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Leonid Bunimovich

Publications and source records attributed to Leonid Bunimovich.

At least 19 recordsLinked to original sources
arXiv2025

For how long time evolution of chaotic or random systems can be predicted

Traditionally, Probability theory was dealing with limit theorems where 'limit" means that time tends to infinity. Questions about finite time dynamics (evolution) were always considered as, although important for practical applications, but untreatable rigorously (mathematically). The same attitude was in the theory of strongly chaotic dynamical systems, which evolve similarly to stochastic processes. However, a natural question on dependence of the process of escape on a position of a "hole" in the state (phase) space, which was never asked in mathematical theory of open dynamical systems, opened up a new direction of research, which was dealing with finite time predictions of evolutions of such systems. It turned out, that transport of orbits in the phase space of the "most strongly chaotic" dynamical systems has three different stages. In the first stage there is a hierarchy of the first hitting probabilities, that shows which parts of the phase space the orbits of a system, which is an equilibrium state, will be more likely to visit the first. A principal (and the most important for applications) question was how the length of this interval changes with more refinement observations of the positions of the orbits in the phase space. Surprisingly, it turned out that the length of the time interval, where finite time predictions are possible, increases (rather to be shrinking), which, at the first sight, seems to be natural. However, this increase of the length of the time interval, where finite time predictions are possible, was rather slow (just linear) with respect to the growth of precision (partition of the phase space) of observations. In the present paper it is proved (by totally different technique) that this growth is actually exponential.

nlin.CD↗
arXiv2024

Antigenic Cooperation in Viral Populations: Redistribution of Loads Among Altruistic Viruses and Maximal Load per Altruist

The paper continues the study of the phenomenon of local immunodeficiency (LI) in viral cross-immunoreactivity networks, with a focus on the roles and interactions between altruistic and persistent viral variants. As always, only the state of stable (i.e. observable) LI is analysed. First, we show that a single altruistic viral variant has an upper limit for the number of persistent viral variants that it can support. Our findings reveal that in viral cross-immunoreactivity networks, altruistic viruses act essentially autonomously from each other. Namely, connections between altruistic viruses do not change neither their qualitative roles, nor the quantitative values of the strengths of their connections in the CRNs. In other words, each altruistic virus does exactly the same actions and with the same strengths with or without presence of other altruistic viruses. However, having more altruistic viruses allows to keep sizes of populations of persistent viruses at the higher levels. Likewise, the strength of the immune response against any altruistic virus remains at the same constant level regardless of how many persistent viruses this altruistic virus supports, i.e. shields from the immune response of the host's immune system. It is also shown that viruses strongly compete with each other in order to become persistent in the state of stable LI. We also present an example for a CRN with stable LI that only consists of persistent viral variants.

q-bio.PE↗
arXiv2024

Back to Boundaries in Billiards

We prove Poisson limit laws for open billiards where the holes are on the boundaries of billiard tables (rather than some abstract holes in the phase space of a billiard). Such holes are of the main interest for billiard systems, especially for applications. Sinai billiards with or without a finite horizon, diamond billiards, and semi-dispersing billiards, as well as focusing billiards with slow decay of correlations, are considered.

math.DS↗
arXiv2023

Antigenic cooperation in Viral Populations: Transformation of Functions of Intra-Host Viral Variants

In this paper we study intra-host viral adaptation by antigenic cooperation - a mechanism of immune escape that serves as an alternative to the standard mechanism of escape by continuous genomic diversification and allows to explain a number of experimental observations associated with the establishment of chronic infections by highly mutable viruses. Within this mechanism, the topology of a cross-immunoreactivity network forces intra-host viral variants to specialize for complementary roles and adapt to host's immune response as a quasi-social ecosystem. Here we study dynamical changes in immune adaptation caused by evolutionary and epidemiological events. First, we show that the emergence of a viral variant with altered antigenic features may result in a rapid re-arrangement of the viral ecosystem and a change in the roles played by existing viral variants. In particular, it may push the population under immune escape by genomic diversification towards the stable state of adaptation by antigenic cooperation. Next, we study the effect of a viral transmission between two chronically infected hosts, which results in merging of two intra-host viral populations in the state of stable immune-adapted equilibrium. In this case, we also describe how the newly formed viral population adapts to the host's environment by changing the functions of its members. The results are obtained analytically for minimal cross-immunoreactivity networks and numerically for larger populations.

