Coexisting attractors, crisis route to chaos in a novel 4D fractional-order system and variable-order circuit implementation

In this paper, a novel 4D fractional-order chaotic system is proposed, and the corresponding dynamics are systematically investigated by considering both fractional-order and traditional system parameters as bifurcation parameters. When varying the traditional system parameters, this system exhibits some conspicuous characteristics. For example, four separate single-wing chaotic attractors coexist, and they will pairwise combine, resulting in a pair of double-wing attractors. More distinctively, by choosing the specific control parameters, transitions from a four-wing attractor to a pair of double-wing attractors to four coexisting single-wing attractors are observed, which means that the novel fractional-order system experiences an unusual and striking double-dip symmetry recovering crisis. However, numerous studies have shown that the fractional differential order has an important effect on the dynamical behavior of a fractional-order system. However, these studies are based only on numerical simulations. Thus, the design of a variable fractional-order circuit to investigate the influence of the order on the dynamical behavior of the fractional-order chaotic circuit is urgently needed. Varying with the order, coexisting period-doubling bifurcation modes appear, which suggests that the orbits have transitions from a coexisting periodic state to a coexisting chaotic state. A variable fractional-order circuit is designed, and the experimental observations are found to be in good agreement with the numerical simulations.