We describe dependencies in bivariate time series by appearance of equal or similar recurrence patterns. Various recurrence concepts and parameters are discussed. They are applied to detect synchronization in model systems of coupled oscillators and coupling in EEG data.

In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, w...

A new family of generalized Stirling and Bell numbers is introduced by considering powers (V U)n of the noncommuting variables U, V satisfying UV = V U +hV s. The case s = 0 (and h = 1) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they a...

We study almost automorphic solutions of recurrence relations with values in a Banach space V for quasilinear almost automorphic difference systems. Its linear part is a constant bounded linear operator Λ defined on V satisfying an exponential dichotomy. We study the existence of almost automorphic solutions of the non-homogeneous linear difference equation and to quasilinear difference equatio...

We study a system of delay difference equations modeling four-dimensional discrete-time delayed neural networks with no internal decay. Such a discrete-time system can be regarded as the discrete analog of a differential equation with piecewise constant argument. By using semicycle analysis method, it is shown that every bounded solution of this discrete-time system is eventually periodic. The ...

A recurrence relation between equal mass two-loop sunrise diagrams differing in dimensionality by 2 is derived and it's solution in terms of Gauss' 2 F 1 and Appell's F 2 hypergeometric functions is presented. For arbitrary space-time dimension d the imaginary part of the diagram on the cut is found to be the 2 F 1 hypergeometric function with argument proportional to the maximum of the Kibble ...

Consider a stochastic difference equation xi+1 = Gi [xi, xi−1, . . . , x0] + fi [xi, xi−1, . . . , x0] + k0 ∑ k=0 σi−k [xi−k, xi−k−1, . . . , x0] ξi+1−k, (1) with the Volterra type nonlinear main term G and the Volterra type noise f + ∑k0 k=0 σi−kξi+1−k. Functions Gi, fi, σi are random while ξi is a martingale-difference. In general, k0 ≥ 1 so Eqn (1) cannot be considered as a stochastic equati...

We present a version of Opial’s inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential equations and difference equations as special cases. Various extensions of our inequality are offered as well.

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation p t ψ ◦ x x Δ Δ q t f ◦ x 0 on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference...

We characterize, under some technical assumptions and up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. Our method is based on differential equations and the obtained measures are particular Beta distributions. We actually recover the free Meixner family of probability distributions so that our met...