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Leonid A. Bunimovich

Publications and source records attributed to Leonid A. Bunimovich.

17 recordsLinked to original sources
arXiv2026

Which subsets and when orbits prefer to visit in multidimensional non-Markovian non-uniformly expanding systems

The paper addresses some basic questions about finite time dynamics for slowly mixing non-uniformly expanding dynamical systems. It is concerned with transport in phase spaces of such systems, and analyzes which subsets and when the orbits prefer to visit. An asymptotic expansion with explicit coefficients for the decay of polynomial escape rates is obtained. Applications to a large class of (non)Markovian non-uniformly expanding systems are considered.

math.DS↗
arXiv2026

Statistical properties for irregular observables in slowly mixing hyperbolic systems

We prove various statistical properties (i.e., decay of correlations, central limit theorems with convergence rates, maximal large deviations and almost sure invariance principles) for non-smooth observables in the form of indicators functions in polynomially mixing hyperbolic dynamical systems. Such results were not known (or even expected to hold) for slowly mixing hyperbolic systems, although they are important for applications to physics and other sciences.

math.DS↗
arXiv2026

Statistics of Weakly Chaotic Systems

One of the major breakthroughs in science of the last (20th) century was building a bridge between the worlds of stochastic (random) systems and deterministic (dynamical) systems. It was started by the celebrated 1958 paper by A.N.Kolmogorov \cite{Kolmo}, who called this new theory (and actually a new way of thinking about deterministic systems) stochasticity of dynamical systems. Later, this name was essentially replaced by a short (sexier but more vague ``chaos theory"). Kolmogorov's discovery demonstrated that the time evolution of deterministic systems could be indistinguishable from the evolution of purely random (stochastic) systems. Moreover, it has been later established that typical deterministic systems are chaotic. However, as well as stochastic systems, which could be more or less random (from random processes with independent values to random processes with long memory) , chaotic dynamical systems could be strongly or weakly chaotic. Actually, the majority of chaotic systems are weakly chaotic, especially those that are relevant models for various real-world processes. Naturally, the theory of weakly chaotic systems is less developed because it deals with more complicated problems than the studies of strongly chaotic systems. We present here a brief review of this theory, including recently published and some new results.

math.DS↗
arXiv2025

Dynamics of the Kac Ring Model with switching scatterers

We introduce a generalized version of the Kac ring model in which particles are of two types, black and white. Black particles modify the environment through which all particles move, thereby inducing indirect and potentially long-range interactions among them. Unlike the inert scatterers of Kac's original model, the scatterers in our setting possess internal states that change upon interaction with black particles and can be interpreted as energy levels of the environment. This makes the model self-consistent, as it incorporates a form of particle interactions, mediated by the environment, that drives the system toward some kind of stationary state. Although indirect and long-range interactions do not necessarily promote thermodynamic states, interactions are necessary for energy to be shared among the elementary constituents of matter, enabling the establishment of equipartition, which is a prerequisite for defining temperature. Therefore, our model is one step forward in this direction, elucidating the role of interactions and energy exchange. We prove that any initial state of the system converges to a time periodic state (i.e. a phase space orbit) and describe basins of attraction for some of such asymptotic periodic states.

cond-mat.stat-mech↗
arXiv2023

Estimates of constants in the limit theorems for chaotic dynamical systems

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems.

math.DS↗
arXiv2021

Elliptic Flowers: simply connected billiard tables with chaotic or non-chaotic flows moving around chaotic or non-chaotic cores

We introduce a class of billiards with chaotic unidirectional flows (or non-chaotic unidirectional flows with "vortices") which go around a chaotic or non-chaotic "core", where orbits can change their orientation. Moreover, the corresponding billiard tables are simply connected in difference with many attempts to build billiards with interesting and/or exotic dynamics by putting inside billiard tables various "scatterers" with funny shapes. Therefore the billiards in this new class are amenable to experimental studies in physics labs as well as to the rigorous mathematical ones, which may shed a new light on understanding of classical and quantum dynamics of Hamiltonian systems.

math.DS↗
arXiv2020

Bridge to Hyperbolic Polygonal Billiards

It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov- Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical polygonal billiard is equivalent to a mathematical (point particle) semi-dispersing billiard. We also conjecture that in fact typical physical billiard in polygon is ergodic under the same conditions.

