Parrondo's paradox for chaos control and anticontrol of fractional-order systems

We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems. The generalization is implemented by applying a parameter switching (PS) algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N ≥ 2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words "winning" and "loosing" in the classical Parrondo's paradox with "order" and "chaos, respectively, the PS algorithm leads to the generalized Parrondo's paradox: chaos₁ + chaos₂ + ••• + chaos_N = order and order₁ + order₂ + ••• + order_N = chaos. Finally, the concept is well demonstrated with the results based on the fractional-order Chen system.