Standard Map

$\begin{figure}\begin{center}\BoxedEPSF{standard_map050.epsf scaled 320}\BoxedEPSF{standard_map100.epsf scaled 320}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{standard_map150.epsf scaled 320}\BoxedEPSF{standard_map200.epsf scaled 320}\end{center}\end{figure}$

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

$\displaystyle I_{n+1}$	$\textstyle =$	$\displaystyle I_n+K\sin\theta_n$	(1)
$\displaystyle \theta_{n+1}$	$\textstyle =$	$\displaystyle \theta_n+I_{n+1} = I_n+\theta_n+K\sin\theta_n,$	(2)

where

and $\theta$ are computed mod $2\pi$ and

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

$\begin{displaymath} \delta I = B e^{-AK^{-1/2}}. \end{displaymath}$

(3)

Numerical experiments give $A\approx 5.26$ and $B\approx 240$ . 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		${1\over 34}$	0.029411764
Italians		-	0.65
Greene	$\approx$	-	0.971635406
MacKay and Pearson		${63\over 64}$	0.984375000
Mather		${4\over 3}$	1.333333333

Fixed Points are found by requiring that

$\displaystyle I_{n+1}$	$\textstyle =$	$\displaystyle I_n$	(4)
$\displaystyle \theta_{n+1}$	$\textstyle =$	$\displaystyle \theta_n.$	(5)

The first gives $K\sin\theta_n=0$ , so $\sin\theta_n=0$ and

$\begin{displaymath} \theta_n =0,\pi. \end{displaymath}$

(6)

The second requirement gives

$\begin{displaymath} I_n+K\sin\theta_n=I_n=0. \end{displaymath}$

(7)

The Fixed Points are therefore $(I,\theta) =(0,0)$ and $(0, \pi)$ . In order to perform a Linear Stability analysis, take differentials of the variables

$\displaystyle dI_{n+1}$	$\textstyle =$	$\displaystyle dI_n+K\cos\theta_n\,d\theta_n$	(8)
$\displaystyle d\theta_{n+1}$	$\textstyle =$	$\displaystyle dI_n+(1+K\cos\theta_n)\,d\theta_n.$	(9)

In Matrix form,

$\begin{displaymath} \left[{\matrix{ \delta I_{n+1}\cr \delta\theta_{n+1}\cr}}\r... ...ght]\left[{\matrix{\delta I_n\cr \delta \theta_n \cr}}\right]. \end{displaymath}$

(10)

The Eigenvalues are found by solving the Characteristic Equation

$\begin{displaymath} \left\vert\matrix{ 1-\lambda & K\cos\theta_n\cr 1 & 1+K\cos\theta_n-\lambda\cr}\right\vert=0, \end{displaymath}$

(11)

$\begin{displaymath} \lambda^2-\lambda(K\cos\theta_n+2)+1 = 0 \end{displaymath}$

(12)

$\begin{displaymath} \lambda_\pm = {\textstyle{1\over 2}}[K\cos\theta_n+2\pm\sqrt{(K\cos\theta_n+2)^2-4}\,]. \end{displaymath}$

(13)

For the Fixed Point $(0, \pi)$ ,

$\displaystyle \lambda^{(0,\pi)}_\pm$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}[2-K\pm\sqrt{(2-K)^2-4}\,]$
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(2-K\pm\sqrt{K^2-4K}\,).$	(14)

The Fixed Point will be stable if $\vert\Re(\lambda^{(0,\pi)})\vert<2.$ Here, that means

$\begin{displaymath} {\textstyle{1\over 2}}\vert 2-K\vert<1 \end{displaymath}$

(15)

$\begin{displaymath} \vert 2-K\vert<2 \end{displaymath}$

(16)

$\begin{displaymath} -2<2-K<2 \end{displaymath}$

(17)

$\begin{displaymath} -4<-K<0 \end{displaymath}$

(18)

so $K\in [0,4)$ . For the Fixed Point (0, 0), the Eigenvalues are

$\displaystyle \lambda^{(0,0)}_\pm$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}[2+K\pm\sqrt{(K+2)^2-4}\,]$
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(2+K\pm\sqrt{K^2+4K}\,).$	(19)

If the map is unstable for the larger Eigenvalue, it is unstable. Therefore, examine $\strut\lambda^{(0,0)}_+$ . We have

$\begin{displaymath} {1\over 2}\left\vert{2+K+\sqrt{K^2+4K}\,}\right\vert<1, \end{displaymath}$

(20)

$\begin{displaymath} -2<2+K+\sqrt{K^2+4K}<2 \end{displaymath}$

(21)

$\begin{displaymath} -4-K<\sqrt{K^2+4K}<-K. \end{displaymath}$

(22)

But

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

References

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