In this paper we develop boundary value methods for detecting Sacker-Sell spectra in discrete time dynamical systems. The algorithms are advancements of earlier methods for computing projectors of exponential dichotomies. The first method is based on the projector residual P 2 − P . If this residual is large, then the difference equation has no exponential dichotomy. Further criterions for dete...

The second order hypergeometric q-difference operator is studied for the value c = −q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space l(Z). The operator has deficiency indices (1, 1) and we describe as explicitly as possible the spectral resolutions of the self-adjoint extensions. This gives rise to one-parameter o...

For two-dimensional (2-D) digital filters implemented by a block recursive equation, explicit relations between their frequency characteristics and those of scalar filter are obtained. Specifically, these include the relation between the discrete-time Fourier transform (DTFT) of the block recursive equation and that of the scalar 2-D difference equation, and the relation between the block matri...

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the finite delay Volterra difference equation x(t + 1) = a(t)x(t) + t−1 ∑ s=t−r b(t, s)x(s). Also, by displaying a slightly different Lyapunov functional we obtain conditions that guarantee the instability of the zero solution. The highlight of the paper is relaxing the condi...

The mathematics of time scales has recently received much attention and holds great promise in a number of areas. In this paper we propose a new area of mathematics, namely the theory of stochastic dynamic equations, which unifies the theories of stochastic differential and difference equations. We give an example involving stochastic dynamic equations, namely an equation modeling a stock price...

The lattice Boussinesq equation (BSQ) is a three-component difference-difference equation defined on an elementary square of the 2D lattice, having 3D consistency. We write the equations in the Hirota bilinear form and construct their multisoliton solutions in terms of Casoratians, following the methodology in our previous papers. In the construction it turns out that instead of the usual discr...

The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(x) = a(x)+T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size x. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high proba...

We study the traffic states and fundamental diagram of vehicular traffic controlled by a series of traffic lights using a deterministic cellular automaton CA model. The CA model is not described by a set of rules but is given by a difference equation. The vehicular traffic varies highly with both signal’s characteristics and vehicular density. The dependence of fundamental diagram on the signal...

We consider the system of Volterra integro-dynamic equations x(t) = A(t)x(t) + ∫ t t0 B(t, s)x(s)∆s and obtain necessary and sufficient conditions for the uniform stability of the zero solution employing the resolvent equation coupled with the variation of parameters formula. The resolvent equation that we use for the study of stability will have to be developed since it is unknown for time sca...

We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential stability of the zero solution of the delay difference equation x(t + 1) = a(t)x(t) + b(t)x(t− h). The highlight of the paper is the relaxing of the condition |a(t)| < 1. An instability criteria for the zero solution is obtained. Moreover, we will provide an example, in which we show that our theorems provide...