math.DS↗
arXiv2020

Ehrenfest Wind-Tree Model is Dynamically Richer Than the Lorentz Gas

We consider a physical Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. We demonstrate that a physical periodic Wind-Tree model is dynamically richer than a physical or mathematical periodic Lorentz gas. Namely, the physical Wind-Tree model may have diffusive behavior as the Lorentz gas does, but it has more superdiffusive regimes than the Lorentz gas. The new superdiffusive regime where the diffusion coefficient D(t)~(ln t)^2 of dynamics seems to be never observed before in any model.

math.DS↗
arXiv2020

Collision of a Hard Ball with Singular Points of the Boundary

Recently were introduced physical billiards where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional), i.e. the particle may collide with these singular points. Here, we consider the collision of a hard ball with a visible singular point and demonstrate that the motion of the smooth ball after collision with a visible singular point is indeed the one that was used in the studies of physical billiards. So such collision is equivalent to the elastic reflection of hard ball's center off a sphere with the center at the singular point and the same radius as the radius of the moving particle.

math.DS↗
arXiv2020

Why Escape Is Faster Than Expected

We consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which orbits escape. We compare various estimates of the escape rate which correspond to a physical picture of leaking in the entire phase space. Moreover, we uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Exact computations are present for the skewed tent map and Arnold's cat map.

math.DS↗
arXiv2019

Early-Time Exponential Instabilities in Non-Chaotic Quantum Systems

The vast majority of dynamical systems in classical physics are chaotic and exhibit the butterfly effect: a minute change in initial conditions can soon have exponentially large effects elsewhere. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quantum interference washes out classical chaos. We demonstrate that quantum mechanics can also play the opposite role and generate exponential instabilities in classically non-chaotic systems within this early-time window. Our calculations employ the out-of-time-ordered correlator (OTOC) -- a diagnostic that reduces to the Lyapunov exponent in the classical limit, but is well defined for general quantum systems. Specifically, we show that a variety of classically non-chaotic models, such as polygonal billiards, whose classical Lyapunov exponents are always zero, demonstrate a Lyapunov-like exponential growth of the OTOC at early times with Planck's-constant-dependent rates. This behavior is sharply contrasted with the slow early-time growth of the analog of the OTOC in the systems' classical counterparts. These results suggest that classical-to-quantum correspondence in dynamics is violated in the OTOC even before quantum interference develops.

quant-ph↗
arXiv2019

Few Islands Approximation of Hamiltonian System with divided Phase Space

It is well known that typical Hamiltonian systems have divided phase space consisting of regions with regular dynamics on KAM tori and region(s) with chaotic dynamics called chaotic sea(s). This complex structure makes rigorous analysis of such systems virtually impossible and significantly complicates numerical exploration of their dynamical properties. In this paper we outline a new approach for the analysis of Hamiltonian systems with divided phase space. These systems are approximated by a sequence of Hamiltonian systems having an increasing (but finite) number of KAM islands. The islands in the approximating systems are sub-islands of the islands in the initial system with an infinite number of KAM-islands. We apply this approach to two-dimensional billiards and demonstrate that it works. In particular the statistical characteristics of the approximating systems tend to the ones for the whole system when the number of islands in the approximating systems grows. Therefore our approach opens up a new way for numerical and analytical studies of the dynamics of Hamiltonian systems with divided phase space.

nlin.CD↗
arXiv2017

Uncovering Hierarchical Structure in Social Networks using Isospectral Reductions

We employ the recently developed theory of isospectral network reductions to analyze multi-mode social networks. This procedure allows us to uncover the hierarchical structure of the networks we consider as well as the hierarchical structure of each mode of the network. Additionally, by performing a dynamical analysis of these networks we are able to analyze the evolution of their structure allowing us to find a number of other network features. We apply both of these approaches to the Southern Women Data Set, one of the most studied social networks and demonstrate that these techniques provide new information, which complements previous findings.

cs.SI↗
arXiv2008

Track billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the segments and the arcs, the billiards considered have non-zero Lyapunov exponents almost everywhere. These results are then extended to a similar class of of 3-dimensional billiards. Finally, we find that for some subclasses of track billiards, the mechanism generating hyperbolicity is not the defocusing one that requires every infinitesimal beam of parallel rays to defocus after every reflection off of the focusing boundary.

math.DS↗
arXiv2006

Escape from a circle and Riemann hypotheses

We consider open circular billiards with one and with two holes. The exact formulas for escape are obtained which involve the Riemann zeta function and Dirichlet L-functions. It is shown that the problem of finding the exact asymptotics in the small hole limit for escape in some of these billiards is equivalent to the Riemann hypothesis.

math.DS↗
arXiv2005

One-Particle and Few-Particle Billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.

nlin.CD↗