Mathematics - Dynamical Systems Publications (50)

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Mathematics - Dynamical Systems Publications

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear models with application in estimation and control of Boolean processes based on noisy and incomplete measurements. The optimal minimum mean square error (MMSE) algorithms for POBDS state estimation, namely, the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS), are intractable in the case of large systems, due to computational and memory requirements. To address this, we propose approximate MMSE filtering and smoothing algorithms based on the auxiliary particle filter (APF) method from sequential Monte-Carlo theory. Read More


Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants $\lambda\in {\Bbb R}$ and $C>0$ such that for any $n\in {\Bbb N}$ and any $i_1, \ldots, i_n \in \{1,\ldots, k\}$, either $M_{i_1} \cdots M_{i_n}={\bf 0}$ or $C^{-1} e^{\lambda n} \leq \| M_{i_1} \cdots M_{i_n} \| \leq C e^{\lambda n}$, where $\|\cdot\|$ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. Read More


We present theoretical and experimental studies on pattern formation with bistable dynamical units coupled in a star network configuration. By applying a localized perturbation to the central or the peripheral elements, we demonstrate the subsequent spreading, pinning, or retraction of the activations; such analysis enables the characterization of the formation of stationary patterns of localized activity. The results are interpreted with a theoretical analysis of a simplified bistable reaction-diffusion model. Read More


We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. Read More


We prove almost sure invariance principle, a strong form of approximation by Brownian motion, for non-autonomous holomorphic dynamical systems on complex projective space $\Bbb{P}^k$ for H\"older continuous and DSH observables. Read More


In recent work it was shown how recursive factorisation of certain QRT maps leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a fifth-order recurrence with the Laurent property. Here we recursively factorise the 12-parameter symmetric QRT map, given by a second-order recurrence for a dependent variable $u_n$, to obtain a system of three coupled recurrences which possesses the Laurent property. As degenerate special cases, we derive systems of two coupled recurrences corresponding to the 5-parameter additive and multiplicative symmetric QRT maps. Read More


We use a computer aided proof to rigorously show the existence of noise induced order in the model of chaotic chemical reactions where it was first discovered numerically by Matsumoto and Tsuda in 1983. We show that in this random dynamical system the increase of noise causes the Lyapunov exponent to decrease from positive to negative, stabilizing the system. The method is based on a certified approximation of the stationary measure in the $L^1$ norm. Read More


The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers of a subset S of a semisimple linear Lie group G (e. Read More


In this note we present some one-parameter families of homogeneous self-similar measures on the line such that - the similarity dimension is greater than $1$ for all parameters and - the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-$\alpha$ projections of the natural measure of the Sierpi\'nski carpet. We present more general one-parameter families of self-similar measures $\nu_\alpha$, such that the set of parameters $\alpha$ for which $\nu_\alpha$ is singular is a dense $G_\delta$ set but this "exceptional" set of parameters of singularity has zero Hausdorff dimension. Read More


Z^d-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green-Kubo's formula is invariant under induction. Read More


According to a conjecture of Lindenstrauss and Tsukamoto, a topological system $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^d)^\mathbb{Z},$shift) if its mean dimension and periodic dimension verify mdim$(X,T)Read More


We study the parameter plane of one-dimensional dynamically defined slices of meromorphic transcendental maps of finite type for which infinity is not an asymptotic value. More precisely we investigate {\em shell components} in these parameter planes which are components of parameter values for which a free asymptotic value is attracted to an attracting periodic orbit. Among other things, we prove that, in contrast to slices of parameter spaces for rational maps, where one sees Mandelbrot-like components that have a unique {\em center}, shell components do not have centers. Read More


What is chaos ? Despite several decades of research on this ubiquitous and fundamental phenomenon there is yet no agreed-upon answer to this question. Recently, it was realized that all stochastic and deterministic differential equations, describing all natural and engineered dynamical systems, possess a topological supersymmetry. It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. Read More


We discuss, in the context of energy flow in high-dimensional systems and Kolmogorov-Arnol'd-Moser (KAM) theory, the behavior of a chain of rotators (rotors) which is purely Hamiltonian, apart from dissipation at just one end. We derive bounds on the dissipation rate which become arbitrarily small in certain physical regimes, and we present numerical evidence that these bounds are sharp. We relate this to the decoupling of non-resonant terms as is known in KAM problems. Read More


