On a solvable class of nonlinear difference equations of fourth order

We consider a class of nonlinear difference equations of the fourth order, which extends some equations in the literature. It is shown that the class of equations is solvable in closed form explaining theoretically, among other things, solvability of some previously considered very special cases. We also present some applications of the main theorem through two examples, which show that some results in the literature are not correct

On a family of nonlinear difference equations of the fifth order solvable in closed form

We present some closed-form formulas for the general solution to the family of difference equations $x_{n+1} = \Phi^{-1}\left(\Phi(x_{n-1})\frac{{\alpha} \Phi(x_{n-2})+{\beta} \Phi(x_{n-4})}{{\gamma} \Phi(x_{n-2})+{\delta} \Phi(x_{n-4})}\right),$ for $n\in{\mathbb N}_0$ where the initial values $x_{-j}$, $j = \overline{0, 4}$ and the parameters ${\alpha}, {\beta}, {\gamma}$ and ${\delta}$ are real numbers satisfying the conditions ${\alpha}^2+{\beta}^2\ne 0,$ ${\gamma}^2+{\delta}^2\ne 0$ and $\Phi$ is a function which is a homeomorphism of the real line such that $\Phi(0) = 0,$ generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.

Some open problems in low dimensional dynamical systems

The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years during my research. I believe that it is worth to think about them and, hopefully, it will be possible either to solve some of the problems or to make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials or some recreational problems with a dynamical component

On Global Attractivity of a Class of Nonautonomous Difference Equations

We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given by y n p ry n−s / q φ n y n−1 , y n−2 , . . . , y n−m y n−s , n ∈ N 0 , with p ≥ 0, r, q > 0, s, m ∈ N and positive initial values, and present some sufficient conditions for the parameters and maps φ n : R m → R , n ∈ N 0 , under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al

Robust Adaptive Stabilization of Linear Time-Invariant Dynamic Systems by Using Fractional-Order Holds and Multirate Sampling Controls

This paper presents a strategy for designing a robust discrete-time adaptive controller for stabilizing linear time-invariant LTI continuous-time dynamic systems. Such systems may be unstable and noninversely stable in the worst case. A reduced-order model is considered to design the adaptive controller. The control design is based on the discretization of the system with the use of a multirate sampling device with fast-sampled control signal. A suitable on-line adaptation of the multirate gains guarantees the stability of the inverse of the discretized estimated model, which is used to parameterize the adaptive controller. A dead zone is included in the parameters estimation algorithm for robustness purposes under the presence of unmodeled dynamics in the controlled dynamic system. The adaptive controller guarantees the boundedness of the system measured signal for all time. Some examples illustrate the efficacy of this control strategy