Homework 7

For this homework, you can submit up to 3 times.

Q7.1 $\quad$ Stable and unstable manifold of Henon map (Example 15.5)

In previous homework, we have studied how Poincare section points stretch and fold along an almost 1-dimensional curve, and how the return map captures the essential dynamics of the system. Now we turn to the 2-dimensional Henon map $(x_{n+1},y_{n+1}) = (1-ax^2_n+b y_n,x_n)$ to study the stretch and fold mechanism, which occurs in any bounded chaotic system. Stable and unstable manifolds of an equilibrium are topologically invariant objects that help us visualize how the flow is structured close to this equilibrium. If you start with a small ball of initial points centered around the fixed point $x_0$, and iterate the map, the ball will be stretched along the unstable manifold $W^u$ and squashed along the stable manifold $W^s$. But it cannot be stretched to infinity if the system is bounded; thus at some point, it will fold back. The stable manifold is harder to plot than the unstable manifold, but if we reverse the dynamics, the stable manifold becomes unstable manifold, so for a low-dimensional map the same technique can be used to obtain stable manifold. Please confirm that the backward Henon map is $(x_{n-1},y_{n-1}) = (y_n, - b^{-1}(1-ay^2_n - x_n)) \,.$ In this homework, we fix the map parameters to $a=6$, $b=-1$. You may have noticed that for $b=-1$, the backward map is the same as the forward map with exchange of coordinates $x$ and $y$. Understanding this observation will help you understand your result.

Now let's turn to the code. Please set 'case = 1' to validate your implementation of Henon map. Note in case1 we use a different set of parameters. Do not be confused. This set of parameters is only used for validation purpose. For other cases that follow, $a=6$, $b=-1$ are used. You are supposed to get a figure similar to fig(a) at the bottom of this page. Also make sure your implementation for backward iteration and the stability matrix is correct.

Set 'case = 2', and read the instructions in the code comments to understand the stable/unstable manifold algorithm. You should get the stable and unstable manifolds as shown in fig(b). Point 'C' is on the diagonal line $y = x$. What is the x-coordinate of 'C' ? ( at least 2 significant digits should be accurate)

Q7.2 $\quad$ Henon map continued -- iteration of region 0BCD

In case2, the stable and unstable manifolds enclose a non-wandering region 0BCD. Along curve 0B, state points are stretching, while along curve 0D, state points are contracting. Denote this region as $M.$. Now perform one forward iteration of $M.$, namely $f(M.)$, we get two future strips $M_{.0}$ and $M_{.1}$, which are the two magenta parts intersected in the OBCD region in fig(c). The lower strip is $M_{.0}$ while the upper strip is $M_{.1}$. Similarly, performing backward iteration of $M.$ : $f^{-1}(M.)$ gives us two green strips in region OBCD in fig(c). The left strip is $M_{0.}$ while the right strip is $M_{1.}$. Overlapping of these 4 strips produces 4 different small regions $M_{0.0}$, $M_{1.0}$, $M_{1.1}$, $M_{0.1}$ from 0 to B to C to D. You have a clear view of these four regions in fig(d).

Now set 'case = 3', try to reproduce fig (c). As you can see in fig (c), the border of magenta region consists of 4 parts: 2 arc-curves and 2 short and almost straight lines. However, we are more interested in the inner arc-curve since it helps us partition the state space, as you can see from fig (d). Which side is the pre-image of this inner arc-curve, namely, the pre-image of the magenta curve in fig (d)?

Q7.3 $\quad$ Henon map continued -- periodic orbits

In case 3, you worked with region 0BCD, and figured out which sides are the pre-images of the inner arc-curves in fig (d). Denote the pre-images for magenta and blue inner arc-curves in fig (d) as $S_1$ and $S_2$ respectively. Iterating $S_1$ and $S_2$ forward and backward for one step, we obtain intersection regions $M_{0.0}$, $M_{1.0}$, $M_{1.1}$, $M_{0.1}$ as shown in fig (d). If we iterate $S_1$ forward and $S_2$ backward for two steps, we can get 16 small intersection regions as shown in fig (e), which consists of 4 groups and each group is close to one vertex of region 0BCD.  Please set 'case = 4' to reproduce fig (e). Choose a point in the top left region, the one that is the closest to point D. Use it as an initial guess to find a periodic orbit with period 4. What is the x-coordinate of this periodic point ?

Q7.4 $\quad$ Henon map continued -- the number of intersection regions

If we iterate region 0BCD forward 4 times, and iterate 0BCD backward 3 times, how many intersection regions will we get ? Needless to say, the answer better be a positive integer.