A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics

We establish a version of the Grobman-Hartman theorem in Banach spaces for nonuniformly hyperbolic dynamics. We also consider the case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. More precisely, we consider sequences of Lipschitz maps Am+fm such that the linear parts Am admit a nonuniform exponential dichotomy, and we establish the existence of a unique sequence of topological conjugacies between the maps Am+fm and Am. Furthermore, we show that the conjugacies are Hölder continuous, with Hölder exponent determined by the ratios of Lyapunov exponents with the same sign. To the best of our knowledge this statement appeared nowhere before in the published literature, even in the particular case of uniform exponential dichotomies, although some experts claim that it is well known in this case. We are also interested in the dependence of the conjugacies on the perturbations fm: we show that it is Hölder continuous, with the same Hölder exponent as the one for the conjugacies. We emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial. In particular, we need to consider several additional Lyapunov norms.     

A homogenous random fractal model for time series produced by constant energy molecular dynamics simulations

In the present work we present an analysis of time series of instantaneous temperature and pressure produced during microcanonical (constant energy) molecular dynamics (MD). Simulations were applied to a nickel oxide grain boundary for a temperature range from about 0.15T_m up to about 0.80T_m, T_m being the melting point of the system. We performed a series of analysis for these time series including test for randomness, power spectrum, Hurst exponent, structure function test and test for multifractality. The obtained results show evidence of an homogenous random fractal model. Pressure time series presents 1/f^α noise over the whole range of frequencies of the system while temperature time series presents a white noise behavior. The origins of this observed behavior are discussed. A comparison also is made with results already obtained from constant temperature MD where the temperature time series present a two-regime behavior: white noise at low frequencies and 1/f^α at high frequencies with α increasing as a function of temperature. The origins of this difference in the behavior are discussed.     

The dynamics of pest control pollution model with age structure and time delay

In this paper, by using pollution model and impulsive delay differential equation, we investigate the dynamics of a pest control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. We assume only the pests are affected by pesticide. We show that there exists a global attractive pest-extinction periodic solution when the periodic natural enemies release amount μ₁ and pesticide input amount μ₂ are larger than some critical value. Further, the condition for the permanence of the system is also given. By numerical analyses, we also show that constant maturation time delay, pulse pesticide input and pulse releasing of the natural enemies can bring obvious effects on the dynamics of system. We believe that the results will provide reliable tactic basis for the practical pest management.     

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Nonexistence of Markovian time dynamics for graphical models of correlated default

Filiz et al. (in arXiv:0809.1393 (2008)) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph G=(V,E), where the vertices V correspond to the firms and the edges E describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be rewritten as a standard Ising model on the graph by identifying the set of defaulting firms in the default model with the set of sites in the Ising model for which the {±1}-valued spin is +1. We ask whether there is a suitable continuous-time Markov chain (X _t ) _t??0 taking values in the subsets of V such that X ₀=?, X _r ?X _s for r??s (that is, once a firm defaults, it stays in default), the distribution of X _T for some fixed time T is the one given by the default model, and the distribution of X _t for other times t is described by a probability distribution in the same family as the default model. In terms of the equivalent Ising model, this corresponds to asking if it is possible to begin at time 0 with a configuration in which every spin is ?1 and then flip spins one at a time from ?1 to +1 according to Markovian dynamics so that the configuration of spins at each time is described by some Ising model and at time T the configuration is distributed according to the prescribed Ising model. We show for three simple but financially natural special cases that this is not possible outside of the trivial case where there is complete independence between the firms.     

Global analysis of an environmental disease transmission model linking within-host and between-host dynamics

In this paper, a multi-scale mathematical model for environmentally transmitted diseases is proposed which couples the pathogen-immune interaction inside the human body with the disease transmission at the population level. The model is based on the nested approach that incorporates the infection-age-structured immunological dynamics into an epidemiological system structured by the chronological time, the infection age and the vaccination age. We conduct detailed analysis for both the within-host and between-host disease dynamics. Particularly, we derive the basic reproduction number R₀ for the between-host model and prove the uniform persistence of the system. Furthermore, using carefully constructed Lyapunov functions, we establish threshold-type results regarding the global dynamics of the between-host system: the disease-free equilibrium is globally asymptotically stable when R₀ < 1, and the endemic equilibrium is globally asymptotically stable when R₀ > 1. We explore the connection between the within-host and between-host dynamics through both mathematical analysis and numerical simulation. We show that the pathogen load and immune strength at the individual level contribute to the disease transmission and spread at the population level. We also find that, although the between-host transmission risk correlates positively with the within-host pathogen load, there is no simple monotonic relationship between the disease prevalence and the individual pathogen load.     

