Retrieval of episodic memory is a dynamical process in the large scale brain networks. In social groups, the neural patterns, associated with 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 minds. 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 the 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.

A method for studying the behavior of the elements of dynamical networks is introduced. We measure the amount of instability stored at each element according to the value of the mean complexity related to this element. Elements with close values of the mean complexity can be unified into complexity clusters; elements with the smallest values of complexities form dynamical hubs. The effectiveness of the method is manifested by its successive application to networks of coupled Lorenz systems. (C) 2017 Elsevier B.V. All rights reserved.

We continue to apply the notion of mean complexities to study dynamical networks. We show that the mean complexities can help to single out the nodes with similar features (and dynamical behavior) and to reveal some properties of the topology of the networks. We found that the nodes with the same degree (number of connections) have equal values of the mean complexities in the regime of full binding. At the same time, the mean complexities of nodes with different degree follow a descending order with respect to the degree.

In the article we obtain estimates for the amount of points that go trough a hole in the phase space of an Anosov system, provided that this hole is an element of a Markov partition. Moreover, we describe invariant sets that remain to stay out of the hole for all instants of time and estimate the survival probability. For algebraic automorphisms of torus these estimates become exact formulas.

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,., p, have two-dimensional unstable manifolds that contain orbits connecting each O-k to the next two equilibrium points Ok+1 and Ok+2 in the chain (Op+1 = O-1). We show that the union of these equilibria and their unstable manifolds form a two-dimensional surface with a boundary that is homeomorphic to a cylinder if p is even and a Mobius strip if p is odd. If, further, each equilibrium in the chain satisfies a condition called 'dissipativity', then this surface is asymptotically stable.

Temporal order memories are critical for everyday animal and human functioning. Experiments and our own experience show that the binding or association of various features of an event together and the maintaining of multimodality events in sequential order are the key components of any sequential memories-episodic, semantic, working, etc. We study a robustness of binding sequential dynamics based on our previously introduced model in the form of generalized Lotka-Volterra equations. In the phase space of the model, there exists a multi-dimensional binding heteroclinic network consisting of saddle equilibrium points and heteroclinic trajectories joining them. We prove here the robustness of the binding sequential dynamics, i.e., the feasibility phenomenon for coupled heteroclinic networks: for each collection of successive heteroclinic trajectories inside the unified networks, there is an open set of initial points such that the trajectory going through each of them follows the prescribed collection staying in a small neighborhood of it. We show also that the symbolic complexity function of the system restricted to this neighborhood is a polynomial of degree L - 1, where L is the number of modalities.

We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy;

This special issue presents a selection of papers from the conference “Dynamics, Bifurcations and Strange Attractors” dedicated to the memory of Leonid Pavlovich Shilnikov (1934–2011) to commemorate his contributions to the theory of dynamical systems and bifurcations. The conference was held at the Lobachevsky State University of Nizhny Novgorod, Russia, on 1–5 July 2013. The conference was attended by 155 participants from all over the world, who contributed to the three focal topics: bifurcations and strange attractors; dynamical systems with additional structures (Hamiltonian, time-reversible, etc.); applications of dynamical systems. The topics were chosen in confluence with pivotal contributions by L. P. Shilnikov to the fields. The speakers presented their current research and outlined future directions in both theory and frontier applications. The organizers of the conference are grateful to its sponsors: Russian Foundation of Basic Research, D. Zimin’s Russian Charitable Foundation “Dynasty,” R&D company Mera-NN, and K. V. Kirsenko (Russia), as well as Office of Naval Research (USA) and its officers, Drs. M. Harper (UK) and M. Shlesinger (USA). We thank the Editors-in-Chief of the International Journal of Bifurcations and Chaos: Ron Chen and Leon Chua for having the proceedings published here. L. P. Shilnikov served on the Editorial Board of the journal from the time it was founded. Our dear friend, mentor and fellow researcher, L. P. Shilnikov conceptualized the theory of global bifurcations of high-dimensional systems and was one of the founders of the mathematical theory of dynamical chaos. He built a profound research school in the city of Nizhny Novgorod (Gorky formerly) — the Shilnikov School that continues to this day. His works greatly influenced the overall development of the mathematical theory of dynamical systems as well as nonlinear dynamics, in general. Shilnikov’s findings have been included in most textand reference books, and are used worldwide by mathematics students and nonlinear dynamists to study the qualitative theory of dynamical systems and chaos. The elegance and completeness of his results let them reach “the heart of the matter,” and provide applied researchers with an in-depth mathematical understanding of the outcomes of natural experiments. The popularity and appreciation are due to the “living classic” status attained by Professor Shilnikov over several decades of his life through continuous hard work on bifurcation theory of multidimensional dynamical systems, mathematical chaos theory and theory of strange attractors. L. P. Shilnikov was born in Kotelnich, Kirov region of Russia on December 17, 1934. After graduating from a local high school in 1952, he became a student in the Department of Physics and Mathematics at Gorky State University. After graduation in 1957, he continued his PhD studies at the same university. He defended his PhD thesis “On birth of stable periodic orbits from singular trajectories” in 1962, it focused on the multidimensional generalization of basic homoclinic bifurcations, which were originally discovered and studied for systems on a plane by A. A. Andronov and E. A. Leontovich in the early 1930s.

