Gram-Schmidt orthogonalization procedure has been selected, using both numerical solution for an initial fractional dynamical system by using variational equations. The Wolff algorithm also makes it possible to plot the spectrum of Lyapunov exponents as a function of control parameters for the initial dynamical system. It has been shown that some spectra of Lyapunov exponents contain positive values indicating the existence of chaotic modes, which are also confirmed by the corresponding phase trajectories.