[2.15] What is the minimum phase space dimension for chaos? This is a slightly confusing topic, since the answer depends on the type of system considered. First consider a flow (or system of differential equation s). In this case the Poincar□-Bendixson theorem tells us that there is no c haos in one or two-dimensional phase spaces. Chaos is possible in three-dime nsional flows--standard examples such as the Lorenz equations are indeed thr ee-dimensional, and there are mathematical 3D flows that are provably chaoti c (e.g. the 'solenoid'). Note: if the flow is non-autonomous then time is a phase space coordinate, s o a system with two physical variables + time becomes three-dimensional, and chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.) For maps, it is possible to have chaos in one dimension, but only if the map is not invertible. A prominent example is the Logistic map x' = f(x) = rx(1-x). This is provably chaotic for r = 4, and many other values of r as well (see e.g. Devaney). Note that every point x < f(1/2) has two preimages, so this m ap is not invertible. For homeomorphisms, we must have at least two-dimensional phase space for ch aos. This is equivalent to the flow result, since a three-dimensional flow g ives rise to a two-dimensional homeomorphism by Poincar□ section (see [2.7] ). Note that a numerical algorithm for a differential equation is a map, becaus e time on the computer is necessarily discrete. Thus numerical solutions of two and even one dimensional systems of ordinary differential equations may exhibit chaos. Usually this results from choosing the size of the time step too large. For example Euler discretization of the Logistic differential equ ation, dx/dt = rx(1-x), is equivalent to the logistic map. See e.g. S. Ushik i, "Central difference scheme and chaos," Physica 4D (1982) 407-424. -- 在细雨的午後 书页里悉哩哩地传来 " 周期3 = ? " 然而我知道 当我正在日耳曼深处的黑森林 继续发掘海森堡未曾做过的梦时 康德的诺言早已远离......... 远来的传教士静静地看着山涧不断反覆叠代自己的 过去 现在 和 未来 於是仅以 一颗量子浑沌 一本符号动力学 祝那发生在周一下午的新生 --

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