Discrete Dynamics and Difference Equations

The following sections are included:

• Preface

• Contents

We study some dynamical properties of the Newton maps associated to real quintic polynomial equations of one variable. Using Tschirnhaus transformations and topological conjugation, we suppose the equation has been reduced to the form p_c = x⁵ − cx+1 = 0 where c ∈ ℝ. Then we use symbolic dynamics and in particular the construction of kneading sequences which allow to know the dynamical behavior of the Newton map associated to the map p_c.

We investigate the dynamics of the Nash better response map for a family of games with two players and two strategies. This family contains the games of Coordination, Stag Hunt and Chicken. Each map is a piecewise rational map of the unit square to itself. We describe completely the dynamics for all maps from the family. All trajectories converge to fixed points or period 2 orbits. We create tools that should be applicable to other systems with similar behavior.

This paper is concerned with the attractive cycles of an artificial neural network consisting of n neurons situated on the vertices of a regular polygon and connected by its edges. The weight matrix and the bias are dependent on two paramters α and β. By elementary analysis, we are able to give a complete account on the relations between the parameters and the existence and stability of cycles in these networks.

The goal of this work is to characterize the fine structures of fractals that are in the frontier between order and chaos or whose dynamics are chaotic.

We built basics of the qualitative theory of the continuous-time difference equations x(t + 1) = f(x(t)), t ∈ ℝ⁺, with the method of going to the infinite-dimensional dynamical system induced by the equation. For a study of this system we suggest an approach to analyzing the asymptotic dynamics of general nondissipative systems on continuous functions spaces. The use of this approach allows us to derive properties of the solutions from that of the ω-limit sets of trajectories of the corresponding dynamical system. In particular, typical continuous solutions are shown to tend (in Hausdorff metric for graphs) to upper semicontinuous functions whose graphs are, in wide conditions, fractal; there may exist especially nonregular solutions described asymptotically exactly by random processes. We introduce the notion of self-stochasticity in deterministic systems — a situation when the global attractor contains random functions. Substantiated is a scenario for a spatial-temporal chaos in distributed parameters systems with regular dynamics on attractor: The attractor consists of cycles only and the onset of chaos results from the very complicated structure of attractor “points” which are elements of some function space (different from the space of smooth functions). We develop a method to research into boundary value problems for partial differential equations, that bases on their reduction to difference equations.

We present an adapted version of the classical Lyapunov-Schmidt reduction method to study, for some given integer q ≥ 1, the bifurcation of q-periodic orbits from fixed points in discrete autonomous systems. The approach puts some particular emphasis on the ℤ_q-equivariance of the reduced problem. We also discuss the relation with normal form theory, consider special cases such as equivariant, reversible or symplectic mappings, and obtain some results on the stability of the bifurcating periodic orbits. We conclude with an application of the approach to the generic bifurcation of q-periodic orbits for q ≥ 3, and showing how for q ≥ 5 Arnol'd tongues appear as an immediate consequence of the ℤ_q-equivariance.

We present an one-to-one correspondence between (i) Pinto's golden tilings; (ii) smooth conjugacy classes of golden diffeomorphism of the circle that are fixed points of renormalization; (iii) smooth conjugacy classes of Anosov difeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugated to the Anosov automorphism G(x, y) = (x + y, x); (iv) solenoid functions. As we will explain, the solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences.

We consider stochastic difference equation

We establish results on asymptotic stability and instability of the trivial solution x_n ≡ 0. We also show, that when f and g have polynomial behavior in zero neighborhood, the rate of decay of x_n is approximately polynomial: we find α > 0 such that x_n decays faster than n^−α+ε but slower than n^−α−ε for any ε > 0.

We construct a discrete-time stochastic process that mimics the incidence of finite-time explosion in the solutions of a scalar nonlinear stochastic differential equation.

We associate to cellular automata elementary rules a class of interval maps defined in [0,1]. The considered configuration space is the space of the one-sided sequences in the set {0,1} and with the appropriate choice of the update procedure the interval maps do not depend on the boundary conditions. We study the rule 184 and obtain a family of transition matrices that characterizes the dynamics of the cellular automaton. We show that these matrices can be obtained recursively by an algorithm that depends on the local rule.

