Entropy and Mode Locking

The next series of examples deals with a standard route to chaos described by the standard model. In this case a nonlinear dynamical system first undergoes a Hopf bifurcation. The resulting torus become increasingly wrinkled. On the way to chaos mode locking first appears (analogous to period doubling), and then chaotic behavior occurs. Applying the Birman-Williams theorem to this scenario, we find that the branch line is not a segment of $R^1$ but rather the circle $S^1$. Mode locking occurs when the mapping $S^1 \rightarrow S^1$ is still invertible. When it loses invertibility, chaos appears.

Invertibility is lost when the circle folds over on itself during the return map. Because of the boundary conditions ($S^1$ is topologically different from $R^1$), two folds must occur. The flow from $S^1$ to its folded over image is described by a three branch manifold. Branches $L$ and $R$ are orientation-preserving. On branch $L$ the rotation angle increases by less than $2\pi$, on branch $R$ it increases by more than $2\pi$. Branch $C$ occurs between the two folds and is orientation reversing.

Chaos appears when two Arnol'd tongues begin to overlap. Arnol'd tongues are described by rational fractions $p/q$, where $q$ is the number of times the closed orbit goes around the torus in the long direction and $p$ is the number of times it goes around in the short direction. Then $q$ is the period and $p$ is the winding number.

The symbol sequence of the saddle node pair in the Arnol'd tongue $p/q$ is $W(1)W(2) \cdots W(q)$, where

$$W(i) = [i\times \frac{p}{q}] - [(i-1)\times \frac{p}{q}]= \l... ...w \left( \begin{array}{c} W(i)=L \\ W(i)=R \end{array} \right)$$

where $[x]$ is the integer part of $x$. For $p/q=3/5$, $W(1)W(2) W(3) W(4)W(5)\rightarrow LRLRR$. The partner orbit is obtained by replacing the penultimate symbol by $C$ (for example, $LRLCR$).

We describe the chaotic behavior when tongues described by rational fractions $p/q$ and $p'/q'$ begin to overlap, with $p/q < p'/q'$ and $pq'-p'q = \pm 1$. At this point the behavior is chaotic and the grammar contains three words. These are:

$A$	The symbol sequence for the left hand tongue $p/q$
$B$	The symbol sequence for the right hand tongue $p'/q'$
$C$	The partner of $B$

Not every symbol sequence is allowed, for $C$ must be preceeded by $A$.

Example 5: Determine the equation which defines the topological entropy for the chaotic attractor formed when the tongues $p/q$ and $p'/q'$ cross. The Transition Matrix is

$$\begin{array}{c} A \\ B \\ C \end{array} \hspace{0.5cm} \lef... ...{ccc} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \end{array} \right]$$

Applying this information to Equ (4.19), we find

$$\det \left[ \begin{array}{ccc} {1 \over X^q} -1 & {1 \over ... ... \over X^{q'}} & {1 \over X^{q'}} & -1 \end{array} \right] = 0$$

This reduces to

$$X^{q+q'} - X^q - X^{q'}-1 = 0$$

Example 6: Use Equ (4.18) to obtain the same result. Since $A$ must proceed $C$, the grammar has three symbols $A$, $B$, and $AC$ with periods $q$, $q'$, and $q+q'$ and no constraints. The secular equation is

$${1 \over X^{q}} + {1 \over X^{q'}} + {1 \over X^{q+q'}}=1$$

which is equivalent to the result above.

Example 7: Compute the topological entropy for the strange attractors which occur when the tongues $p/q$ and $p'/q'$ just overlap, for the pairs $(\frac{1}{2},\frac{2}{3})$, $(\frac{1}{2},\frac{3}{5})$, $(\frac{3}{5},\frac{2}{3})$. Solution:

$p/q$	$p'/q'$	$A$	$B$	$X_0$	$h_T$
			$C$
$\frac{1}{2}$	$\frac{2}{3}$	$LR$	$LRR$	1.429108	0.357051
			$LCR$
$\frac{1}{2}$	$\frac{3}{5}$	$LR$	$LRLRR$	1.307395	0.268037
			$LRLCR$
$\frac{3}{5}$	$\frac{2}{3}$	$LRLRR$	$LRR$	1.252073	0.224801
			$LCR$

Example 8: Compute the topological entropy for the low period Arnol'd tongues for which $pq'-qp'=\pm 1$. Solution: To period ten, here they are. The entries for which $q$ and $q'$ have a common factor are left blank.

	3	4	5	6	7	8	9	10
2	0.357051		0.268037		0.219131		0.187366
3		0.253442	0.224801		0.186002	0.172048		0.150507
4			0.196620	0.178525	0.164136		0.142458	0.134018
5				0.160664	0.148188	0.137920	0.129277
6					0.135847	0.126721	0.119017	0.112400
7						0.117680	0.110713	0.104716
8							0.103803	0.098310
9								0.092856