A 2-D Map, also called the Taylor-Greene-Chirikov Map in some of the older literature.
where
and
are computed mod
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}](s3_514.gif) |
(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 |
![$>$](s3_517.gif) |
![${1\over 34}$](s3_518.gif) |
0.029411764 |
Italians |
![$>$](s3_517.gif) |
- |
0.65 |
Greene |
![$\approx$](s3_519.gif) |
- |
0.971635406 |
MacKay and Pearson |
![$<$](s3_520.gif) |
![${63\over 64}$](s3_521.gif) |
0.984375000 |
Mather |
![$<$](s3_520.gif) |
![${4\over 3}$](s3_522.gif) |
1.333333333 |
Fixed Points are found by requiring that
The first gives
, so
and
![\begin{displaymath}
\theta_n =0,\pi.
\end{displaymath}](s3_527.gif) |
(6) |
The second requirement gives
![\begin{displaymath}
I_n+K\sin\theta_n=I_n=0.
\end{displaymath}](s3_528.gif) |
(7) |
The Fixed Points are therefore
and
. In order to perform a
Linear Stability analysis, take differentials of the variables
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}](s3_535.gif) |
(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}](s3_536.gif) |
(11) |
so
![\begin{displaymath}
\lambda^2-\lambda(K\cos\theta_n+2)+1 = 0
\end{displaymath}](s3_537.gif) |
(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}](s3_538.gif) |
(13) |
For the Fixed Point
,
The Fixed Point will be stable if
Here, that means
![\begin{displaymath}
{\textstyle{1\over 2}}\vert 2-K\vert<1
\end{displaymath}](s3_543.gif) |
(15) |
![\begin{displaymath}
\vert 2-K\vert<2
\end{displaymath}](s3_544.gif) |
(16) |
![\begin{displaymath}
-2<2-K<2
\end{displaymath}](s3_545.gif) |
(17) |
![\begin{displaymath}
-4<-K<0
\end{displaymath}](s3_546.gif) |
(18) |
so
. For the Fixed Point (0, 0), the Eigenvalues are
If the map is unstable for the larger Eigenvalue, it is unstable. Therefore,
examine
. We have
![\begin{displaymath}
{1\over 2}\left\vert{2+K+\sqrt{K^2+4K}\,}\right\vert<1,
\end{displaymath}](s3_552.gif) |
(20) |
so
![\begin{displaymath}
-2<2+K+\sqrt{K^2+4K}<2
\end{displaymath}](s3_553.gif) |
(21) |
![\begin{displaymath}
-4-K<\sqrt{K^2+4K}<-K.
\end{displaymath}](s3_554.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-26