Nontrivial Collective Behaviour and Chaos in Inhomogeneous Coupled

Nontrivial Collective Behaviour and Chaos in Inhomogeneous Coupled Map Lattices A. Loskutov, S. Rybalko, A. K. Prokhorov, E. M. Belova Physics Faculty Moscow State University 1

Some publications • • A. Loskutov, S. D. Rybalko, D. N. Udin and K. A. Vasiliev. Model of a spatially inhomogeneous one-dimensional active medium. Theor. and Math. Physics, 2000, v. 124, No 3, p. 1286 -1297. A. Loskutov, S. D. Rybalko and A. K. Prokhorov. Dynamics of parametrically excited maps. - Moscow Univ. Phys. Bull. , 2002, No 4, p. 3 -12. A. Loskutov, A. K. Prokhorov and S. D. Rybalko. Analysis of inhomogeneous chains of coupled quadratic maps. - Theor. and Math. Physics, 2002, v. 132, No 1, p. 980 -996. A. Loskutov. Synergetics and nonlinear dynamics: New Ideas. - In: The Book of Lectures on "Fractals and Applied Synergetics". Russian Academy of Sciences Press, Moscow, 2000. 2

Main results 1. A new method of effective local analysis for coupled map dynamics is proposed. In contrast to the previously suggested methods, it allows us visually investigate the time and spatial evolutions of a distributed medium described by a set of maps. 2. The efficiency of the method is demonstrated by the examples of models of diffusively coupled quadratic maps. 3. Analysis of the linear chain in the presence of space defects reveals some new global phenomena in the behaviour. 3

A family of coupled maps To describe the lattice dynamics it is necessary to take into account space and time actions. Formalizations: 4

Types of coupling Diffusive coupling: Open-flow coupling: 5

System of two quadratic maps with diffusive coupling 4 Bifurcation diagram: 2 0 -2 6 -4 0 1 2 3 4

System of two quadratic maps with open-flow type of coupling 7

Coupled map lattices Many of the parameters that should be calculated for dynamical systems (e. g. , metric entropy, the spectrum of Lyapunov exponents, the correlation decay, etc. ) reflect the asymptotic behaviour of the system, i. e. at But it is more important to find the characteristics determining the local properties of these systems and their time evolution. 8

Local criterion for coupled-map dynamics x 1, x 2, …, x. T T=nt A: x 1, …, xt B 1: x 1, …, xt B 2: x 1, x 2, …, xt B 3: x 1, x 2, x 3, …, xt … Bt/2: x 1, x 2, x 3, …, xt/2, x 1, x 2, x 3, …, xt 9

Open-flow model a Boundary condition: x=0 S 1, S 2 are the areas of existence of stable solutions Z is the area of existence of the zigzag solution 10

Open-flow model a = 1. 7 (ε = 0. 11595) 11

Ring lattice Periodic boundary conditions a=1. 44 eps=0. 1 12

Ring lattice: fully developed turbulence a = 1. 88 (ε = 0. 3) 13

Ring lattice: super transient behaviour a = 1. 8 (ε = 0. 3) 14

Ring lattice with space defects Case 1 Case 2 Case 3 15

Inhomogeneous lattices Case 1. Ring lattice with defects after 100 K iterations a 1=1. 44 (order) N=93; a 2=1, 97 (chaos) N=7. eps=0. 7 16

Ring lattices with periodic defects a 1 = 1. 8, a 2 = 1. 97 (ε = 0. 7) Random initial conditions 17 Other initial conditions

“Brownian” particles in the homogeneous lattice a = 1. 8 (ε = 0. 1) 18

Ring lattices with a single defect a 1=1. 8 (all elements except for i=75), a 2 = 1. 99 in i = 75 (ε = 0. 1) (after 5 · 10^7 preliminary iterations) 19

Concluding remarks v A new method for the analysis for coupled map chains is developed. It permits to visualize the behaviour of the individual elements and the dynamics of the entire system as a whole. v. The behaviour of both homogeneous systems of diffusively coupled one-dimensional quadratic maps and systems with spatial inhomogeneity have been investigated. v. It is shown that the presence of different types of inhomogeneities in the model can substantially change the behaviour of the entire system. v In particular, for some values of the parameters and diffusion coefficients, the dynamics of a inhomogeneous chain with alternating defects is synchronized for any initial conditions. It should be noted that homogeneous systems with parameter values for which this effect takes place, exhibit space–time 20 chaos.