In this letter we obtain sharp estimates on the growth rate of solutions to a nonlinear ODE with a nonautonomous forcing term. The equation is superlinear in the state variable and hence solutions exhibit rapid growth and finite-time blow-up. The importance of ODEs of the type considered here stems from the key role they play in understanding the asymptotic behaviour of more complex systems involving delay and randomness. Read More


We study fractal properties of invariant graphs of hyperbolic and partially hyperbolic skew product diffeomorphisms in dimension three. We describe the critical (either Lipschitz or at all scales H\"older continuous) regularity of such graphs. We provide a formula for their box dimension given in terms of appropriate pressure functions. Read More


System of differential equations describing the initial stage of the capture of oscillatory systems into the parametric autoresonance is considered. Of special interest are solutions whose amplitude increases without bound with time. The possibility of capture the system into the autoresonance is related with the stability of such solutions. Read More


We provide new examples of translation actions on locally compact groups with the "local spectral gap property" introduced in \cite{BISG15}. This property has applications to strong ergodicity, the Banach-Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\text{Isom}(\mathbb{R}^d)$ of orientation-preserving isometries of the euclidean space $\mathbb{R}^d$, for $d \geq 3$. Read More


We show that practical uniform global asymptotic stability (pUGAS) is equivalent to existence of a bounded uniformly globally weakly attractive set. This result is valid for a wide class of robustly forward complete distributed parameter systems, including time-delay systems, switched systems, many classes of PDEs and evolution differential equations in Banach spaces. We apply this criterion to show that existence of a non-coercive Lyapunov function ensures pUGAS of robustly forward complete systems. Read More


The chaos control problem of continuous time Rabinovich chaotic system is addressed. An instantaneous control input has been designed using predictive control principle to guarantee the convergence of the chaotic trajectory towards an unstable equilibrium point. Numerical simulations are presented to verify all the theoretical analyses. Read More


This paper gives a classification of partially hyperbolic systems in dimension 3 which have at least one torus tangent to the center-stable bundle. Read More


Let $X$ be a compact metrizable group and $\Gamma$ a countable group acting on $X$ by continuous group automorphisms. We give sufficient conditions under which the dynamical system $(X,\Gamma)$ is surjunctive, i.e. Read More


We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of {\epsilon}-rotation sets. These are obtained by replacing orbits with {\epsilon}-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as {\epsilon} decreases to zero. Read More


For a pinched Hadamard manifold $X$ and a discrete group of isometries $\Gamma$ of $X$, the critical exponent $\delta_\Gamma$ is the exponential growth rate of the orbit of a point in $X$ under the action of $\Gamma$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $\Gamma_0$ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients $\Gamma_0/\Gamma$. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest. Read More


An elliptic Newton flow is a dynamical system that can be interpreted as a continuous version of Newton's iteration method for finding the zeros of an elliptic function f. Previous work focusses on structurally stable flows (i.e. Read More


We consider a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a class of partially hyperbolic skew products, whose behavior on the preserved fibers is dominated by the expansion of the base map. In particular we apply the results to power law mixing toral extensions. Read More


We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincar\'e flow and a generalization of Ma\~n\'e-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities. Read More


We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $\theta$ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan $\theta$ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations. Read More


We consider a system of nonlinear partial differential equations that describes an age-structured population inhabiting several temporally varying patches. We prove existence and uniqueness of solution and analyze its large-time behavior in cases when the environment is constant and when it changes periodically. A pivotal assumption is that individuals can disperse and that each patch can be reached from every other patch, directly or through several intermediary patches. Read More


We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem. Read More


We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time. Read More


The purpose of this paper is to point out a new connection between information theory and dynamical systems. In the information theory side, we consider rate distortion theory, which studies lossy data compression of stochastic processes under distortion constraints. In the dynamical systems side, we consider mean dimension theory, which studies how many parameters per second we need to describe a dynamical system. Read More


We prove, in particular, that for each invertible $\mu$-preserving transformation $T$, a measurable set $A$ with $\mu(A)>0$ and an integer $\ell\geq 1$ there exists an infinite subset of positive integers $\cN_A\subset\bbN$ with uniformly bounded gaps such that \[ \liminf_{N\to\infty,\, N\in\cN_A}\frac 1N\sum_{n=1}^N\mu\big(\cap_{j=0}^\ell (T^{-jn}A\cap T^{-j(N-n)}A)\big)>0. \] As a corollary we obtain that any subset of integers with positive upper density contains "many" arithmetic progressions with both steps $n$ and $N-n$ with $N$ belonging to infinite sets of integers with bounded gaps. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemer\' edi theorem. Read More


