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We show that a reservoir computer is an effective tool for model-free prediction of extreme events in deterministic chaotic systems. This prediction allows us to suppress unwanted extreme events, by applying weak control perturbations to the system at times preceding expected extreme events. The effectiveness of such a prediction and prevention strategy is demonstrated for a system of globally coupled FitzHugh-Nagumo neurons and for a system of two almost identical unidirectionally coupled chaotic oscillators.

The dynamics behaviors of coupled Saito's oscillators are investigated intensively. Various phenomena are explored such as antisynchronization (AS), hysteresis, bistability, riddled basin and coexistence of chaotic AS and lag AS. With increasing coupling intensity, the coupled chaotic oscillators undergo a transition from phase synchronization to AS. With decreasing coupling intensity, they will transit from AS to antiphase synchronization. The necessary condition for AS is explored and the stability of AS is studied.

We developed the control technique for nonlinear oscillators when simple feedback alters the oscillation energy. The control does not require any computation nor knowledge of the system equations. Depending on the control perturbation parameters, two scenarios take place: (i) changing (linearly or nonlinearly) the oscillator damping, and (ii) suppressing (enhancing) the driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks is considered.

We have observed anti-synchronization phenomena in coupled identical chaotic oscillators. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. We have qualitatively analyzed its base mechanism by using the dynamics of the difference and the sum of the relevant variables in coupled chaotic oscillators. Near the threshold of the synchronization and anti-synchronization transition, we have obtained the novel characteristic relation.

This work provides a methodology for the control of complex non-linear systems by compensating for the non-linear part using support vector machines (SVM) and subsequently developing simple linear feedback law for control. The method tested for the benchmark Rossler and Lorenz chaotic oscillators shows excellent performance.

The synchronization transition is described for a system of two asymmetrically coupled chaotic oscillators. Such a system can represent the two-cluster state in a large ensemble of globally coupled oscillators. It is shown that the transition can be typically mediated by a transcritical transversal bifurcation. The latter has a hard brunch that dominates the global dynamics, so that the synchronization transition is normally hard. For a particular example of coupled logistic maps a diversity of transition scenaria includes both local and global riddling. In the case of small non-identity of the interacting systems the riddling is shown to turn into an exterior or interior crisis.

A nonlinear adaptive control strategy for a class of chaotic oscillators is proposed. The class of chaotic systems include both externally and parametrically excited systems, such as the Duffing and the Van de Pol oscillators. By assuming that the exact model of the system is not known, the controller employs a nonlinear observer to track the states and disturbances of the nonlinear system, and constructs a state-error-like feedback law. Simulations are provided to illustrate the effectiveness of the controller.

The synchronization of two different chaotic oscillators is studied. First, this problem is understood in terms of chaos suppression. Next, a robust asymptotic controller is used in the chaos control. The robust controller comprises an uncertainty estimator and a linearizing control law. In this way, synchronization is obtained despite the master model being strictly different from the slave model. In fact, the controller does not know the master oscillator model.

A time delay coordinates (TDC) strategy to control a class of chaotic oscillators is studied. The class of chaotic systems includes both externally and parametrically excited systems, such as the Duffing and the Coulombic oscillators. By assuming that the exact model of the system is not known and that position is the only state available for measurements, the controller comprises a linearizing-like feedback and an uncertainty estimator. Simulations are provided to illustrate the performance of the controller.