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Retrieval of episodic memory is a dynamical process in the large scale brain networks. In social groups, the neural patterns, associated to specific events directly experienced by single members, are encoded, recalled and shared by all participants. Here we construct and study the dynamical model for the formation and maintaining of episodic memory in small ensembles of interacting brains. We prove that the unconventional dynamical attractor of this process - the nonsmooth heteroclinic torus - is structurally stable within the Lotka-Volterra-like sets of equations. Dynamics on this torus combines absence of chaos with asymptotic instability of every separate trajectory; its adequate quantitative characteristics are length-related Lyapunov exponents. Variation of the coupling strength between the participants results in different types of sequential switching between metastable states; we interpret them as stages in formation and modification of the episodic memory., Comment: 16 pages, 14 figures

A phenomenon of weak transient chaos is discussed that is caused by sub-exponential divergence of trajectories in the basin of a non-chaotic attractor. Such a regime is not easy to detect, because conventional characteristics, such as the largest Lyapunov exponent is non-positive. Here we study, how such a divergence can be exposed and detected. First, we show that weak transient chaos can be exposed if a small random perturbation is added to the system, leading to positive values of the largest Lyapunov exponent. Second, we introduce an alternative definition of the Lyapunov exponent, which allows us to detect weak transient chaos in the deterministic unperturbed system. We show that this novel characteristic becomes positive, reflecting transient chaos. We demonstrate this phenomenon and its detection using a master-slave system where the master possesses a heteroclinic cycle attractor, while the slave is the Van-der-Pol-Duffing oscillator possessing a stable limit cycle.

We study a simple dynamical model exhibiting sequential dynamics. We show that in this model there exist sets of parameter values for which a cyclic chain of saddle equilibria, $O_k$, $k=1, \ldots, p$, have two dimensional unstable manifolds that contain orbits connecting each $O_k$ to the next two equilibrium points $O_{k+1}$ and $O_{k+2}$ in the chain ($O_{p+1} = O_1$). We show that the union of these equilibria and their unstable manifolds form a $2$-dimensional surface with boundary that is homeomorphic to a cylinder if $p$ is even and a M\"{o}bius strip if $p$ is odd. If, further, each equilibrium in the chain satisfies a condition called ``dissipativity," then this surface is asymptotically stable.

The (e,n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval n. Behavior of the (e, n)-complexity functions as n goes to infinity is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of (e,n)-separated sets. We study such measures. In particular, we prove that they are invariant if the (e,n)-complexity function grows subexponentially. Keywords: Topological entropy, complexity functions, separability.

The ability of nonlinear dynamical systems to process incoming information is a key problem of many fundamental and applied sciences. Information processing by computation with attractors (steady states, limit cycles and strange attractors) has been a subject of many publications. In this paper we discuss a new direction in information dynamics based on neurophysiological experiments that can be applied for the explanation and prediction of many phenomena in living biological systems and for the design of new paradigms in neural computation. This new concept is the Winnerless Competition (WLC) principle. The main point of this principle is the transformation of the incoming identity or spatial inputs into identity-temporal output based on the intrinsic switching dynamics of the neural system. In the presence of stimuli the sequence of the switching, whose geometrical image in the phase space is a heteroclinic contour, uniquely depends on the incoming information. The key problem in the realization of the WLC principle is the robustness against noise and, simultaneously, the sensitivity of the switching to the incoming input. In this paper we prove two theorems about the stability of the sequential switching and give several examples of WLC networks that illustrate the coexistence of sensitivity and robustness., Comment: 31 pages, 8 figures, in press for International Journal of Bifurcation and Chaos (2004) Vol. 14 (4)

Generalized synchronization of chaos is a type of cooperative behavior in directionally-coupled oscillators that is characterized by existence of stable and persistent functional dependence of response trajectories from the chaotic trajectory of driving oscillator. In many practical cases this function is non-differentiable and has a very complex shape. The generalized synchrony in such cases seems to be undetectable, and only the cases, in which a differentiable synchronization function exists, are considered to make sense in practice. We show that this viewpoint is not always correct and the non-differentiable generalized synchrony can be revealed in many practical cases. Conditions for detection of generalized synchrony are derived analytically, and illustrated numerically with a simple example of non-differentiable generalized synchronization., Comment: 8 pages, 8 figures, submitted to PRE