q-bio.PE↗
arXiv2022

Detecting intrinsic global geometry of an obstacle via layered scattering

Given a closed $k$-dimensional submanifold $K$, incapsulated in a compact domain $M \subset \mathbb E^n$, $k \leq n-2$, we consider the problem of determining the intrinsic geometry of the obstacle $K$ (like volume, integral curvature) from the scattering data, produced by the reflections of geodesic trajectories from the boundary of a tubular $ε$-neighborhood $\mathsf T(K, ε)$ of $K$ in $M$. The geodesics that participate in this scattering emanate from the boundary $\partial M$ and terminate there after a few reflections from the boundary $\partial \mathsf T(K, ε)$. However, the major problem in this setting is that a ray (a billiard trajectory) may get stuck in the vicinity of $K$ by entering some trap there so that this ray will have infinitely many reflections from $\partial \mathsf T(K, ε)$. To rule out such a possibility, we modify the geometry of a tube $\mathsf T(K, ε)$ by building it from spherical bubbles. We need to use $\lceil \dim(K)/2\rceil$ many bubbling tubes $\{\mathsf T(K, ε_j)\}_j$ for detecting certain global invariants of $K$, invariants which reflect its intrinsic geometry. Thus the words "layered scattering" in the title. These invariants were studied by Hermann Weyl in his classical theory of tubes $\mathsf T(K, ε)$ and their volumes.

math.DS↗
arXiv2021

Poisson Approximations and Convergence Rates for Hyperbolic Dynamical Systems

We prove the asymptotic functional Poisson laws in the total variation norm and obtain estimates of the corresponding convergence rates for a large class of hyperbolic dynamical systems. These results generalize the ones obtained before in this area. Applications to intermittent solenoids, Axiom A systems, Hénon attractors and to billiards, are also considered.

math.DS↗
arXiv2020

Local Immunodeficiency: Role of Neutral Viruses

This paper analyzes the role of neutral viruses in the phenomenon of local immunodeficiency. We show that, even in the absence of altruistic viruses, neutral viruses can support the existence of persistent viruses, and thus local immunodeficiency. However, in all such cases neutral viruses can maintain only bounded (relatively small) concentration of persistent viruses. Moreover, in all such cases the state of local immunodeficiency could only be marginally stable, while it is known that altruistic viruses can maintain stable local immunodeficiency. We also present an absolutely minimal cross-immunoreactivity network where a stable and robust state of local immunodeficiency can be maintained. It is now a challenge to synthetic biology to build such small networks with stable local immunodeficiency. Another important challenge for biology is to understand which types of viruses can play a role of persistent, altrustic and neutral ones, and whether a role which a given virus plays depends on the structure (topology) of a given cross-immunoreactivity network.

math.DS↗
arXiv2019

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

In this paper we study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

math.CO↗
arXiv2019

Spectral and Dynamic Consequences of Network Specialization

One of the hallmarks of real networks is their ability to perform increasingly complex tasks as their topology evolves. To explain this, it has been observed that as a network grows certain subsets of the network begin to specialize the function(s) they perform. A recent model of network growth based on this notion of specialization has been able to reproduce some of the most well-known topological features found in real-world networks including right-skewed degree distributions, the small world property, modular as well as hierarchical topology, etc. Here we describe how specialization under this model also effects the spectral properties of a network. This allows us to give conditions under which a network is able to maintain its dynamics as its topology evolves. Specifically, we show that if a network is intrinsically stable, which is a stronger version of the standard notion of global stability, then the network maintains this type of dynamics as the network evolves. This is one of the first steps toward unifying the rigorous study of the two types of dynamics exhibited by networks. These are the \emph{dynamics of} a network, which is the study of the topological evolution of the network's structure, modeled here by the process of network specialization, and the \emph{dynamics on} a network, which is the changing state of the network elements, where the type of dynamics we consider is global stability. The main examples we apply our results to are recurrent neural networks, which are the basis of certain types of machine learning algorithms.

nlin.AO↗
arXiv2019

Graph Hausdorff dimension, Kolmogorov complexity and construction of fractal graphs

