[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. -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --

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