![]() The important point here is that, because of chaos, one is able to produce an infinite number of desired dynamical behaviors (either periodic and not periodic) using the same chaotic system, with the only help of tiny perturbations chosen properly. Indeed, if it is true that a small perturbation can give rise to a very large response in the course of time, it is also true that a judicious choice of such a perturbation can direct the trajectory to wherever one wants in the attractor, and to produce a series of desired dynamical states. This fact has suggested the idea that the critical sensitivity of a chaotic system to changes (perturbations) in its initial conditions may be, in fact, very desirable in practical experimental situations. If one switches on the stabilizing perturbations, the trajectory moves to the neighborhood of the desired periodic orbit that can now be stabilized. The idea of controlling chaos is then when a trajectory approaches ergodically a desired periodic orbit embedded in the attractor, one applies small perturbations to stabilize such an orbit. Secondly, the dynamics in the chaotic attractor is ergodic, which implies that during its temporal evolution the system ergodically visits small neighborhood of every point in each one of the unstable periodic orbits embedded within the chaotic attractor.Ī relevant consequence of these properties is that a chaotic dynamics can be seen as shadowing some periodic behavior at a given time, and erratically jumping from one to another periodic orbit. In other words, the skeleton of a chaotic attractor is a collection of an infinite number of periodic orbits, each one being unstable. Firstly, there is an infinite number of unstable periodic orbits embedded in the underlying chaotic set. ![]() Besides their critical sensitivity to initial conditions, chaotic systems exhibit two other important properties. Indeed, the prediction trajectory emerging from a bonafide initial condition and the real trajectory emerging from the real initial condition diverge exponentially in course of time, so that the error in the prediction (the distance between prediction and real trajectories) grows exponentially in time, until making the system's real trajectory completely different from the predicted one at long times.įor many years, this feature made chaos undesirable, and most experimentalists considered such characteristic as something to be strongly avoided. They can be found in meteorology, solar system, heart and brain of living organisms and so on.ĭue to their critical dependence on the initial conditions, and due to the fact that, in general, experimental initial conditions are never known perfectly, these systems are instrinsically unpredictable. Many natural phenomena can also be characterized as being chaotic. They can be found, for example, in Chemistry (Belouzov–Zhabotinski reaction), in Nonlinear Optics (lasers), in Electronics (Chua–Matsumoto circuit), in Fluid Dynamics (Rayleigh–Bénard convection), etc. However, only in the last thirty years, experimental observations have pointed out that, in fact, chaotic systems are common in nature. The fact that some dynamical model systems showing the above necessary conditions possess such a critical dependence on the initial conditions was known since the end of the last century. The necessary requirements for a deterministic system to be chaotic are that the system must be nonlinear, and be at least three dimensional. ![]() This property implies that two trajectories emerging from two different closeby initial conditions separate exponentially in the course of time. The book is a timely and comprehensive reference guide for graduate students, researchers, and practitioners in the areas of chaos theory and intelligent control.A deterministic system is said to be chaotic whenever its evolution sensitively depends on the initial conditions. Not only does it provide the readers with chaos fundamentals and intelligent control-based algorithms it also discusses key applications of chaos as well as multidisciplinary solutions developed via intelligent control. ![]() The book furthercovers fuzzy logic controllers, evolutionary algorithms, swarm intelligence, and petri nets among other topics. Topics include fractional chaotic systems, chaos control, chaos synchronization, memristors, jerk circuits, chaotic systems with hidden attractors, mechanical and biological chaos, and circuit realization of chaotic systems. Written by eminent scientists and active researchers andusing a clear, matter-of-fact style, it covers advanced theories, methods, and applications in a varietyof research areas, and explains key concepts in modeling, analysis, and control of chaotic and hyperchaotic systems. ![]() The book reports on the latest advances in and applications of chaos theory and intelligent control. ![]()
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