Mathematics - Dynamical Systems Publications (50)


Mathematics - Dynamical Systems Publications

For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Read More

We show that every invertible strong mixing transformation on a Lebesgue space has strictly over-recurrent sets. Also, we give an explicit procedure for constructing strong mixing transformations with no under-recurrent sets. This answers both parts of a question of V. Read More

The Kechris-Pestov-Todorcevic correspondence connects extreme amenability of non-Archimedean Polish groups with Ramsey properties of classes of finite structures. The purpose of the present paper is to recast it as one of the instances of a more general construction, allowing to show that Ramsey-type statements actually appear as natural combinatorial expressions of the existence of fixed points in certain compactifications of groups, and that similar correspondences in fact exist in various dynamical contexts. Read More

Using the mould formalism introduced by Jean Ecalle, we define and study the geometric complexity of an isochronous center condition. The role played by several Lie ideals is discussed coming from the interplay between the universal mould of the correction and the different Lie algebras generated by the comoulds. This strategy enters in the general program proposed by J. Read More

The main purpose of this paper is to provide a summary of the fundamental methods for analyzing delay differential equations arising in biology and medicine. These methods are employed to illustrate the effects of time delay on the behavior of solutions, which include destabilization of steady states, periodic and oscillatory solutions, bifurcations, and stability switches. The biological interpretations of delay effects are briefly discussed. Read More

We apply lattice points counting results of Gorodnik and Nevo to solve a shrinking target problem in the setting of geodesic flows on hyperbolic manifolds of finite volume. Read More

The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. Read More

The paper concerns with Wilson-Cowan neural model with impulses. The main novelty of the study is that besides the traditional singularity of the model, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. Read More

We introduce a new isomorphism-invariant notion of entropy for measure preserving actions of arbitrary countable groups on probability spaces, which we call cocycle entropy. We develop methods to show that cocycle entropy satisfies many of the properties of classical amenable entropy theory, but applies in much greater generality to actions of non-amenable groups. One key ingredient in our approach is a proof of a subadditive convergence principle which is valid for measure-preserving amenable equivalence relations, going beyond the Ornstein-Weiss Lemma for amenable groups. Read More

Consider a nonuniformly expanding dynamical system $T \colon M \to M$. Suppose that $v \colon M \to \mathbb{R}^d$ is a bounded observable, and $\mu$, $\rho$ are probability measures on $M$. Let $X_n$ and $Y_n$ be two discrete time random processes, corresponding to Birkhoff sums of $v$ on the probability spaces $(M, \mu)$ and $(M, \rho)$. Read More

We ask under what conditions on the function $f$, and a set of maps $\mathcal T$, it is the case that $f$ is a coboundary for some map in $\mathcal T$. We also consider for a function $f$, and a set of maps $\mathcal T$, when we have $f$ being a coboundary for all the maps in $\mathcal T$. Read More

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\varphi: \mathcal{Z}_{\omega}(f) \to \mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. Read More

Of late, there has been intense interest in the realization of topological phases in very experimentally accessible classical systems like mechanical metamaterials and photonic crystals. Subjecting them to a time-dependent driving protocol further expands the diversity of possible topological behavior. We introduce a very realistic experimental proposal for a mechanical Floquet Chern insulator using a lattice of masses equipped with time-varying electromagnets. Read More

We describe the equivalence classes of germs of generic 1-parameter families of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is under conjugacy by holomorphic change of coordinate and parameter. We provide a description of the modulus space and (almost) unique normal forms. Read More

We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous states. As a consequence of the underlying Hamiltonian structure there exist unusual behaviours compared with networks of coupled limit cycle oscillators or activator-inhibitor systems. Read More

In this work we review a class of deterministic nonlinear models for the propagation of infectious diseases over contact networks with strongly-connected topologies. We consider network models for susceptible-infected (SI), susceptible-infected-susceptible (SIS), and susceptible-infected-recovered (SIR) settings. In each setting, we provide a comprehensive nonlinear analysis of equilibria, stability properties, convergence, monotonicity, positivity, and threshold conditions. Read More

We present an adaptation of Stein's method of normal approximation to the study of both discrete- and continuous-time dynamical systems. We obtain new correlation-decay conditions on dynamical systems for a multivariate central limit theorem augmented by a rate of convergence. We then present a scheme for checking these conditions in actual examples. Read More

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Sim\'anyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. Read More

