[2.3] What is a dynamical system? A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical ru le which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Mathematically, a dyn amical system is described by an initial value problem. Dynamical systems are "deterministic" if there is a unique consequent to eve ry state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution (the "perfect" coin toss has two c onsequents with equal probability for each initial state). Most of nonlinear science--and everything in this FAQ--deals with deterministic systems. A dynamical system can have discrete or continuous time. The discrete case i s defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from th e initial state z_0 at the next time value. The continuous case is defined b y a "flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to give a differential equation, dz/dt = F(z). In this case we call F(z) a "vector field," it gives a vector pointing in the direction of the velocity at every point in phase space. -- 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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