Coarse-grained entropy in dynamical systems

Let M be the phase space of a physical system. Consider the dynamics, determined by the invertible map T: M → M, preserving the measure µ on M. Let ν be another measure on M, dν = ρdµ. Gibbs introduced the quantity s(ρ) = ?∝ρ log ρdµ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (fine-grained) entropy the so called coarse-grained entropy. First we obtain a formula for the difference between the coarse-grained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information. Then we consider the behavior of the coarse-grained entropy as a function of time. The dynamics transforms ν in the following way: ν → ν _n , dν _n = ρ ○ T ^?n dµ. Hence, we obtain the sequence of densities ρ _n = ρ ○ T ^?n and the corresponding values of the Gibbs and the coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map T. Finally, we give numerical calculation of the coarse-grained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements.     

Ordered K-groups associated to substitutional dynamics

The Matsumoto K₀-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C^∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.     

Actions of large semigroups and random walks on isometric extensions of boundaries

Let G be a real algebraic semi-simple group, X an isometric extension of the flag space of G by a compact group C. We assume that G is topologically transitive on X. We consider a closed sub-semigroup T of G and a probability measure μ on T such that T is Zariski-dense in G and the support of μ generates T. We show that there is a finite number of T-invariant minimal subsets in X and these minimal subsets are the supports of the extremal μ-stationary measures on X. We describe the structure of these measures, we show the conditional equidistribution on C of the μ-random walk and we calculate the algebraic hull of the corresponding cocycle. A certain subgroup generated by the “spectrum” of T can be calculated and plays an essential role in the proofs.     

The timing and deterrence of terrorist attacks due to exogenous dynamics

In this paper, we develop a model for the timing and deterrence of terrorist attacks due to exogenous dynamics. The defender moves first and the attacker second in a two-stage game which is repeated over T periods. We study the effects of dynamics of several critical components of counter-terrorism games, including the unit defence costs (eg, immediately after an attack, the defender would easily acquire defensive funding), unit attack costs (eg, the attacker may accumulate resources as time goes), and the asset valuation (eg, the asset valuation may change over time). We study deterministic dynamics and conduct simulations using random dynamics. We determine the timing of terrorist attacks and how these can be deterred.     

p-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ? _p , the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.     

Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

We study increasing sequences of positive integers (nk)_k?1 with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with supk?1‖Tnk‖<∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (nkα)_k?1, α∈R, or on the growth of nk+1/nk. Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C₀-semigroups is also discussed.     

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 ? β)]^?1 n log n. For β = 1, we prove that the mixing time is of order n ^3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).     

Quantum model for the price dynamics: The problem of smoothness of trajectories

We apply methods of quantum mechanics to mathematical modelling of price dynamics in a financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Our model is a quantum-like model of the financial market, cf. with works of W. Segal, I.E. Segal, E. Haven. In this paper we study the problem of smoothness of price-trajectories in the Bohmian financial model. We show that even the smooth evolution of the financial pilot wave ψ(t,x) (representing expectations of traders) can induce jumps of prices of shares.     

Generalized Browder's and Weyl's theorems for Banach space operators

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of σ(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f∈H((T)), the space of functions analytic on an open neighborhood of σ(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f∈H(σ(T)).     

Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain

In this paper, we study the global dynamics of a class of differential equations with temporal delay and spatial non-locality in an unbounded domain. Adopting the compact open topology, we describe the delicate asymptotic properties of the nonlocal delayed effect and establish some a priori estimate for nontrivial solutions which enables us to show the permanence of the equation. Combining these results with a dynamical systems approach, we determine the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker?s birth function and Mackey-Glass?s hematopoiesis function, we obtain threshold results for the global dynamics of these two models. We explain why our results on the global attractivity of the positive equilibrium in C₊?{0} under the compact open topology becomes invalid in C₊?{0} with respect to the usual supremum norm, and we identify a subset of C₊?{0} in which the positive equilibrium remains attractive with respect to the supremum norm.     

Uniformity and inexact version of a proximal method for metrically regular mappings

We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0∈T(x).     

Approximation with the output of linear control systems

In this paper we give a new proof that for controllable and observable linear systems every L²[0,T] function can be approximated in the L²[0,T] sense with an output function generated by an L²[0,T] input function. We also give a new characterization of how continuous functions on [0,T] are uniformly approximated by an output generated by a continuous input function. The relative degree of the transfer function of the system determines those functions that can be approximated. We further show that if the initial data is allowed to vary then every continuous function is uniformly approximated by outputs generated by continuous functions.