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

“ELECTRICAL NEURONAL OSCILLATIONS AND COGNITION

(ENOC)”

COST ACTION B27Presentation for the 1st Working Group

meetings (MK)

• Theoretical considerations

• 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

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.

Lorenz attractor

Template for Lorenz attractor

=16, b=4, r=45.92

bzxydtdz

xzyrxdtdy

xydtdx

)(

Chua’s spiral type attractor

Template for Chua’s attractor

b=100/7, n=-8/7, m=-5/7, =9

2/|)1||1)(|()(

))((

xxmnmxxh

bzdtdz

zyxdtdy

xhxydtdx

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.

Z e

g(x)

T

T1 DtX

X 1

3. Generalized synchronization (GS) of unidirectionally coupled dynamical systems

• Two unidirectionally coupled dynamical systems: x = f(x) and y = g(y,h(x)) , where xRn and yRm, are said to posses the property of GS if there is a function H:RnRm, a manifold M={(x,y) | y=H(x)} and a subset BRn Rm with MB, such that all the trajectories of the coupled system with initial conditions in B approach M as time goes to infinity.

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.

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) R0

+ , such that:

(i) d(x) = 0 iff x; and(ii) For each aM(m) and each x(n),

d(x)d(ua), where the sum is over all the uM(k) such that there is a vM(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.

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.