The ergodicity hypothesis for the system of three (or more) falling balls, with upward decreasing masses, is known for almost three decades. For three falling balls, the model has non-zero Lyapunov exponents for every point in phase space. The usual approach for proving ergodicity for such systems is to verify the five conditions of the local ergodic theorem and then use a transitivity argument to obtain the existence of only one ergodic component. Read More


The polar vortices play a crucial role in the formation of the ozone hole and can cause severe weather anomalies. Their boundaries, known as the vortex `edges', are typically identified via methods that are either frame-dependent or return non-material structures, and hence are unsuitable for assessing material transport barriers. Using two-dimensional velocity data on isentropic surfaces in the northern hemisphere, we show that elliptic Lagrangian Coherent Structures (LCSs) identify the correct outermost material surface dividing the coherent vortex core from the surrounding incoherent surf zone. Read More


The Boltzmann equation is an integro-differential equation which describes the density function of the distribution of the velocities of the molecules of dilute monoatomic gases under the assumption that the energy is only transferred via collisions between the molecules. In 1956 Kac studied the Boltzmann equation and defined a property of the density function that he called the "Boltzmann property" which describes the behavior of the density function at a given fixed time as the number of particles tends to infinity. The Boltzmann property has been studied extensively since then, and now it is simply called chaos, or Kac's chaos. Read More


We study the indices of the geodesic central configurations on $\H^2$. We then show that central configurations are bounded away from the singularity set. With Morse's inequality, we get a lower bound for the number of central configurations on $\H^2$. Read More


This is a survey on the Darboux theory of integrability for polynomial vector fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We also provide new results about the maximum number of parallels and meridians that a polynomial vector field $\X$ on $\sss^n$ can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field $\Y$ in $\R^n$ can have in function of the degree of $\Y$. Read More


We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is $0$. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. Read More


We study the convergence and differentiability properties of series $$V_{\alpha,\beta}(n,t)=\sum_{p \leq n}p^{-\alpha}\exp(2\pi i p^{\beta}t),\quad t \in [0,1),\;n \in \mathbb{N}$$ defined over prime numbers $p$ and depending on parameters $\alpha,\beta \in \mathbb{R}_{>0}$. The visible fractal nature of the graphs as a function of $\alpha,\beta$ is analyzed in terms of H\"{o}lder continuity, self similarity and fractal dimension. These trigonometric sums visualize the distribution of primes: a prime number $p$ divides a natural number $n$ with probability $np^{-1}$. Read More


In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Read More


In this paper we prove that the well-known quasi-steady state approximations, commonly used in enzyme kinetics, which can be interpreted as the reduced system of a differential system depending on a perturbative parameter, according to Tihonov theory, are asymptotically equivalent to the center manifold of the system. This allows to give a mathematical foundation for the application of a mechanistic method to determine the center manifold of (at this moment, still simple) enzyme reactions. Read More


In this paper, a novel scheme for synchronizing four drive and four response systems is proposed by the authors. The idea of multi switching and dual combination synchronization is extended to dual combination-combination multi switching synchronization involving eight chaotic systems and is a first of its kind. Due to the multiple combination of chaotic systems and multi switching the resultant dynamic behaviour is so complex that, in communication theory, transmission and security of the resultant signal is more effective. Read More


We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evolution equations: the latter have typical features of diffusive processes, but cannot be well-posed on graphs with unbounded degree. Read More


In this work, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the discrete optimization problem into continuous spaces. We explore multiple embedding techniques -- namely, the construction of dynamical flows on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. In particular, we find that the Procrustes approximation provides a competitive biasing approach for the Lin--Kernighan heuristic. Read More


We show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is well-behaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the norm-square of a moment map on an affine variety. Read More


This paper takes the first step towards the development of a control framework for underactuated flying humanoid robots. We assume that the robot is powered by four thrust forces placed at the robot end effectors, namely the robot hands and feet. Then, the control objective is defined as the asymptotic stabilization of the robot centroidal momentum. Read More


Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let $X$ be a compact totally disconnected space and $f:X\to X$ a continuous map. Read More