The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hidden attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lya-

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We study the dynamics of nonlinear oscillators under mean-field diffusive coupling. We observe that this form of coupling leads to amplitude death via a synchronization transition in the parameter space of the coupling strength and mean-field control parameter. A general criterion for amplitude death for any given dynamical system with mean-field diffusion is obtained, and these dynamical transitions are characterized using various indices such as average phase difference, Lyapunov exponents, and average amplitude. This behavior is analyzed in the parameter plane by numerical studies of specific cases of the Landau-Stuart limit-cycle oscillator, and Rössler, Lorenz, FitzHugh-Nagumo excitable, and Chua systems.

We propose a general strategy to stabilize the fixed points of nonlinear oscillators with augmented dynamics. By using this scheme, either the unstable fixed points of the oscillatory system or a new fixed point of the augmented system can be stabilized. The Lyapunov exponents are used to study the dynamical properties. This scheme is illustrated with a chaotic Lorenz oscillator coupled through an external linear dynamical system. The experimental demonstration of the proposed scheme to stabilize the fixed points is also presented.

Amplitude death (AD) in hidden attractors is attained with a scheme of linear augmentation. This linear control scheme is capable of stabilizing the system to a fixed point state even when the original system does not have any fixed point. Depending on the control parameter, different routes to AD such as boundary crises and Hopf bifurcation are observed. Lyapunov exponent and amplitude index are used to study the dynamical properties of the system.