We study non-linear difference equations X_n+1 = F(X_n), where X_n belongs to a certain matrix algebra A, and F is a polynomial map defined on A. We are interested in analyzing the type of periodic orbits and their stability. We also will discuss the dependence of the dynamical behavior on parameters in different situations, since we can consider the parameters to be in the algebra A or in some sub-algebra of A. We study the concrete cases when F is a quadratic map and A is M₂(ℝ), or some sub-algebra of M₂(ℝ).

The Beverton–Holt equation, usually treated as a rational difference equation, is shown in fact to be a logistic difference equation. Based on a crucial transformation connected to logistic equations, an elementary proof of the Cushing–Henson conjectures is given.

We establish some sufficient conditions for the global attractivity of the multiplicative difference equation with variable nonnegative ω-periodic positive coefficients

We study the boundedness nature and the periodic character of solutions, of nonautonomous rational difference equations including Pielou's equation.

We derive a threshold value for the coupling strength in terms of the topological entropy, to achieve synchronization of two coupled piecewise linear maps, for the unidirectional and for the bidirectional coupling. We prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the bidirectional coupling of two identical chaotic Duffing equations is given.

Iteration of smooth maps appears naturally in the study of continuous difference equations and boundary value problems. Moreover, it is a subject that may be studied by its own interest, generalizing the iteration theory for interval maps. Our study is motivated by the works of A. N. Sharkovsky et al.^1,3, E. Yu. Romanenko et al.², S. Vinagre et al.⁴ and R. Severino et al.⁵ . We study families of discrete dynamical systems of the type (Ω,f), where Ω is some class of smooth functions, e.g., a sub-class of C^r(J, ℝ), where J is an interval, and f is a smooth map f : ℝ → ℝ. The action is given by φ ↦ f ○ φ. We analyze in particular the case when f is a family of quadratic maps. For this family we analyze the topological behaviour of the system and the parameter dependence on the spectral decomposition of the iterates.

In this work we derive the formula which connects the number of focal points in (0, N + 1] for conjoined bases of symplectic difference systems. The similar result is obtained for the number of focal points in [0, N + 1). We prove that the number of focal points in (0, N + 1] and [0, N + 1) for the principal solutions at i = 0 and i = N + 1 coincides. The consideration is based on the new concept of the comparative index.

We have introduced the notion of conductance in discrete dynamical systems using the known results from graph theory applied to systems arising from the iteration of continuous functions. The conductance allowed differentiating several systems with the same topological entropy, characterizing them from the point of view of the ability of the system to go out from a small subset of the state space. There are several other definitions of conductance and the results differ from one to another. Our goal is to understand the meaning of each one concerning the dynamical behaviour in connection with the decay of correlations and mixing time. Our results are supported by computational techniques using symbolic dynamics, and the tree-structure of the unimodal and bimodal maps.

This paper is to present a model of spatial equilibrium using a nonlinear generalization of Markov-chain type model, and to show the dynamic stability of a unique equilibrium. Even at an equilibrium, people continue to migrate among regions as well as among agent-types, and yet their overall distribution remain unchanged. The model is also adapted to suggest a theory of traffic distribution in a city.

We exploit ideas of nonlinear dynamics in a complex non-deterministic dynamical setting. Our object of study is the observed riverflow time series of the Portuguese Paiva river whose water is used for public supply. The Ruelle-Takens delay embedding of the daily riverow time series revealed an intermittent dynamical behavior due to precipitation occurrence. The laminar phase occurs in the absence of rainfall. The nearest neighbor method of prediction revealed good predictability in the laminar regime, but we warn that this method is misleading in the presence of rain. We present some new insights between the quality of the prediction in the laminar regime, the embedding dimension, and the number of nearest neighbors considered.

In this paper we survey Reid roundabout theorems for time scale symplectic systems (). These theorems list equivalent conditions for the positivity and nonnegativity of the quadratic functional associated with (). The Reid roundabout theorems in this paper do not impose any normality assumption. We also show that Jacobi systems for nonlinear time scale control problems naturally lead to time scale symplectic systems, and that such a system consists of the Hamiltonian equations corresponding to the weak maximum principle for the quadratic functional .

