Mathematics - Dynamical Systems Publications (50)

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Mathematics - Dynamical Systems Publications

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis (TICA), dynamic mode decomposition (DMD), and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods. Read More


We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space. Read More


We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines on which the projection of the measure is singular. In fact the set of such directions is residual. For the corresponding self-similar set $K$ we obtain that for almost every affine line $\ell$ the Hausdorff measure, at the critical dimension, of the intersection of $K$ with $\ell$ is equal to zero. Read More


We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic models of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. Read More


We study the self-semiconjugations of the Tent-map $f:\, x\mapsto 1-|2x-1|$ for $x\in [0,\, 1]$. We prove that each of these semi-conjugations $\xi$ is piecewise linear. For any $n\in \mathbb{N}$ we denote $A_n = f^{-n}(0)$ and describe the maps $\psi:\, A_n\rightarrow [0,\, 1]$ such that $\psi\circ f = f\circ \psi$. Read More


An SEIRS epidemic with disease fatalities is introduced in a growing population (modelled as a super-critical linear birth and death process). The study of the initial phase of the epidemic is stochastic, while the analysis of the major outbreaks is deterministic. Depending on the values of the parameters, the following scenarios are possible. Read More


In classical shallow water wave (SWW) theory, there exist two integrable one-dimensional SWW equation [Hirota-Satsuma (HS) type and Ablowitz-Kaup-Newell-Segur (AKNS) type] in the Boussinesq approximation. In this paper, we mainly focus on the integrable SWW equation of AKNS type. The nonlocal symmetry in form of square spectral function is derived starting from its Lax pair. Read More


The brain processes visual inputs having structure over a large range of spatial scales. The precise mechanisms or algorithms used by the brain to achieve this feat are largely unknown and an open problem in visual neuroscience. In particular, the spatial extent in visual space over which primary visual cortex (V1) performs evidence integration has been shown to change as a function of contrast and other visual parameters, thus adapting scale in visual space in an input-dependent manner. Read More


We present a new stabilised and efficient high-order nodal spectral element method based on the Mixed Eulerian Lagrangian (MEL) method for general-purpose simulation of fully nonlinear water waves and wave-body interactions. In this MEL formulation a standard Laplace formulation is used to handle arbitrary body shapes using unstructured - possibly hybrid - meshes consisting of high-order curvilinear iso-parametric quadrilateral/triangular elements to represent the body surfaces and for the evolving free surface. Importantly, our numerical analysis highlights that a single top layer of quadrilaterals elements resolves temporal instabilities in the numerical MEL scheme that are known to be associated with mesh topology containing asymmetric element orderings. Read More


The present paper deals with a prey-predator model with prey refuge proportion to both species and independent harvesting of each species. Our study shows that using refuge as control, it can break the limit circle of the system and reach the required state of equilibrium level. It is established the optimal harvesting policy. Read More


Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie-Gower type for which we consider that prey reproduces mush faster than predators. This naturally leads to introduce a small parameter $\epsilon$ which gives rise to a slow-fast system. Read More


For finitely generated groups $H$ and $G$, we observe that $H$ admits a translation-like action on $G$ implies there is a regular map, which was introduced in Benjamini, Schramm and Tim\'{a}r's joint paper, from $H$ to $G$. Combining with several known obstructions to the existence of regular maps, we have various applications. For example, we show that the Baumslag-Solitar groups do not admit translation-like actions on the classical lamplighter group. Read More


We prove the almost sure invariance principle for H\"older continuous observables on Young towers with exponential tails with rate $o(n^{1/p})$ for every $p$. As a part of our method, we show that Young towers can always be constructed to have zero distortion. Read More


In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting $\Gamma \times S^1$-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted $\Gamma\times S^1$-degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group $\Gamma=D_8$. Read More


In this paper we prove that translation structures for which the corresponding vertical translation flows is weakly mixing and disjoint with its inverse, form a $G_\delta$-dense set in every non-hyperelliptic connected component of the moduli space $\mathcal M$. This is in contrast to hyperelliptic case, where for every translation structure the associated vertical flow is isomorphic to its inverse. To prove the main result, we study limits of the off-diagonal 3-joinings of special representations of vertical translation flows. Read More


Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by planar orientation preserving homeomorphisms with linear diagonalizable conjugating element. That is, we consider orientation preserving homeomorphisms $f, h: \mathbb{R}^2 \to \mathbb{R}^2$ such that $h f h^{-1} = f^n$, for some $ n\geq 2$ and we assume that $h$ is a linear diagonalizable transformation. Read More


In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Read More


We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We prove more generally that this result holds for all products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S. We deduce a W* strong rigidity theorem for irreducible actions of such product groups and we prove strong solidity of the associated locally compact group von Neumann algebras. Read More


In [2], the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov diffeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmueller polynomial corresponding to those surface diffeomorphisms. Read More


We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\geq 2$. According to \cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\Lambda$ of the action, included in $Fix(f)$. Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\Lambda$ and any point $x\in\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Read More


We study rational step function skew products over certain rotations of the circle proving ergodicity and bounded rational ergodicity when rotation number is a quadratic irrational. The latter arises from a consideration of the asymptotic temporal statistics of an orbit as modelled by an associated affine random walk. Read More


In this article we describe some new examples of correspondence between Diophantine approximation and homogeneous dynamics, by characterizing two kinds of exceptional orbits of geodesic flow associated with the Modular surface. The characterization uses a two parameter family of continued fraction expansion of endpoints of the lifts to the hyperbolic plane of the corresponding geodesics. Read More


We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathbb{C}_p$. Each such function has a unique fixed point. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. Read More


For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, \omega)$ over a domain $U \subset \mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that the domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^\infty(U)$-module structure of $A^2(U, \omega)$. Read More


As defined by W. Thurston, the core entropy of a polynomial is the entropy of the restriction to its Hubbard tree. For each d >= 2, we study the core entropy as a function on the parameter space of polynomials of degree d, and prove it varies continuously both as a function of the combinatorial data and of the coefficients of the polynomials. Read More


We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. Read More


In this paper we present a new class of dynamical system without equilibria which possesses a multi scroll attractor. It is a piecewise-linear (PWL) system which is simple, stable, displays chaotic behavior and serves as a model for analogous non-linear systems. We test for chaos using the 0-1 Test for Chaos of Ref. Read More


We describe the shrinking target set for the Bedford-McMullen carpets, with targets being either cylinders or geometric balls. Read More


In this paper we explore a characterisation of relative equilibria in terms of sectional curvatures of the Jacobi-Maupertuis metric. We consider the planar $N$-body problem with an attractive $1/r^{\alpha}$ potential for general masses. Let $q(t)$ be a relative equilibria, we show that the sectional curvature is zero along $q(t)$, for a certain set of planes containing $\dot{q}(t)$, if and only if $\alpha=2$. Read More


We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Read More


We prove the existence of automorphisms of $\mathbb C^2$ having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times \mathbb C^\ast$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Read More


The concept of dynamical compensation has been recently introduced to describe the ability of a biological system to keep its output dynamics unchanged in the face of varying parameters. Here we show that, according to its original definition, dynamical compensation is equivalent to lack of structural identifiability. This is relevant if model parameters need to be estimated, which is often the case in biological modelling. Read More


In this article, we use $\lambda$-sequences to derive common fixed points for a family of self-mappings defined on a complete $G$-metric space. We imitate some existing techniques in our proofs and show that the tools emlyed can be used at a larger scale. These results generalise well known results in the literature. Read More


D. Ruelle considered a general setting where he is able to describe a formulation of the concept of Gibbs state based on conjugating homeomorphism in the so called Smale spaces. On this setting he shows a relation of KMS states of $C^*$-algebras and equilibrium probabilities of Thermodynamic Formalism. Read More


Chemical reactions modeled by ordinary differential equations are finite-dimensional dissipative dynamical systems with multiple time-scales. They are numerically hard to tackle -- especially when they enter an optimal control problem as "infinite-dimensional" constraints. Since discretization of such problems usually results in high-dimensional nonlinear problems, model (order) reduction via slow manifold computation seems to be an attractive approach. Read More


Message-passing methods provide a powerful approach for calculating the expected size of cascades either on random networks (e.g., drawn from a configuration-model ensemble or its generalizations) asymptotically as the number $N$ of nodes becomes infinite or on specific finite-size networks. Read More


