We present the existence of resonance and beats in open and forced chemical oscillatory systems using a superimposed sinusoidal modulation on the inflow rates of the reagents. We demonstrate control over the periodicity of the forced oscillations and show that the time period of beats follows the relation known for forced physical oscillators. Based on experimental results and numerical model simulations, we could show that resonance and beats are internal properties of chemical oscillatory systems. A forced open chemical oscillatory system is a counterpart of the forced oscillators known form the classical mechanics (e.g., driven pendulum), in which instead of applying a periodic external driving force, the periodically changing chemical potential drives the open oscillatory systems.

(joint work with Hugh Shearer Lawson, Gábor Holló, Róbert Horváth)

Let us consider scalar delay differential equations of the form x'(t)=-ax(t)+f(x(t-1)), where a>0 and f is a nondecreasing C1-function. This talks gives an overview of the periodic orbits and the global attractor. 

After showing some well-known results of Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, I introduce the notion of large-amplitude periodic (LAP) orbits. First we discuss the bifurcation and the existence of a pair of LAP orbits. Then we describe the geometric properties of the unstable set of a specific LAP orbit in detail. Complicated configurations of LAP orbits appear when the dynamical system has several unstable equilibria – we also consider this case. These are joint works with Tibor Krisztin and Szandra Beretka.

No preliminary knowledge of delay equations is presumed.

This study is motivated by the recent achievements in fractional calculus and its numerous applications related to anomalous (super) diffusion. Let us consider a fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition. It is self-adjoint but nonlocal. The nonlocal problems are computationally expensive. Several different techniques were recently proposed to localize the nonlocal elliptic operator, thus increasing the space dimension of the original computational domain. 

An alternative approach [1,2,3] is discussed in this talk. Let $\cal A$ be a properly scaled symmetric and positive definite (SPD) sparse  matrix, arising from finite element or finite difference discretization of the initial (standard, local) diffusion problem. A method for solving algebraic systems of linear equations involving $\cal A^\alpha$, $0 < \alpha < 1$, is presented. The solution methods are based on best uniform rational approximations (BURA) of the scalar function $t^{\alpha}$, $0\le t\le 1$. The method has exponential convergence rate with respect to the degree of rational approximation $k$. The error estimates of the last variant of BURA methods are robust with respect to the spectral condition number $\kappa (\cal A)$. A stabilized modification of the Remez algorithm is developed to compute the BURA of $t^{\alpha}$. Although the fractional power of $\cal A$ is a dense matrix, the algorithm has complexity of order $O(N)$, where $N$ is the number of unknowns. At this point we assume that some solver of optimal complexity (say multigrid or multilevel) is used for the involved systems with matrices ${\cal A} + d_j \cal I$, $d_j \ge 0$, $j=1, \dots, k$. The comparative numerical tests confirm the advantages of the BURA method.

Acknowledgement: This research has been partially supported by the Bulgarian NSF Grant DN12/1.