Chaos behavior can be observed in a diverse variety of real phenomena. The attractors (n-scrolls, multi-scroll, or infinitely many scrolls) generated are classified as: Hidden and Self-Excited
Since Leonov et al. published their seminal paper, the attractors in dynamical systems have been categorized as self-excited attractors and hidden attractors. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, the most of common examples of integer-order chaotic flows showing self-excited attractors are Lorenz, Chen, Rössler, and Lü systems, among many others. Conversely, an attractor is called hidden if its basin of attraction does not intersect with small neighborhoods of the unstable equilibrium. Additionally, the attractors in dynamical systems with no-equilibrium, with curves and surfaces of equilibria, and with stable equilibria also belong to the category of hidden attractors.
In this research, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained.
Hidden attractors are very important in engineering applications because they allow the study and understanding of the unexpected and potentially disastrous responses of the dynamical systems to perturbations, for instance, in mechanical structures, like a bridge or airplane wings, aircraft control systems, PLL circuits, drilling systems with induction motors, and secure communication schemes. Hence, numerous integer-order chaotic flows with hidden attractors have been proposed.
However, it should be noted that most of the studies about hidden attractors have mainly concentrated on continuous-time dynamical systems of integer-order. In recent years, fractional calculus has received much attention due to fractional derivatives providing more accurate models than their integer-order counterparts.
Therefore, the research effort oriented to hidden attractors in fractional-order dynamical systems is vital to understand this exciting and still less-explored subject of importance. In the literature, few works have reported hidden attractors in fractional-order dynamical systems with one stable equilibrium, with no-equilibria, with a line or surfaces of equilibria, or even in fractional-order hyperchaotic systems.
However, those fractional order systems generate only one family of hidden attractors, i.e., line, surface, stable, and without equilibrium. A remaining research question is whether fractional-order dynamical systems whose dynamics can generate both self-excited and hidden attractors could exist.
For details consult:
Munoz-Pacheco, J. M., Zambrano-Serrano, E., Volos, C., Jafari, S., Kengne, J., & Rajagopal, K. (2018). A new fractional-order chaotic system with different families of hidden and self-excited attractors. Entropy, 20(8), 564.