1. “ELECTRICAL NEURONAL OSCILLATIONS AND COGNITION (ENOC)” COST ACTION B27 Presentation for the 1 st Working Group meetings (MK) Theoretical considerations

2. In this presentation we will give a short description of the theoretical mathematical results and interests of the researchers from Macedonia, closely related to this action

3. 1. Topological description of chaotic attractor with spiral structure. The template for a chaotic dynamical system describes the topological properties of the periodic orbits embedded in the attractor. A template, i.e. branched 2-manifilod is found for a chaotic attractor with spiral structure.

4. Lorenz attractor

5. Template for Lorenz attractor

6.  =16, b=4, r=45.92

7. Chua’s spiral type attractor

8. Template for Chua’s attractor

9. b=100/7, n=-8/7, m=-5/7,  =9

10. 2. Control of trajectories in chaotic dynamical system using small perturbations. For a given chaotic system x= f(x) and given point z, using an algorithm, a function g(x) with previously chosen small values, can be found, such that for a given trajectory of the system, after the adding the small perturbation g(x) to the system, the new trajectory of the system x = f(x) + g(x) will be targeted toward a previously chosen small neighborhood of the point z.

12. 3. Generalized synchronization (GS) of unidirectionally coupled dynamical systems Two unidirectionally coupled dynamical systems: x = f(x) and y = g(y,h(x)), where x  R n and y  R m, are said to posses the property of GS if there is a function H:R n  R m, a manifold M={(x,y) | y=H(x)} and a subset B  R n  R m with M  B, such that all the trajectories of the coupled system with initial conditions in B approach M as time goes to infinity.

13. Necessary and sufficient conditions for the occurrence of GS are given in terms of asymptotic stability. Also, the robustness of the synchronization, i.e. the stability of the synchronization manifold M under small perturbations will be examined.

14. 4. Vector valued structures An attemption will be made to apply the results in the area of vector valued, i.e. (n,m) algebraic structures, and generalized metrics, to the objectives of this action. Let  be an (n,m)-equivalence on a set M. A map d:M (n)  R 0 +, such that:

15. (i) d(x) = 0 iff x  ; and (ii) For each a  M (m) and each x  (n), d(x)  d(ua), where the sum is over all the u  M (k) such that there is a v  M (m) with uv=x in M (n) ; is said to be an (n,m,  )-metric on M, and the pair (M,d) is said to be (n,m,  )-metric space. An example of (3,1,D)-metric is the area of triangles in the plane.

16. References D. Gligoroski, D. Dimovski, V.Urumov: Control in multidimenzional chaotic systems by small pertutrbations, Physical Review E, V.51, N.3; 1995, 1690-1694 L.Kocarev, Z. Tasev, D. Dimovski; Topological description of a chaotic attractor with spiral srtructure, Physics letters A, 190, 1994, 399-402 L.Kocarev, U.Parlitz: Generalized Synchronization, Predictability, and Equivalence of Unidirectionally Coupled Dynamical Systems, Physical Review Letters, V.76, N.11, 1996, 1816-1819 D. Dimovski; Generalized metrics - (n,m,r)-metrics, Mat. Bilten, 16, Skopje, 1992, 73-76.