Ogorzalek M.J., Galias Z.

Effects of Noise in Pattern Formation in Large Arrays of
Coupled Chaotic Oscillators
M.J. Ogorzalek, Z. Galias
University of Mining and Metallurgy, Krakòw, Poland

Pattern formation and spatio-temporal dynamics of oscillator arrays have been extensively studied recently [1], [4].Many interesting kinds of cooperative behavior have been reported in arrays of chaotic oscillators [2], [3]. Among these phenomena are: creation of target waves, autowave-like behavior and spatial memory effects. The networks studied in [2] and [3] were considered to be noise-free - just responding to external stimulation which caused development of dynamic phenomena. It seems that the effects of noise cannot be neglected in such networks. Previous research results [5] have shown that the dynamic behavior observed in a network of oscillators can be very sensitive to noise -- in particular due to attractor crowding (and shrinking of their respective basins of attraction) in large networks spontaneous switchings between observed attractors can be observed in the presence of noise. Intuitively, another effect of existing noise can be expected in arrays of chaotic oscillators - as all the cells operate in an unstable mode one can expect spontaneous development of chaotic oscillations due to inevitable noise and destruction of patterns existing in a noise-free network. This second kind of effects is studied in detail in the present paper. We investigate how noise level and interconnection strength changes the observed phenomena.

The effects of our extensive simulation studies allowed us to draw the following conclusions:

• We confirmed that in the case of weak local coupling between the chaotic cells the noise modeled by small random initial conditions introduced in the network of chaotic elements has a destructive effect. Even very small level of noise causes propagation failure of waves and eventually leads to complete disorder.
• In the case of strong coupling (above the ``mutual resonance level'') astonishingly enough the noise has very little influence. The auto-wave phenomena are prelevant and only very high level of noise destroys the spatial structure and spatial memory effect.

We hope that our study will open new research opportunities and enable explanation of mechanism existing in the brain. Finding the parameters responsible for self-organization, pattern creation and memory effects could offer also new signal processing applications.

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