[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.
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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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