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Crisis-induced Intermittency in Coupled Chaotic Maps
Gouhei Tanaka
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Mathematical models represented by dynamical systems often exhibit a coexistence state of multiple solutions. For instance, multiple regimes of different dynamics can coexist in biological systems [6,17]. In artificial neural networks for associative memories, multiple memories are embedded in coexisting attractors [7]. Other multistable systems include electrical systems [10] and optical systems [19]. In such a multistable system, an observable attractor is different depending on the initial condition. Variation of a system parameter in a multistable system often leads to a sudden change of attractors in size and/or in shape. It is typically a global bifurcation (crisis) involving intermittent behaviors [4,5].
In this document we consider coupled chaotic maps with a coexistence state of multiple chaotic attractors. The phase space is divided into several basins of attraction. The boundary separating the basins of attraction is called a basin boundary. It is demonstrated that a basin boundary with fractal nature provides a clue to understanding an onset of intermittency through numerical computation of its fractal dimension. This approach is useful for comprehending a global bifurcation in high-dimensional dynamical systems where observation of basin structure is rather hard.
Following an introduction of 2D coupled chaotic maps, we show a sudden onset of intermittent behaviors induced by a crisis with variation of a system parameter. Observation of qualitative changes in basin structure provides information on the crisis inducing intermittency. It is illustrated that variation of fractal dimension of the basin boundary can quantitatively characterize the changes in the basin structure and failure of computation of the fractal dimension implies the crisis [16]. This idea is applied to an analysis of a global bifurcation inducing itinerant memory dynamics in a chaotic neural network [18].