PROPERTIES OF ONE-DIMENSIONAL ASYMMETRIC MAPS

Abstract

There exists, hitherto unknown, a class of maps with precision-dependent periods. Each of these maps has a true period which is infinite. However, the actual value can never be determined computationally unless a computer with an infinite word length is available. The computed period is always fictitious whose value increases with increasing arithmetic precision. In this sense, the map is said to have a relative, apparently real period (RARP). Examples of RARP maps have been given, together with numerical and analytical demonstrations to confirm their RARP behaviour. All these maps possess at least a discontinuity, indicating that a discontinuity is probably necessary for RARP behaviour to exist. The logistic map with a sectional discontinuity at x = 0 has been studied numerically. In the periodic region, the periods of the cycles fall into six different first-level inverse cascades, where an inverse cascade is actually a decreasing arithmetic progression. Within each inverse cascade, if our Rule I holds, then between any two consecutive terms of the cascade, there exist new cycles with periods that can be predicted by our summation rule. These new cycles form higher-level inverse and direct cascades. On the other hand, if our Rule II holds within an inverse cascade, no new cycles exist between any two consecutive terms. In the chaotic region, there exist windows of stable orbits and sometimes they obey the modified summation rule. When the sectional discontinuity of the logistic map lies at any general point x = xd, we find that the main characteristics of the map with xd = 0, such as the occurrence of inverse cascades, the validity of Rule I, Rule II and the summation rule, are commonly observed in the map with xd ? 0. However, there are differences as well such as the number of inverse cascades and the routes to chaos. The modified summation rule holds only for some values of xd ?0. As xd varies, the accumulation point aacc of the inverse cascade also changes. This functional dependence of aacc on xd has been explained, and further, analytical expressions for aacc which yield values identical to the computed results have been derived. Knowing that bifurcations within an inverse cascade occur whenever one of the cycle elements approaches xd, a method for verifying the values of the bifurcation points within any cascade has been described, which involves the formulation of a polynomial equation. In the linear-logistic map, it is observed that at every other period-doubling, the old cycle just before the bifurcation is superstable instead of critically stable. This behaviour has been accounted for. By analysing the graphs of the Lyapunov exponent against the parameter r of the map, we classify the stable cycles in the chaotic region into six different patterns with explicitly-defined rules. Each pattern consists of at least an arithmetic progression, with the terms intermingled with chaos. One pattern is self-similar, another exhibits the period-adding phenomenon while all of them are related to each other through their dependence on the same variables. Using these patterns, the existence of stable cycles in some domains of r can be predicted.