In this paper, we define the notion of reconstruction that maps the dynamic model of a mass action chemical reaction network (CRN) to another system through a positive diagonal matrix. These two network systems will share some common properties, including the set of equilibria and stability behavior (if some moderate conditions also hold). We prove that for those mass action systems having complex balanced reconstructions, the local asymptotic stability of equilibrium points can be reached if the initial state is selected judiciously and the generalized Gibbs' free energy is defined acting as the Lyapunov function. Read More


We extend the classical Fuller index, to prove what may be considered to be one "corrected" version of the original Seifert conjecture, on the existence of periodic orbits for vector fields on $S ^{2k+1} $. We also give some generalizations and extensions of this theorem for certain other smooth manifolds. As one application we show that for a given smooth non-singular vector field $X _{1} $ on $S^{2k+1} $ with no periodic orbits, and any homotopy $\{X _{t} \}$ of smooth non-singular vector fields, with $X _{0} $ the Hopf vector field, $\{X _{t}\}$ has a sky catastrophe, which is a kind bifurcation originally discovered by Fuller. Read More


We consider the stability of nonlinear traveling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. Read More


The main goal of the paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear memory function monomials of different orders (from zero up to the chain order). Read More


We show in this paper a sufficient condition for the existence of solution, the synchronized and the periodic locked state in abstract mean field models or interconnected systems. This condition is true for a small perturbation independently of the number of oscillators. We show in addition a numerical example of linear mean field system. Read More


We consider the problem of a slender rod slipping along a rough surface. Painlev\'e showed that the governing rigid body equations for this problem can exhibit multiple solutions (the {\it indeterminate} case) or no solutions at all (the {\it inconsistent} case), provided the coefficient of friction $\mu$ exceeds a certain critical value $\mu_P$. Subsequently G\'enot and Brogliato proved that, from a consistent state, the rod cannot reach an inconsistent state through slipping. Read More


The plant hormones brassinosteroid (BR) and gibberellin (GA) have important roles in a wide range of processes involved in plant growth and development. The BR signalling pathway acts by altering the phosphorylation state of its transcription factors BZR1/2, whereas the GA signalling pathway acts by reducing the stability of its transcription factor DELLA. Both signalling pathways include a negative feedback, with high levels of BR causing increased repression of key BR-biosynthetic genes mediated by BZR1/2, and high levels of GA causing decreased stability of DELLA, where DELLA is responsible for activating key genes involved in GA biosynthesis. Read More


Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time-parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy. Read More


We study partially hyperbolic sets of C1-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations. A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely. Read More


2017Mar
Affiliations: 1Faculty of Engineering University of Rijeka, Croatia, 2Faculty of Engineering University of Rijeka, Croatia, 3Faculty of Mechanical Engineering and Mathematics, University of California, Santa Barbara

For every non-autonomous system, there is the related family of Koopman operators $\mathcal{K}^{(t,t_0)}$, parameterized by the time pair $(t,t_0)$. In this paper we are investigating the time dependency of the spectral properties of the Koopman operator family in the linear non-autonomous case and we propose an algorithm for computation of its spectrum from observed data only. To build this algorithm we use the concept of the fundamental matrix of linear non-autonomous systems and some specific aspects of Arnoldi-like methods. Read More


We propose and analyze a heterogenous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method (SAM) suggested here may provide approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a computational effort that grows like \(H^{-1}\) (the inverse of the stepsize), uniformly in the forcing frequency \(\Omega\). Read More


Abelian cellular automata (CA) are CA which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images weak *-converge towards the uniform Bernoulli measure (the Haar measure in this setting). We are interested in structural phenomena, i. Read More


Let $(X,\mathfrak{B},\mu)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\nu$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast\nu \rightarrow \mu$ in the weak-$\ast$ topology. Read More


In this paper we provide a method to generate a continuum of limit cycles using a single precomputed exponentially stable limit cycle designed within the Hybrid Zero Dynamics framework. Guarantees for existence and stability of these limit cycles are provided. We derive analytical constraints that ensure boundedness of the state under arbitrary switching among a finite set of limit cycles extracted from the continuum. Read More