arXiv subjects
Publications and source records attributed to Leonid Bunimovich.
We consider open systems generated from one-dimensional maps that admit a finite Markov partition and use the recently developed theory of isospectral graph transformations to estimate a system's survival probabilities. We show that these estimates are better than those obtained through a more direct approach.
It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network. This procedure allows for the introduction of new equivalence relations between networks, where two networks are spectrally equivalent if they can be reduced to the same network. Additionally, using this theory it is possible to establish whether a network, modeled as a dynamical system, has a globally attracting fixed point (is strongly synchronizing). In this paper we further develop this theory of isospectral network transformations and demonstrate that our procedures are applicable to families of parameterized networks and networks of arbitrary size.
Precopulatory courtship is a high-cost, non-well understood animal world mystery. Drosophila's (=D.'s) precopulatory courtship not only shows marked structural similarities with mammalian courtship, but also with human spoken language. This suggests the study of purpose, modalities and in particular of the power of this language and to compare it to human language. Following a mathematical symbolic dynamics approach, we translate courtship videos of D.'s body language into a formal language. This approach made it possible to show that D. may use its body language to express individual information - information that may be important for evolutionary optimization, on top of the sexual group membership. Here, we use Chomsky's hierarchical language classification to characterize the power of D.'s body language, and then compare it with the power of languages spoken by humans. We find that from a formal language point of view, D.'s body language is at least as powerful as the languages spoken by humans. From this we conclude that human intellect cannot be the direct consequence of the formal grammar complexity of human language.
It is well known that the distributions of hitting times in Markov chains are quite irregular, unless the limit as time tends to infinity is considered. We show that nevertheless for a typical finite irreducible Markov chain and for nondegenerate initial distributions the tails of the distributions of the hitting times for the states of a Markov chain can be ordered, i.e., they do not overlap after a certain finite moment of time. If one considers instead each state of a Markov chain as a source rather than a sink then again the states can generically be ordered according to their efficiency. The mechanisms underlying these two orderings are essentially different though.
Following [6,12], we study coupled map networks over arbitrary finite graphs. An estimate from below for a topological entropy of a perturbed coupled map network via a topological entropy of an unperturbed network by making use of the covering relations for coupled map networks is obtained. The result is quite general, particularly no assumptions on hyperbolicity of a local dynamics or linearity of coupling are made.
Let G be an arbitrary finite weighted digraph with weights in the set of complex rational functions. A general procedure is proposed which allows for the reduction of G to a smaller graph with a less complicated structure having the same spectrum as of G (up to some set known in advance). The proposed procedure has a lot of flexibility and could be used e.g. for design of networks with prescribed spectral and dynamical properties.
In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L2(T2). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.
A natural question of how the survival probability depends upon a position of a hole was seemingly never addressed in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period) which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.
We show that under certain simple assumptions on the topology (structure) of networks of strongly interacting chaotic elements a phenomenon of long range action takes place, namely that the asymptotic (as time goes to infinity) dynamics of an arbitrary large network is completely determined by its boundary conditions. This phenomenon takes place under very mild and robust assumptions on local dynamics with short range interactions. However, we show that it is unstable with respect to arbitrarily weak local random perturbations.
We study a class of dynamical systems which generalizes and unifies some models arising in the analysis of switched flow systems in manufacturing. General properties of these dynamical systems, called pseudo billiards, as well as some their perturbations are discussed.
We discuss a notion of phase transitions in multicomponent systems and clarify relations between deterministic chaotic and stochastic models of this type of systems. Connections between various definitions of SRB measures are considered as well.