Let $\Gamma\backslash G$ be a $\mathbb{Z}^d$-cover of a compact rank one homogeneous space, and $\{a_t\}$ a one-parameter diagonalizable subgroup of $G$. We prove the following $\it{local\, mixing\, theorem}$: for any compactly supported measure $\mu$ on $\Gamma\backslash G$ with a continuous density: $$\lim_{t\to \infty} t^{d/2} \int \psi \, d\mu_t=c \int \psi \,dg$$ where $c>0$ is a constant depending only on $\Gamma$. More generally, we establish the local mixing theorem for any $\mathbb{Z}^d$-cover of a homogeneous space $\Gamma_0\backslash G$ with $\Gamma_0$ a convex cocompact Zariski dense subgroup of $G$. Read More

This paper presents full classification of second minimal odd periodic orbits of a continuous endomorphisms on the real line. A $(2k+1)$-periodic orbit ($k\geq 3$) is called second minimal for the map $f$, if $2k-1$ is a minimal period of $f$ in the Sharkovskii ordering. We prove that there are $4k-3$ types of second minimal $(2k+1)$-orbits, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses. Read More

A diffusive predator-prey system with predator interference and Neumann boundary conditions is considered in this paper. We derive some results on the existence and nonexistence of nonconstant stationary solutions. It is shown that there exist no nonconstant stationary solutions when the effect of the predator interference is strong or the conversion rate of the predator is large, and nonconstant stationary solutions emerge when the diffusion rate of the predator is large. Read More

In this note, continuous transitive maps $f$ on the interval $I$ are re-addressed, where $I$ denotes one of the intervals: $(-\infty, \infty)$, $(-\infty, a]$, $[b, \infty)$, $[a, b]$, where $a < b$ are real numbers. Such maps must have a fixed point, say $z$, in the interior of $I$. Some well-known properties of such maps are re-proved in a systematic way according to the following : (1) $f$ moves some point $c \ne z$ away from $z$, i. Read More

Dynamical compensation (DC) has been recently defined as the ability of a biological system to keep its output dynamics unchanged in the face of varying parameters. This concept is purported to describe a design principle that provides robustness to physiological circuits. Here we note the similitude between DC and Structural Identifiability (SI), and we argue that the former can be explained in terms of (lack of) the latter. Read More

For any $2\le r\le \infty$, $n\ge 2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $N$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Gochenko-Shil'nikov-Turaev, Kaloshin, Bonatti-Diaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore for any $2\le r<\infty$ and any $k\ge 0$, we prove the existence of an open set $\hat U$ of $k$-parameter families in $U$ so that for a generic $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the map $f_a$ displays a fast growth of periodic points. Read More

We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface introduced in [DFG], the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. Read More

We prove that every nondegenerate contact form on a closed connected three-manifold, such that the associated contact structure has torsion first Chern class, has either two or infinitely many simple Reeb orbits. By previous results it follows that under the above assumptions, there are infinitely many simple Reeb orbits if the three-manifold is not the three-sphere or a lens space. We also show that for non-torsion contact structures, every nondegenerate contact form has at least four simple Reeb orbits. Read More

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it we further establish the corresponding Wentzell-Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. Read More

In the social, behavioral, and economic sciences, it is an important problem to predict which individual opinions will eventually dominate in a large population, if there will be a consensus, and how long it takes a consensus to form. This idea has been studied heavily both in physics and in other disciplines, and the answer depends strongly on both the model for opinions and for the network structure on which the opinions evolve. One model that was created to study consensus formation quantitatively is the Deffuant model, in which the opinion distribution of a population evolves via sequential random pairwise encounters. Read More

In this article, we investigate the method of upper and lower solutions for Volterra integral equation of the first kind on arbitrary time scale $\mathbb{T}$. We establish some existence results in a certain sector. Moreover, monotone iterative technique is used to obtain maximal and minimal solutions of the considered equation. Read More

We study chemical reaction networks with discrete state spaces, such as the standard continuous time Markov chain model, and present sufficient conditions on the structure of the network that guarantee the system exhibits an extinction event. The conditions we derive involve creating a modified chemical reaction network called a domination-expanded reaction network and then checking properties of this network. We apply the results to several networks including an EnvZ-OmpR signaling pathway in Escherichia coli. Read More

In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method is also presented for stable solutions. Read More