In this paper we introduce and study discrete analogues of Lebesgue and Hausdorff dimensions for graphs. It turned out that they are closely related to well-known graph characteristics such as rank dimension and Prague (or Nešetřil-Rödl) dimension. It allows us to formally define fractal graphs and establish fractality of some graph classes. We show, how Hausdorff dimension of graphs is related to their Kolmogorov complexity. We also demonstrate fruitfulness of this interdisciplinary approach by discovering a novel property of general compact metric spaces using ideas from hypergraphs theory and by proving an estimation for Prague dimension of almost all graphs using methods from algorithmic information theory.

math.CO↗
arXiv2019

Local Immunodeficiency: Minimal Networks and Stability

Some basic aspects of the recently discovered phenomenon of local immunodeficiency \cite{pnas} generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated. We prove that local immunodeficiency (LI) that's stable under perturbations already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large \cite{pnas}, but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topology. In particular, the scale-free property of a CR network, assumed in \cite{pnas}, is not required.

math.DS↗
arXiv2018

Where and When Orbits of Strongly Chaotic Systems Prefer to Go

We prove that transport in the phase space of the "most strongly chaotic" dynamical systems has three different stages. Consider a finite Markov partition (coarse graining) $ξ$ of the phase space of such a system. In the first short times interval there is a hierarchy with respect to the values of the first passage probabilities for the elements of $ξ$ and therefore finite time predictions can be made about which element of the Markov partition trajectories will be most likely to hit first at a given moment. In the third long times interval, which goes to infinity, there is an opposite hierarchy of the first passage probabilities for the elements of $ξ$ and therefore again finite time predictions can be made. In the second intermediate times interval there is no hierarchy in the set of all elements of the Markov partition. We also obtain estimates on the length of the short times interval and show that its length is growing with refinement of the Markov partition which shows that practically only this interval should be taken into account in many cases. These results demonstrate that finite time predictions for the evolution of strongly chaotic dynamical systems are possible. In particular, one can predict that an orbit is more likely to first enter one subset of phase space than another at a given moment in time. Moreover, these results suggest an algorithm which accelerates the process of escape through "holes" in the phase space of dynamical systems with strongly chaotic behavior.

math.DS↗
arXiv2018

How smooth can convex chaotic billiard tables be?

We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to extend standard KAM theory to analyze stability of periodic orbits, because of the low smoothness of the system. In fact, the stadium has a $C^1$ boundary, and we show that $C^2$ smoothing results in appearance of elliptic periodic orbits.

math.DS↗
arXiv2018

On Attractors of Isospectral Compressions of Networks

In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristic (attribute) of nodes (or edges) of networks. Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. It is shown that any orbit of such dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. Such attractor is a smaller network where a chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral contractions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that spectrally equivalent with respect to some characteristic networks could be non-spectrally equivalent for another characteristic of nodes (edges). These results suggest a new constructive approach to analysis of networks structures and to comparison of topologies of different networks.

math.DS↗
arXiv2018

Generalized Eigenvectors of Isospectral Transformations,Spectral Equivalence and Reconstruction of Original Networks

Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors.We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks.

math.DS↗
arXiv2016

Mechanisms for Network Growth that Preserve Spectral and Local Structure

We introduce a method that can be used to evolve the topology of a network in a way that preserves both the network's spectral as well as local structure. This method is quite versatile in the sense that it can be used to evolve a network's topology over any collection of the network's elements. This evolution preserves both the eigenvector centrality of these elements as well as the eigenvalues of the original network. Although this method is introduced as a tool to model network growth, we show it can also be used to compare the topology of different networks where two networks are considered similar if their evolved topologies are the same. Because this method preserves the spectral structure of a network, which is related to the network's dynamics, it can also be used to study the interplay of network growth and function. We show that if a network's dynamics is intrinsically stable, which is a stronger version of the standard notion of stability, then the network remains intrinsically stable as the network's topology evolves. This is of interest since the growth of a network can have a destabilizing effect on the network's dynamics, in general. In this sense the methods developed here can be used as a tool for designing mechanisms of network growth that ensure a network remains stabile as it grows.

nlin.AO↗
arXiv2015

On another edge of defocusing: hyperbolicity of asymmetric lemon billiards

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables--asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.

math.DS↗