In this paper we consider a two species population model based on the discretisation of the original Lotka-Volterra competition equations. We analyse the global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the case where the interspecific competition is considered to be “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated using the method of Critical Curves.

In this work we discuss the structure of ordinal pattern distributions obtained from orbits of dynamical systems. In particular, we consider the extreme cases of systems with a singular pattern distribution and of realizing each ordinal pattern of any order, respectively. Finally, we review results relating the Kolmogorov-Sinai entropy and the topological entropy of one-dimensional dynamical systems to the richness of the underlying ordinal pattern distribution.

The paper deals with the asymptotic estimate of solutions of a difference equation, which arises as a discretization of pantograph equation with a forcing term. The term with delayed argument is approximated via linear interpolation between the closest mesh points. The derived asymptotic estimate is compared with the estimate corresponding to the continuous counterpart.

The classical theory of k-th order linear functional and difference equations is obtained as a special case of the theory developed here for the k-th order functional equation model which generalizes the first order model f ○ ϕ − Af = B. The function ϕ belongs to a special space S of continuous strictly monotonic functions equipped with a group multiplication ○. The equation's domain can be a finite interval, a half-line or the real line.

Announced here are the main theorems on solution structure for k-th order homogeneous linear functional equations in S.

The Marx model for the profit rate r depending on the exploitation rate e and on the organic composition of the capital k is studied using a dynamical approach. Supposing both e(t) and k(t) are continuous functions of time we derive a law for r(t) in the long term. Depending upon the hypothesis set on the growth of k(t) and e(t) in the long term, r(t) can fall to zero or remain constant. This last case contradicts the classical hypothesis of Marx stating that the profit rate must decrease in the long term. Introducing a discrete dynamical system in the model and, supposing that both k and e depend on the profit rate of the previous cycle, we get a discrete dynamical system for r, r_n+1 = f_a (r_n), which is a family of unimodal maps depending on the parameter a. In this map we can have a fixed point when a is small and, when we increase a, we get a cascade of period doubling bifurcations leading to chaos. When a is very big, the system has again periodic stable orbits of period five and, finally, period three.

Sometimes we obtain attractive results when associating facts to simple elements. The goal of this work is to introduce a possible alternative in the study of the dynamics of rational maps , making some associations with the matrix of its coefficients. Calculating the numerical range W(A), the numerical radii r(A) and , the boundary of the numerical range ∂W(A), powers and iterations, we found relations very interesting, specially with the entropy of this maps.

We consider the difference equations of the form

We classify the set of possible nonoscillatory solutions of the above equations according to their asymptotic behavior. Some oscillation results for equation with maximum function are also obtained.

In this work is studied the oscillatory behavior of the delay differential difference equation of mixed type

The dynamic contact problem in linear viscoelasticity will be discretized with inite element method and finite difference method. The concept of the a priori stability estimation for dynamic frictional contact will be analyzed by using the local split of the Coulomb model. The split will be implemented such that the global balance of energy will be preserved in the case of perfect stick, while in the case of slip an algorithmically consistent approximation will be produced pointwise on the contact interface.

Necessary and sufficient conditions in terms of the integral of the coefficient are derived under which positive decreasing solutions of a half-linear dynamic equation are (normalized) regularly varying of a known index.

We analyze the behavior of the solutions for initial value problems associated to certain fuzzy differential equations, paying special attention to its advantages and disadvantages in relation with the modelization of real phenomena subject to uncertainty and comparing the results obtained with the corresponding results in the ordinary case.

We consider a third order nonlinear difference equation

The sufficient conditions under which for any nonzero constant there exists a solution of the above equation which tends to this constants, are obtained. Also, the sufficient conditions for the existence a solution of the considered equation which can be written in the following form

Here u, v and w are three linearly independent solution of equation

It is known that solutions of certain classes of linear hyperbolic systems with nonlinear boundary conditions and consistent initial conditions can be written via the iteration of a map of the interval. In this work we characterize the solutions of such problems, with a vortex as initial condition and the iteration of a bimodal map of the interval, using the bimodal topological invariants.

In this paper, by solving the (2+1)-dimensional discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for several (2+1)-dimensional differential-difference hierarchies and obtain the corresponding conserved densities and associated fluxes.