We derive sufficient conditions for the existence of an invariant set in an absorbing region homeomorphic to the product of a multidimensional torus and a ball. This set consists of low dimensional tori labeled by symbolic sequences. It may appear as a result of the breakdown of an attracting multidimensional torus. Trajectories on the set manifest chaotic behavior for some angular coordinates and may behave regularly for others, i.e. the dynamics on the set is of the chimera state type.

Combining the results of brain imaging and nonlinear dynamics provides a new hierarchical vision of brain network functionality that is helpful in understanding the relationship of the network to different mental tasks. Using these ideas it is possible to build adequate models for the description and prediction of different cognitive activities in which the number of variables is usually small enough for analysis. The dynamical images of different mental processes depend on their temporal organization and, as a rule, cannot be just simple attractors since cognition is characterized by transient dynamics. The mathematical image for a robust transient is a stable heteroclinic channel consisting of a chain of saddles connected by unstable separatrices. We focus here on hierarchical chunking dynamics that can represent several cognitive activities. Chunking is the dynamical phenomenon that means dividing a long information chain into shorter items. Chunking is known to be important in many processes of perception, learning, memory and cognition. We prove that in the phase space of the model that describes chunking there exists a new mathematical object — heteroclinic sequence of heteroclinic cycles — using the technique of slow–fast approximations. This new object serves as a skeleton of motions reflecting sequential features of hierarchical chunking dynamics and is an adequate image of the chunking processing.

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

Recent results of imaging technologies and non-linear dynamics make possible to relate the structure and dynamics of functional brain networks to different mental tasks and to build theoretical models for the description and prediction of cognitive activity. Such models are non-linear dynamical descriptions of the interaction of the core components—brain modes—participating in a specific mental function. The dynamical images of different mental processes depend on their temporal features. The dynamics of many cognitive functions are transient. They are often observed as a chain of sequentially changing metastable states. A stable heteroclinic channel (SHC) consisting of a chain of saddles—metastable states—connected by unstable separatrices is a mathematical image for robust transients. In this paper we focus on hierarchical chunking dynamics that can represent several forms of transient cognitive activity. Chunking is a dynamical phenomenon that nature uses to perform information processing of long sequences by dividing them in shorter information items. Chunking, for example, makes more efficient the use of short-term memory by breaking up long strings of information (like in language where one can see the separation of a novel on chapters, paragraphs, sentences, and finally words). Chunking is important in many processes of perception, learning, and cognition in humans and animals. Based on anatomical information about the hierarchical organization of functional brain networks, we propose a cognitive network architecture that hierarchically chunks and super-chunks switching sequences of metastable states produced by winnerless competitive heteroclinic dynamics.

This important collection presents recent advances in nonlinear dynamics including analytical solutions, chaos in Hamiltonian systems, time-delay, uncertainty, and bio-network dynamics. Nonlinear Dyna

An approach is presented for analyzing the dynamics of excitatory neural networks. It is based on the reduction of dynamics of networks to the dynamics of cellular automata (CA) on the graph of synaptic connections. We illustrate our approach on an example of network of Morris-Lecar neurons coupled by chemical synapses with short-term plasticity.

Following on previous work, we discuss a scenario in which two neural ensembles have a master–slave arrangement. We will consider a more particular case in which the master system exhibits transient dynamics due to a stable heteroclinic sequence (SHS) and the slave system has attractors in the form of limit cycles. We will give sufficient conditions that guarantee the existence of a SHS in the form of a tube in the full phase space, together with an open stable heteroclinic channel that surrounds it. We present a numerical observation of chaotic-like transient behaviour different from well-known transient chaos.

Умер Леонид Павлович Шильников. . .Тяжело на душе и обидно.Особенно нам, его ученикам и соратникам.Уходит эпоха создания новой парадигмы в теории дифференциальных уравнений -хаотической динамики.До последнего дня его жизни он был весь в науке, это была его жизнь.Мы обсуждали задачи, подходы, направления, в которых нужно двигаться.Вспоминает Л. М. Лерман: «Когда я ему сказал, что у меня начало получаться то, что мы обсуждали и что он предложил сделать, он ответил: «Ты мне пролил бальзам вот сюда» -и показал слабой рукой на сердце.И это было за неделю до его ухода. . .» Он был настоящий большой ученый.Это понимали все, кто с ним когда-либо общался.Мы это наблюдали много раз, причем как у нас здесь в России и бывшем Союзе, так и за границей.Он не говорил по-английски, но его понимали: язык математики интернационален.А если не понимали, то только потому, что он опережал в идеях.Вот что написал, узнав о его смерти, профессор Г. Николис из Свободного университета Брюсселя, с которым у ЛП были давние научные связи: «Это печальное

A repressilator is a synthetic regulatory network that produces self-sustained oscillations. We analyze the evolution of the oscillatory solution in the repressilator model. We have established a connection between the evolution of the oscillatory solution and formation of a heteroclinic cycle at infinity. The convergence of the limit cycle to the heteroclinic cycle occurs very differently compared to the well-studied cases. The transition studied here presents a new bifurcation scenario.