This note studies the asymptotic behaviour of linear and nonlinear stochastic difference equations whose structure is motivated by a financial market model. The asymptotic results show that the models can produce behaviour consistent with random walk efficient markets as well as bubbles or crashes.

This contribution investigates the discrete analogue of the Emden-Fowler differential equation. The main attention is paid to the asymptotic behavior of solution of this equation. This work takes up again the research started in paper [4], where the authors have already presented some asymptotic estimates of the solution of the considered equation. The aim of this paper is to improve these estimates.

The paper deals with the existence of the so-called intermediate solutions for the nonlinear difference equation with deviating argument. The roles of the nonlinearity and deviating argument are discussed and illustrated by examples.

Recent results in the oscillation theory of symplectic difference systems are presented. A particular attention is devoted to Sturmian and oscillation theorems for these systems and for the so-called reversed symplectic systems. Some open problems and conjectures are formulated.

In this paper, we discuss the existence of periodic solutions of the periodic difference equation $$ x(n + 1) = f(n, x(n)),\ \ n \in \mathbf{Z} $$ and the periodic difference equation with finite delay $$ x(n + 1) = f(n, x_n),\ \ n \in \mathbf{Z}, $$ where $x$ and $f$ are $d$-vectors, and $\mathbf{Z}$ denotes the set of integers. We show the existence of periodic solutions by using Browder's fixed point theorem, and illustrate an example by using a boundedness result due to Shunian Zhang.

The existence of almost periodic solutions of a nonlinear Volterra difference equation with infinite delay is obtained by using some restrictive conditions of the equation.

We are concerned with dynamic optimization processes from a viewpoint of Golden optimality. A path is called Golden if any state moves to the next state repeating the same Golden section in each transition. A policy is called Golden if it, together with a relevant dynamics, yields a Golden path. The problem is whether an optimal path/policy is Golden or not. This paper minimizes a quadratic criterion and maximizes a square-root criterion over an infinite horizon. We show that a Golden path is optimal in both optimizations. The Golden optimal path is obtained by solving a corresponding Bellman equation for dynamic programming. This in turn admits a Golden optimal policy.

A discrete-time predator-prey model with an age structure in the prey population in which both the predator and the adult prey can consume juvenile prey population is proposed. The fecundity of the adult prey population is assumed to be a constant and the juvenile survival probability is density dependent on its own population size. It is shown that both populations go to extinction if the inherent net reproductive number of the prey is less than 1. Both populations can survive if the prey's inherent net reproductive number is larger than one and the predator's reproductive number is greater than 1 when the prey population is stabilized in the steady state fashion.

Sufficient conditions have been recently given for a class of ergodic maps of an interval onto itself: $I= [0, 1] \subset R \to I$ and its associated binary function to generate a sequence of independent and identically distributed (i.i.d.) binary random variables. Jacobian elliptic Chebyshev map, its derivative and second derivative induce Jacobian elliptic space curve. A mapping of the space curve with its coordinates, e.g., $X$, $Y$ and $Z$, onto itself is introduced which defines 3 projective onto mappings, represented in the form of rational functions of $\{x_n, y_n, z_n\}_{n=0}^{\infty}$. Such mappings with their absolutely continuous invariant measures as functions of elliptic integrals and their associated binary function can generate a 3-dimensional sequence of i.i.d. binary random vectors.

In this paper, we explore the effects of temporary policy in a dynamic general equilibrium by using Neumeyer and Yano's money-in-utility model. Even under dynamic interaction between consumers through markets, the impact of temporary policy both on the present and future economy is very small if the long-run interest rate level is low, which coincides with the intuitive explanation by the permanent income hypothesis.

Using Markov partitions and algebraic graph theory we introduce, in the context of discrete dynamical systems, some laws which characterize the nonlinear dynamics of iterated maps of the interval. In the Markov digraphs we assume that each directed edge has a weight associated to it, given by the Markov invariant measure. This system of weights produces a diffusion process determined by a transition matrix. In this setting, we define a current and a potential which are dynamical invariants.

We investigate global behavior of solutions of a nonlinear difference system $$ x_{n+1} = px_n (1 + y_n),\quad y_{n+1} = qy_n (1 + x_n),\quad n = 0, 1, 2, \dots, $$ where parameters $p$, $q$ and initial values $x_0$, $y_0$ are positive. We give sufficient conditions for every solution of the system to be unbounded and sufficient conditions for the global stable manifold of the positive equilibrium to exist, which is a unbounded separatrix for the system. Some related conjectures are also given.

