“Standard Map”的概念、定义、习题解题方法及翻译-数学辞典

单词

Standard Map

释义

Standard Map

A 2-D Map, also called the Taylor-Greene-Chirikov Map in some of the older literature.

			(1)
			(2)

where

and

are computed mod

and

is a Positive constant. An analytic estimate of the width of theChaotic zone (Chirikov 1979) finds

(3)

Numerical experiments give

and

. The value of

at which global Chaos occurs has been bounded by various authors. Greene's Method is the most accurate method so far devised.

Author	Bound	Fraction	Decimal
Hermann			0.029411764
Italians		-	0.65
Greene		-	0.971635406
MacKay and Pearson			0.984375000
Mather			1.333333333

Fixed Points are found by requiring that

			(4)
			(5)

The first gives

, so

and

(6)

The second requirement gives

(7)

The Fixed Points are therefore

and

. In order to perform a Linear Stability analysis, take differentials of the variables

			(8)
			(9)

In Matrix form,

(10)

The Eigenvalues are found by solving the Characteristic Equation

(11)

(12)

(13)

For the Fixed Point


			(14)

The Fixed Point will be stable if

Here, that means

(15)

(16)

(17)

(18)

. For the Fixed Point (0, 0), the Eigenvalues are


			(19)

If the map is unstable for the larger Eigenvalue, it is unstable. Therefore, examine

. We have

(20)

(21)

(22)

But

, so the second part of the inequality cannot be true. Therefore, the map is unstable at the FixedPoint (0, 0).

References

Chirikov, B. V. ``A Universal Instability of Many-Dimensional Oscillator Systems.'' Phys. Rep. 52, 264-379, 1979.

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