In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we estimate the constants in Bourgain's proof and subsequently estimate the measure of the parameter set in his result. We show, that it has positive, but not full Hausdorff dimension. Read More

In this note we prove that Moebius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all $C^{1+\alpha}$ surface diffeomorphisms with positive entropy correlate with the Moebius function. Read More

For expanding or hyperbolic dynamical systems, we use upper and lower natural density and Banach density to divide dynamical orbits into several different level sets. Meanwhile, non-recurrence and Birkhoff averages are considered and we obtained simultaneous level sets by mixing them together. By studying the topological entropy via multifractal analysis, we reveal the complexity of each level set. Read More

We obtain a unique, canonical one-to-one correspondence between postcritically minimal rational Newton maps arising from entire maps of the complex plane and postcritically finite Newton maps of polynomials, which preserves the dynamics of Julia sets. This bijection is induced by parabolic surgery developed by P. Ha\"issinsky. Read More

A probabilistic framework is proposed for the optimization of efficient switched control strategies for physical systems dominated by stochastic excitation. In this framework, the equation for the state trajectory is replaced with an equivalent equation for its probability distribution function in the constrained optimization setting. This allows for a large class of control rules to be considered, including hysteresis and a mix of continuous and discrete random variables. Read More

In the present paper, we consider nonlinear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of nonlinear Markov operators is equivalent to their orthogonal preserving property. Read More

We prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property admits off-diagonal asymptotic pairs. Other implications of the pseudo-orbit tracing property for group actions are presented. Using Chung and Li's algebraic characterization of expansiveness, we prove obtain the pseudo-orbit tracing property for a class of expansive algebraic actions. Read More

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two. Read More

In this paper, we study Hyers-Ulam stability for integral equation of Volterra type in time scale setting. Moreover we study the stability of the considered equation in Hyers-Ulam-Rassias sense. Our technique depends on successive approximation method, and we use time scale variant of induction principle to show that equation (1. Read More

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow. Read More

Given a one-dimensional shift $X$, let $|F_X(n)|$ be the number of follower sets of words of length $n$ in $X$, and $|P_X(n)|$ be the number of predecessor sets of words of length $n$ in $X$. We call the sequence $\{|F_X(n)|\}_{n \in \mathbb{N}}$ the follower set sequence of the shift $X$, and $\{|P_X(n)|\}_{n \in \mathbb{N}}$ the predecessor set sequence of the shift $X$. Extender sets are a generalization of follower sets, and we define the extender set sequence similarly. Read More

We generalize the curved $N$-body problem to spheres and hyperbolic spheres whose curvature $\kappa$ varies in time. Unlike in the particular case when the curvature is constant, the equations of motion are non-autonomous. We first briefly consider the analogue of the Kepler problem and then investigate the homographic orbits for any number of bodies, proving the existence of several such classes of solutions on spheres. Read More

We introduce and study properties of phyllotactic and rhombic tilings on the cylin- der. These are discrete sets of points that generalize cylindrical lattices. Rhombic tilings appear as periodic orbits of a discrete dynamical system S that models plant pattern formation by stacking disks of equal radius on the cylinder. Read More

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. Read More

We develop in this paper a general and novel technique to establish quantitative estimates for higher order correlations of group actions. In particular, we prove that actions of semisimple Lie groups, as well as semisimple $S$-algebraic groups and semisimple adele groups, on homogeneous spaces are exponentially mixing of all orders. As a combinatorial application of our results, we prove for semisimple Lie groups an effective analogue of the Furstenberg-Katznelson-Weiss Theorem on finite configurations in small neighborhoods of lattices. Read More

In this paper, we characterize all the irreducible Darboux polynomials and polynomial first integrals of FitzHugh-Nagumo (F-N) system. The method of the weight homogeneous polynomials and the characteristic curves is widely used to give a complete classification of Darboux polynomials of a system. However, this method does not work for F-N system. Read More

Given a real number $ \beta > 1$, we study the associated $ (-\beta)$-shift introduced by S. Ito and T. Sadahiro. Read More

In analogy to the topological entropy for continuous endomorphisms of totally disconnected locally compact groups, we introduce a notion of topological entropy for continuous endomorphisms of locally linearly compact vector spaces. We study the fundamental properties of this entropy and we prove the Addition Theorem, showing that the topological entropy is additive with respect to short exact sequences. By means of Lefschetz Duality, we connect the topological entropy to the algebraic entropy in a Bridge Theorem. Read More