All posts under category "Mathematics / Math.DS"
Let $M$ be a compact manifold and $f:,Mto M$ be a $C^1$ diffeomorphism on $M$. If $mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $mu$ $a.,,e.,,xin M,$ there is a dominated splitting $T_{orb(x)}M=Eoplus F$ on its orbit $orb(x)$, then w
Recently, a flexible and stable algorithm was introduced for the computation of 2D unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem.
Here the Integral Value Transformations (IVTs) are considered to be Discrete Dynamical System map in the spacemathbb{N}_(0). In this paper, the dynamics of IVTs is deciphered through the light of Topological Dynamics.
The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural equation, i.e. when the dependence of the curvature of the trajec
Adaptive systems must strike a balance between prediction and surprise to thrive in uncertain environments. We propose an information-theoretic balance function, $ f(p) = -(1 - p)ln(1 - p) + ln p $, which quantifies the net informational gain from contrasting explained variance $p$ with unexplained
We consider the static and dynamic models of Cournot duopoly with tax evasion. In the dynamic model we introduce the time delay and we analyze the local stability of the stationary state. There is a critical value of the delay when the Hopf bifurcation occurs.
We establish unique continuation for various discrete nonlinear wave equations. For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some arbitrarily small time interval, then they coincide everywhere. Moreover, we establish analogous results for the Toda,
We consider the dynamical behavior of Martin-L'of random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that s
We consider a system of birational functional equations (BFEs) (or finite-difference equations at w=m in Z) for functions y(w) of the form: y(w+1)=F_n(y(w)), y(w):C to C^N, n=deg(F_n(y)), F_n in (bf Bir}(C^N), where the map F_n is a given birational one of the group of all automorphisms of C^N to C^
A topological group $G$ is {em extremely amenable} if every compact $G$-space has a $G$-fixed point. Let $X$ be compact and $Gsubset{mathrm{Homeo}} (X)$. We prove that the following are equivalent: (1) $G$ is extremely amenable; (2) every minimal closed $G$-invariant subset of $exp R$ is a singleton
Assume that a family of stochastic processes on some Polish space $E$ converges to a deterministic process; the convergence is in distribution (hence in probability) at every fixed point in time. This assumption holds for a large family of processes, among which many mean field interaction models an
The periodic Toda lattice with $N$ sites is globally symplectomorphic to a two parameter family of $N-1$ coupled harmonic oscillators. The action variables fill out the whole positive quadrant of $R^{N-1}$. We prove that in the interior of the positive quadrant as well as in a neighborhood of the or
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept o
We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation
We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some textit{finite} precision, we propose a model where spikes are effective at times multiple of a characterist
Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerden's Theorem. We establish several properties of superfilters, which generalize both Ramsey's Theorem and its variant for ultrafilters on the natural numbers. We use them
This paper presents an application of the Infinite Unit Axiom, introduced by Yaroslav Sergeyev, (see [11] - [14]) to the development of one-dimensional cellular automata. This application allows the establishment of a new and more precise metric on the space of definition for one-dimensional cellula
In this paper, a Novel Active Disturbance Rejection Control (N-ADRC) strategy is proposed that replaces the Linear Extended state observer (LESO) used in Conventional ADRC (C-ADRC) with a Nested LESO. In the nested LESO, the inner-loop LESO actively estimates and eliminates the generalized disturban
Motivated by recent progress on the interplay between graph theory, dynamics, and systems theory, we revisit the analysis of chemical reaction networks described by mass action kinetics. For reaction networks possessing a thermodynamic equilibrium we derive a compact formulation exhibiting at the sa
Extending and unifying concepts extensively used in the literature, we introduce the notion of approximable interpolation sets for algebras of functions on locally compact groups, especially for weakly almost periodic functions and for uniformly continuous functions. We characterize approximable int
The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks
In this note we give an example of a one-dimensional manifold with two connected components and a complete metric whose group of isometries has an orbit which is not closed. This answers a question of S. Gao and A. S. Kechris.
We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar'e equations. In this case, the invariant Lagrangian is defined
The Sharing Economy (which includes Airbnb, Apple, Alibaba, Uber, WeWork, Ebay, Didi Chuxing, Amazon) blossomed across the world, triggered structural changes in industries and significantly affected international capital flows primarily by disobeying a wide variety of statutes and laws in many coun
We prove that an arbitrary Poisson structure omega^{ij}(u) and an arbitrary closed 3-form T_{ijk}(u) generate the local Poisson structure A^{ij}(u,u_x) = M^i_s(u,u_x)omega^{sj}(u), where M^i_s(u,u_x)(delta^s_j + omega^{sp}(u)T_{pjk}(u)u^k_x) = delta^i_j, on the corresponding loop space. We obtain al
The behaviour of neurons under the influence of periodic external input has been modelled very successfully by circle maps. The aim of this note is to extend certain aspects of this analysis to a much more general class of forcing processes. We apply results on the fibred rotation number of randomly
In this paper, we mainly study the robust stability of linear continuous systems with parameter uncertainties, a more general kind of uncertainties for system matrices is considered, i.e., entries of system matrices are rational functions of uncertain parameters which are varying in intervals. we pr
One of the key socioeconomic phenomena to explain is the distribution of wealth. Bouchaud and M'ezard have proposed an interesting model of economy [Bouchaud and M'ezard (2000)] based on trade and investments of agents. In the mean-field approximation, the model produces a stationary wealth distribu
A separated net is a set of points which is relatively dense and uniformly discrete (another name for a Delone set). We are dealing with tilings and separated nets in Euclidean spaces and with the question whether a given separated net is biLipschitz to the integer lattice. In this paper we show, as
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