Control of Chaotic and Hyperchaotic Systems

Since the seminal work of Ott, Grebogy, and York [43], who demonstrated the ability to stabilize Unstable Periodic Orbits (UPOs) among those embedded in a chaotic attractor, control of chaotic systems has become an extensive field of research. ¹ Such an interest is, indeed, motivated by many reasons. For instance, by essence, chaotic systems are highly sensitive to 292(tiny) variations of system parameters, initial conditions, and external disturbances. Therefore, in some processes, chaos may lead to harmful or even catastrophic situations ² if not subdued. In such cases, the main purpose then is to reduce the chaotic phenomena as much as possible, by means of intentional and suitable control signals applied to the system. As another motivation, in some practical contexts, bringing of chaotic dynamics into a process or exploiting the chaotic nature of a system may efficiently avoid some costly and painful tasks. ³ For instance, as pointed out by Ott et al. [43], a chaotic attractor is composed of a dense set of UPOs. Then, a key idea is to associate some of these orbits with different tasks to perform. ⁴ Thus, instead of designing, making, and using several devices, the same chaotic system can serve multiple purposes by simply switching, in a controlled manner, among the different orbits of interest. In such a case, the use of chaos then involves stabilizing trajectories of interest, ⁵ while preserving, as much as possible, some of the inherent properties of chaotic (possibly hyperchaotic ⁶ ) systems. Another motivation to deal with chaos control is that introduction of particular control laws or modification of the experimental conditions may lead some nonchaotic systems (initially) to perform undesirable chaotic behaviors (i.e., some transitions from order to chaos, sometimes referred to as “chaotification” [13]). Recovery of original properties such as functionality, stability, … then implies both design and application of suitable control laws (e.g., see in Gills et al. [25], the “green problem” related to laser systems). Finally, owing to their intrinsic properties (such as sensitive dependence on initial conditions, inherent instability, sensitivity to perturbations and disturbances, etc.) the control of chaos appears to be an interesting challenge to achieve high performance near the stability boundaries.