1333 Words6 Pages

the dynamical system into a regular stationary process produces such a sequence, then the dynamical system is called a Bernoulli system.
1.6.1 Deterministic Chaos
Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. Chaotic behavior can be observed in many natural*…show more content…*

There is some controversy over the existence of chaotic dynamics in tectonic plates and in economics [6]. One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population growth under density dependence can lead to chaotic dynamics. Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions. A related field of physics called quantum chaos theory investigates the relationship between chaos and quantum mechanics. The correspondence principle states that classical mechanics is a special case of quantum mechanics, the classical limit. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos. Recently, another field, called relativistic chaos*…show more content…*

This research activity focuses on the developments of theoretical foundations for chaos synchronization based on the experimental studies of synchronization phenomenon in physical, neurobiological and other systems. 1.8.2 Synchronization of Chaotic systems Synchronization of chaotic oscillators is one of the most important achievements of nonlin- ear dynamics within the last few decades was the discovery of complex, chaotic motion in rather simple oscillators. Now this phenomenon is well studied and is a subject of under- graduate and high-school courses; nevertheless some introductory presentation is pertinent. The term “chaotic” means that the long term behavior of a dynamical system cannot be predicted even if there were no natural fluctuations of the systems parameters or influence of a noisy environment. Irregularity and unpredictability result from the internal deter- ministic dynamics of the system, however contradictory this may sound. If we describe the oscillation of dissipative, self-sustained chaotic systems in the phase space, then we find that it does not correspond to such simple geometrical objects like a limit cycle any more, but rather to complex structures that are called strange attractors [8,

There is some controversy over the existence of chaotic dynamics in tectonic plates and in economics [6]. One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population growth under density dependence can lead to chaotic dynamics. Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions. A related field of physics called quantum chaos theory investigates the relationship between chaos and quantum mechanics. The correspondence principle states that classical mechanics is a special case of quantum mechanics, the classical limit. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos. Recently, another field, called relativistic chaos

This research activity focuses on the developments of theoretical foundations for chaos synchronization based on the experimental studies of synchronization phenomenon in physical, neurobiological and other systems. 1.8.2 Synchronization of Chaotic systems Synchronization of chaotic oscillators is one of the most important achievements of nonlin- ear dynamics within the last few decades was the discovery of complex, chaotic motion in rather simple oscillators. Now this phenomenon is well studied and is a subject of under- graduate and high-school courses; nevertheless some introductory presentation is pertinent. The term “chaotic” means that the long term behavior of a dynamical system cannot be predicted even if there were no natural fluctuations of the systems parameters or influence of a noisy environment. Irregularity and unpredictability result from the internal deter- ministic dynamics of the system, however contradictory this may sound. If we describe the oscillation of dissipative, self-sustained chaotic systems in the phase space, then we find that it does not correspond to such simple geometrical objects like a limit cycle any more, but rather to complex structures that are called strange attractors [8,

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