In this paper we obtain some necessary and sufficient conditions for the asymptotic stability of the zero solution of a delay difference system $$ x_n = A (x_{n-k} + x_{n-k-l}),\quad n = 0, 1, 2, \dots, $$ where $A$ is a $2 \times 2$ real constant matrix and $k$ and $l$ are positive integers.

We study the two-sector economy with sector-specific externality following Benhabib, Nishimura and Venditti [2002]. We focus on the external effect of capital-labor ratio and provide the characterization of equilibrium paths in the case Benhabib, Nishimura and Venditti did not discuss explicitly.

The main aim of this paper is to analyse the dynamics of nonlinear discrete-time maps generated by duopoly games in which players have homogeneous expectations and heterogenous nonlinear cost functions. This framework leads to reaction functions that are non-monotonic and asymmetric and, in the particular case of naïve expectations, the model takes the form of an anti-triangular map, $T(x, y) = (f(y), g(x))$ characterized by a rich dynamical behavior, from stable to chaotic Nash equilibria. We also present the computation of topological entropy of this nonlinear Cournot model by using tools from symbolic dynamics and tensor products.

This paper develops a two–period overlapping generation model to examine the behavior of an economy that incorporates endogenous labor–care choice. Assuming a log–linear utility function and a Cobb–Douglas production function, we show that there exists multiple equilibria, comprising a unique trajectory satisfying saddle–path stability and other equilibria, which have an infinite number of converging transition paths.

We establish some results on $\rho$-stabilities of the zero solution of a system of linear Volterra difference equations. Furthermore, for the equations with nonnegative coefficients we give an easy criterion for the $\rho$-UAS of the zero solution.

In this paper, we obtain the necessary and sufficient condition for the asymptotic stability of the linear delay difference equation $$ x_{n+1} - x_{n-1} + p \sum_{j=1}^{N} x_{n-k+(j-1)l} = 0 $$ where $n = 0, 1, 2, \dots,\ p$ is a real number, and $k$, $l$, and $N$ are positive integers such that $k \gt (N - 1)l$.

We consider a two-sector economy with Cobb-Douglas technologies, labor-augmenting global external effects and increasing social returns. We prove the existence of a normalized balanced growth path and we give conditions for the occurrence of sunspot fluctuations that are compatible with both types of capital intensity configuration at the private level provided the elasticity of intertemporal substitution in consumption admits intermediary values. We finally show that the existence of period-two cycles requires the consumption good to be physical capital intensive at the private level.

We obtain some oscillation theorems for linear delay dynamic equations on a time scale. We illustrate the results by a number of examples.

In this paper, we present the strong and the variational form of a dynamic contact problem with friction. We derive the result and obtain an incremental formulation by space discretization with finite element method and by time discretization with finite difference method of the dynamic contact problem. As well we describe the solution strategies for spatially discrete system and the condition for stability of the solution. Difficulties caused by the discontinuity of the Coulomb's friction law (passage from sliding to adhesion), are tried to be passed over using the Newton-Raphson iterative techniques, and their success depends also on the small value of the parameter, from the regularized friction law.

Using an invariant manifold theorem we demonstrate that the dynamics of nonautonomous dissipative delayed difference equations (with delay $M$) is asymptotically equivalent to the long-term behavior of an $N$-dimensional first order difference equation (with $N \leq M)$ – assumed the nonlinearity is small Lipschitzian on the absorbing set. As consequence we obtain a result of Kirchgraber that multi-step methods for the numerical solution of ordinary differential equations are essentially one-step methods, and generalize it to varying step-sizes.

We establish oscillation criteria for the linear dynamic equation $(r(t)y^{\Delta})^{\Delta} + p(t) y^{\sigma} = 0$. These criteria can be understood as an extension of the classical Willett criterion. What is special on these new results is that the constant involved in the criteria, which is equal to the "magic" 1/4 in the differential equations case, is in fact no more constant. In general case, it depends on the asymptotic behavior of the coefficients $p$, $r$, and primarily on the asymptotic behavior of graininess. In addition, we prove that the value of this new "constant" is the best possible.

The paper concerns necessary and sufficient conditions on the fading intensity of a state–independent stochastic perturbation for the asymptotic stability of a linear stochastic Volterra difference equation. In broad terms, it is shown here that the results obtained in the deterministic case are robust to fading stochastic perturbations which are independent of the state, once it is known that these perturbations fade more rapidly than an identifiable critical rate.

