Degenerate difference-differential equations: Algebraic theory

Degenerate KAM theory for partial differential equations

Article history: Received 1 August 2010 Available online 16 November 2010 MSC: 37K55 35L05 This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymp...

Nonstandard finite difference schemes for differential equations

In this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (NSFDs). Numerical examples confirming then efficiency of schemes, for some differential equations are provided. In order to illustrate the accuracy of the new NSFDs, the numerical results are compared with ...

Algebraic Difference Equations

Stability and Bifurcation Analysis of Differential–Difference–Algebraic Equations

This paper treats a nonlinear dynamical system with both continuous-time and discrete-time variables as a differential–difference–algebraic equation (DDA) or a hybrid dynamical system, presents a fundamental analyzing method of such a DDA system for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist four types of generic codimensi...

On the Algebraic Difference Equations

We continue the study of algebraic difference equations of the type un+2un = ψ(un+1), which started in a previous paper. Here we study the case where the algebraic curves related to the equations are quartics Q(K) of the plane. We prove, as in “on some algebraic difference equations un+2un = ψ(un+1) in R∗, related to families of conics or cubics: generalization of the Lyness’ sequences” (2004),...