[2.7] How are maps related to flows (differential equations)?
Every differential equation gives rise to a map, the time one map, defined b
y advancing the flow one unit of time. This map may or may not be useful. If
the differential equation contains a term or terms periodic in time, then t
he time T map (where T is the period) is very useful--it is an example of a
Poincar□ section. The time T map in a system with periodic terms is also ca
lled a stroboscopic map, since we are effectively looking at the location in
phase space with a stroboscope tuned to the period T. This map is useful be
cause it permits us to dispense with time as a phase space coordinate: the r
emaining coordinates describe the state completely so long as we agree to co
nsider the same instant within every period.
In autonomous systems (no time-dependent terms in the equations), it may als
o be possible to define a Poincar□ section and again reduce the phase space
dimension by one. Here the Poincar□ section is defined not by a fixed time
interval, but by successive times when an orbit crosses a fixed surface in
phase space. (Surface here means a manifold of dimension one less than the p
hase space dimension).
However, not every flow has a global Poincar□ section (e.g. any flow with a
n equilibrium point), which would need to be transverse to every possible or
bit.
Maps arising from stroboscopic sampling or Poincar□ section of a flow are n
ecessarily invertible, because the flow has a unique solution through any po
int in phase space--the solution is unique both forward and backward in time
. However, noninvertible maps can be relevant to differential equations: Poi
ncar□ maps are sometimes very well approximated by noninvertible maps. For
example, the Henon map (x,y) -> (-y-a+x^2,bx) with small |b| is close to the
logistic map, x -> -a+x^2.
It is often (though not always) possible to go backwards, from an invertible
map to a differential equation having the map as its Poincar□ map. This is
called a suspension of the map. One can also do this procedure approximatel
y for maps that are close to the identity, giving a flow that approximates t
he map to some order. This is extremely useful in bifurcation theory.
Note that any numerical solution procedure for a differential initial value
problem which uses discrete time steps in the approximation is effectively a
map. This is not a trivial observation; it helps explain for example why a
continuous-time system which should not exhibit chaos may have numerical sol
utions which do--see [2.15].
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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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