The continuous two-dimensional Lotka-Volterra competition model is converted to a discrete version using a noncanonical symplectic numerical method. The local stability of the differential equations and difference equations are analyzed and compared. We found that the numerical method preserves the local dynamics of the continuous model. The local stability criteria are the same between the continuous model and the discrete model. The discrete-time model is dynamically consistent with its continuous counterpart.

In this paper we discuss the Morishima's example, which implies a kind of eventually asymptotical stability of solutions for a difference equation $x(n + 1) = f(x(n))$ for $n = 0, 1, 2, \cdots$. We define new definitions of eventual stability of periodic points in the meaning of the large in the same way as ones of Lakshmikantham et. al. and Yoshizawa. By applying the Lyapunov's second method we give eventual stability criteria in the large of the difference equation. In order to illustrate our main results on eventual stability an example of a set of 2-periodic points for eventual stability is given with an analytical estimation.

An algorithm is proposed for determining the periodic behavior of the common driver of a system of nonautonomous difference equations from observations of the equation trajectories only. Methods of attractor reconstruction are used to build a semiconjugacy to a topological version of the driver system. The algorithm is described in detail and implemented on several examples.

We discuss the occurrence and nature of periodic and chaotic behavior in a class of nonlinear second order difference equations and present criteria for the attractivity of the periodic solutions.

The migration into a natural population of a controlled population, e.g., a transgenic population, is studied using a one island selection-migration model. A 2-dimensional system of nonlinear difference equations describes changes in allele frequency and population size between generations. Biologically reasonable conditions are obtained which guarantee the existence and uniqueness of a polymorphic equilibrium in the cases of complete dominance and no dominance in fitness. This model may provide some useful information about the migration of transgenes into a natural population.

In this paper we consider the May's host parasitoid equation, $$ \tag{1} x_{n+1} = \frac{ax_n^2}{(1 + x_n) x_{n-1}}, \alpha \gt 1. $$ We show that with initial conditions $x_{-1} = x_0 = 1$ there are values of $\alpha$ giving periodic solutions of prime period $n$ for all integers $n \geq 7$. There are no non-equilibrium periodic solutions of periods 2, 3, 4, 5 or 6.

We use some basic results from the theory of variational methods to prove the exisetnce and uniqueness of periodic solutions to second-order nonlinear discrete problems. Our method has no continuous counterpart since it is based on the finite dimension of the corresponding function space. Our main tools include matrix theory and the Fundamental theorem of calculus for Lebesgue Integral.

We usually discuss the time evolution from the present to the future or from the past, precisely, from some fixed initial time in the past, to the present or to the future. But we sometimes consider the time evolution from the remote past to the remote future as in the theory of stationary stochastic processes or dynamical systems. In the present paper we consider time evolutions governed by noise driven automorphism on locally compact abelian groups and give a necessary and sufficient condition for the time evolution to admit remote past. It turns out that to admit the remote past is fairly restrictive.

This paper studies the Friedman rule on the optimal quantity of money in a money in the utility model with habit formation. If habits are formed only in consumption, or if habit formations in both consumption and real money holdings are symmetric, the Friedman rule is optimal. However, if habit forming is assumed in real money holdings, then the Freidman rule may not be optimal.

In this paper we study the bifurcation of subharmonic tori from branches of symmetric relative equilibria in reversible systems with an additional $S^1$-symmetry. The analysis is based on a detailed study of the Poincaré map, which appears to be generated by a reduced vectorfield on the section. The results are applied to a simple example system, but the bifurcation behaviour described by our theoretical results can also be observed in, for example, the spherical pendulum and the Furuta pendulum.

Hantaviruses are a group of viruses that infect wild rodents without causing any apparent illness or disease. New discrete-time models for the spread of hantavirus in a rodent population are formulated and analyzed. The models are structured by the stages of the infection, the stages of development, and the sex of the rodent. The basic reproduction number $\mathcal{R}_0$ is computed for the deterministic model and a condition is given for a simplified model with males only to be permanent. A stochastic model is also formulated. Numerical simulations illustrate the differences between the deterministic and stochastic models and the dynamics in the male and female rodents. It is shown, in the numerical examples, that a transcritical bifurcation occurs at $\mathcal{R}_0 = 1$ and a unique enzootic equilibrium exists when $\mathcal{R}_0 \gt 1$. The sensitivity of the equilibrium values to changes in the parameters is also investigated.