“Circle Map”的概念、定义、习题解题方法及翻译-数学辞典

单词

Circle Map

释义

Circle Map

A 1-D Map which maps a Circle onto itself

(1)

where

is computed mod 1. Note that the circle map has two parameters:

and

can beinterpreted as an externally applied frequency, and

as a strength of nonlinearity. The 1-D Jacobian is

(2)

so the circle map is not Area-Preserving. It is related to the Standard Map

			(3)
			(4)

for

and

computed mod 1. Writing

(5)

gives the circle map with

and

. The unperturbed circle map has the form

(6)

is Rational, then it is known as the map Winding Number, defined by

(7)

and implies a periodic trajectory, since

will return to the same point (at most) every

Orbits. If

is Irrational, then the motion is quasiperiodic. If

is Nonzero,then the motion may be periodic in some finite region surrounding each Rational

. Thisexecution of periodic motion in response to an Irrational forcing is known as ModeLocking.

If a plot is made of vs. with the regions of periodic Mode-Locked parameter spaceplotted around Rational values (Winding Numbers), then the regions are seen to widen upward from 0 at tosome finite width at . The region surrounding each Rational Number is known as an Arnold Tongue. At , the Arnold Tongues are an isolated set of Measure zero. At , they form aCantor Set of Dimension . For , the tongues overlap, and the circle map becomesnoninvertible. The circle map has a Feigenbaum Constant

(8)

See also Arnold Tongue, Devil's Staircase, Mode Locking, Winding Number (Map)

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