Based on our previous study that discovered a new mechanism of the riddled-like basin appearance in piecewise continuous and noninvertible links of two conservative mappings, and the mechanism being the mixing of different attraction basins on a fat-fractal set, which was addressed as the forbidden region net [Phys. Rev. E 72 (2005) 025201(R)], we show numerically that the riddled-like attraction basins and the new mechanism appear in an integrate-and-fire circuit that can be described by piecewise continuous and noninvertible links of two dissipative mappings. This leads to the fact that vast increase of the precision of the initial condition results in practically no improvement in predictability of the iteration destination. Predictability is usually characterized by examining how error probability in the prediction, f(ε), scales with the precision ε under the initial condition. Typically, f(ε)~εα with 0⩽α⩽1 being the uncertainty exponent. For riddled-like basins, α should be zero, which is in good agreement with the numerical simulation results.
This paper reports five different features in piecewise continuous and noninvertible system described by two conservative maps or one conservative map and another dissipative map. The features are as follows: the stochastic web bounded by images of discontinuous borderline is the only chaotic trajectory; phase collapse is caused by the irreversibility which makes some points to have two pre-images; fat fractal forbidden web induced by irreversibility; riddled-like attraction basin in stochastic web and forbidden web, when there are different attractors coexisting; and the impossibility to tell to which attractor an initial condition will approach. In an integrate-and-fire circuit, we find that the features still exist in the circuit system described by two dissipative maps, so they are common features in discontinuous and noninvertible maps.
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