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The problems of synchronization and control of chaotic systems with uncertain parameters are considered as those of nonlinear adaptive control. To solve these problems for continuous-time systems the so called speed-gradient method is applied. As an example the problem of master-slave synchronization of the two forced Duffing's systems is considered.

In this paper, we examine the global dynamics of the complete Owen-Sherratt model describing the tumor-macrophage interactions. We show for this dynamics that there is a positively invariant polytope. We give upper and lower ultimate bounds for densities of cell populations involved in this model. Besides, we derive sufficient conditions under which each trajectory in tends to the mutant cells-free equilibrium point or to the equilibrium point of macrophages in isolation or to the coordinate plane corresponding to the absence of normal tissue cells depending on initial conditions. The biological sense of our results is discussed as well. [ABSTRACT FROM AUTHOR]

When synchronization sets in, coupled systems oscillate in a coherent way. It is possible to observe also some intermediate regimes characterized by incomplete synchrony which are referred to as partial synchronization. The paper focuses on analysis of partial synchronization in networks of linearly coupled oscillators. [ABSTRACT FROM AUTHOR]

Synchronization in identical drive-response systems is a problem that can be cast in an observer design framework. In this paper we extend this approach by studying the analysis/design of partial synchronization by means of observer theory. In doing so, we introduce the concept of partial observer—an observer to reconstruct a part of the system state vector. It is also shown how the observability condition can be utilized to analyze the dynamics in an array of coupled identical systems. [ABSTRACT FROM AUTHOR]

In this Letter we analyze the concept of partial synchronization as observer design. Our aim is to estimate a signal, the function of the coordinates of a particular system that, when considering a specific output, is not necessarily fully observable. [ABSTRACT FROM AUTHOR]

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system's Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates. [ABSTRACT FROM AUTHOR]

This erratum corrects a mistake previously published by the author [A. Y. Pogromsky, Chaos 18, 037107 (2008)]. [ABSTRACT FROM AUTHOR]