Novel Mechanism of Noise-Induced Transitions Between Attractors of a 2D-Discontinuous Piecewise-Smooth Map Arising in Asset Market Modeling

Анотація

We consider a stochastic version of a 2D-discontinuous piecewise-linear map that describes the dynamics of a financial market model with heterogeneous traders, extended by the introduction of noisy signals that influence the market maker’s price-setting. Using Stochastic Sensitivity Function (SSF) analysis, we examine how exogenous noise interacts with coexisting attractors in this class of maps. Three findings emerge: First, we show that the SSF does not diverge near border-collision bifurcation unlike the SSF near bifurcations associated with stability loss, revealing a structural property unique to piecewise-smooth dynamics. Second, we identify and dissect a novel noise-induced transition mechanism that operates between state-space partitions caused by the discontinuity of the map, which we formalize through the concept of d-critical intensity. Third, we demonstrate that this mechanism explains stochastic phenomena previously unaccounted for, such as unidirectional and bidirectional transitions between cycles within their basins, and the emergence of ghost dynamics. These results broaden the scope of SSF analysis, provide new insights into irregular market fluctuations under noise, and may be useful in studying other applied models in the presence of stochastic forces where the dynamics are described by discontinuous piecewise-smooth maps.

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