Mega-Instability: Order Effect on the Fractional Order of Periodically Forced Oscillators - Zainab Dheyaa Ridha; Ali A. Shukur (Fractal and Fractional)
The stability of differential equations is one of the most important aspects to consider in dynamical system theory. Chaotic systems were classified according to stability as multi-stable systems; systems with a single stable equilibrium; bi-stable systems; and, recently, mega-stable systems. Mega-stability refers to the infinity countable nested attractors of a periodically forced non-autonomous system. Many researchers attempted to present a simple mega-stable system. In this paper, we investigated the mega-stability of periodically damped non-autonomous differential systems with the following different order cases: integer and fractional. In the case of the integer order, we generalize the mega-stable system, such that the velocity is multiplied by a trigonometrical polynomial, and we present the necessary and sufficient conditions to generated countable infinity nested attractors. In the case of the fractional order, we obtained that the fractional order of periodically damped non-autonomous differential systems has infinity countable nested unstable attractors for some orders. The mega-instability was illustrated for two examples, showing the order effect on the trajectories. In addition, and to further recent work presenting simple high dimensional mega-stable chaotic systems, we introduce a 4D mega-stable hyperchaotic system, examining chaotic and hyperchaotic behaviors through Lyapunov exponents and bifurcation diagrams.