Difference Equations, Discrete Dynamical Systems and Applications

In this paper, we study the Beverton–Holt equation with periodic inherent growth rate and periodic carrying capacity in the quantum calculus time setting. After a brief introduction to quantum calculus, we solve the Beverton–Holt q-difference equation using the logistic transformation. This leads to a linear q-difference equation where the solution is obtained using variation of parameters. The analysis of the solution aids our investigation of the first and second Cushing–Henson conjectures under the assumption of a periodic growth rate and a periodic carrying capacity. The first Cushing–Henson conjecture holds in the classical sense, which guarantees the existence of a unique periodic solution which is globally attractive. The analysis of the average of the unique periodic solution of the Beverton–Holt q-difference equation yields formulations of modified second Cushing–Henson conjectures.

In 1998 at the ICDEA Poznan the first author talked about pullback attractors of nonautonomous difference equations. That talk was published as [7] in the Journal of Difference Equations & Applications in 2000. Since then the theory of nonautonomous dynamical systems has been the topic of many papers and there are some new developments, in particular concerning the construction of forward nonautonomous attractors, that will be discussed here.

Sacker and Sell showed that if the linear skew product flow generated by a diffeomorphism has the so-called no nontrivial bounded solution property and if the diffeomorphism is chain recurrent on the underlying invariant set in the manifold, then this underlying set must be hyperbolic. The purpose of this note is to point out (what in fact may be well-known but the author has never seen it in print) that the assumption of chain recurrence is not necessary provided the assumption of no nontrivial bounded solution property is strengthened. This additional assumption is more or less the same as the analytic strong transversality property proved by Mañé to be equivalent to structural stability. Actually the result proved here is largely implicit in the results of Sacker and Sell but they did not state such a result explicitly. The proof here uses different techniques from those of Sacker and Sell, being based on ideas of Coppel.

For a class of piecewise linear maps on T on [0, 1] and \(0\le a<b\le 1\), we consider the invariant set \(T_{a,b}:=\bigcap \limits _{n=0}^{\infty }T^{-n}[a,b]\). We obtain a sufficient condition under which the Hausdorff dimension of the set \(T_{a,b}\) is locally constant.

We investigate the relationship between oscillatory properties of half-linear even order difference equations and nonpositivity of the associated energy functionals. We convert the investigated difference equation into a Hamiltonian type difference system and using this transformation we establish our main result which says that the existence of two (or more) generalized zeros of a solution of the investigated difference equation implies that the corresponding energy functional attains a nonpositive value.

The purpose of this paper is to provide an overview of important results in the oscillation and spectral theory of discrete symplectic systems with nonlinear dependence on the spectral parameter, which were recently introduced by the author. In addition, we derive new comparison theorems for finite eigenvalues of such systems.

Cookie-cutter-like dynamic system is an important tool in study of dimensional properties of Sturm Hamiltonian. We prove that for the Cookie-cutter-like dynamic system with unbounded expansion, the properties such as bounded variation, bounded distortion, existence of Gibbs-like measure still hold.

Known as the plague of the Internet age, malware causes mass economic losses. A computer worm is a kind of stand-alone malware which spreads itself to neighboring nodes by exploiting vulnerabilities. Computer worms are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue. In order to establish one such here, the nodes on the network are divided into three compartments: susceptible nodes (S), latent nodes (L) and breaking-out nodes (B). By the compartment method, a discrete model of computer worm prevalence is established. This model includes a reintroduction parameter which models the users’ security awareness. This is a more realistic model of computer worm spread than the ones in literature, and it can be used to understand the influence of security awareness on the propagation of computer worms. To be specific, the dynamics of this model is analyzed by use of the stability theory concerning difference equations. First, the basic reproduction number determining the behavior of worm propagation on the network is calculated. Then, the asymptotic stability of the worm-free equilibrium is proved if the threshold is below unity. Finally, the asymptotic stability of the worm equilibrium is shown by numerical simulations provided the threshold exceeds unity.

In this paper, we study the existence and uniqueness of almost periodic solutions for a class of neutral functional dynamic systems in the sense of Stepanov, that is, it is not necessary to restrict our system to be continuous. Since the discussions aim at dynamic systems, the present paper will involve the Lebesgue measure and Lebesgue integral functions of time scales.

In this paper we present sufficient conditions for the oscillation of all solutions of a linear delay partial difference equation with impulses.

Theoretical framework and an appropriate algorithm is developed to measure the nonstationarity (NS) of data streams. With the nonstationary measure, the properties of stock returns are studied. Three experiments illustrate that: the nonstationarity of stock return can not be diversified with big portfolio; nonstationarity, which can explain the risk premium, is positively related to the investing period.

Convertible bonds are popular financial vehicles for companies and attracting more and more attention. Default risks and Compound stochastic processes with jumps in interest rates make it hard to price convertible bonds and their derivatives. A novel mechanism is introduced to price a compound option on convertible bonds with default risks and stochastic interest rates.

When \(G=\mathbb {Z}^d\), we show that if (X, G) is disjoint from all minimal systems and transitive, then (X, G) is a weakly mixing M-system without nontrivial minimal factor. Moreover, we show that if (X, G) is weakly mixing with dense distal points and with G being Abelian, then (X, G) is disjoint from all minimal systems. These generalize some related results in the case of \(G=\mathbb {Z}\).