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9
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84926901592
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Here and throughout this paper we treat ρz(z) as a continuous function, although there are clearly cases where it is not continuous (for instance, chaotic processes typically have discontinuous density functions). Our results appear to hold for these cases, as well. We have chosen to treat ρz(z) as a continuous function to avoid the more rigorous, but perhaps less familiar, notation employing probability measures.
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14
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84926920602
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If the density function is discontinuous, the integral (6) must be replaced by a Lebesque integral over the probability measure d μz(z).
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19
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84926901588
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The map xn+1=2xnmod1 must be implemented with care. Direct floating-point implementation results in improper convergence to the unstable fixed point at x=0 (for example, in a double-precision implementation with 64 bits of precision, x will converge to zero in about 64 iterations). To avoid this, we employ a well-known binary-shift technique on a finite binary array. At each iteration of the map, a left shift is performed on the array and the least significant element of the array is set to 1 or 0 randomly with equal probability.
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21
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84926882864
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More rigorously, this onset condition must be supplemented with a condition on the correlation properties of the driving sequence lcurl xnrcurl. Examples can be constructed where no intermittent behavior will be observed. One is the case for which a large value of xn on one iterate (expansion) is countered by an approximate reciprocal contraction on the next iterate. Another example is the sequence generated from the map xn+1=xn+ ε mod1, where epsilon is a small irrational number. The correlation condition is not needed for delta-function-correlated random driving (or tent-map driving, which is also delta correlated). For 2x mod1, the exponential decay of correlations is evidently sufficient to guarantee intermittency. A mathematically precise correlation condition that guarantees intermittent behavior for arbitrary driving signals is still outstanding.
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25
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84926939746
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In practice we add a small amount of noise to the map (of magnitude app 10-300) at each iteration. This allows us to carry out simulations just above the critical value a=ac. Such a small amount of noise does not affect the distribution of laminar phases in a fundamental way, although in a future paper we will show that higher noise levels do change the character of the distribution.
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26
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84926901587
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W. Feller, An Introduction to Probability Theory and its Applications, 2nd ed. (Wiley, New York, 1971), Vol. II.
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