Invariant density and finite-time Lyapunov exponents for nonhyperbolic dynamics on Henon attractor
In this project I investigated numerically the relation between the invariant density and finite time Lyapunov exponents and the angle between stable and unstable manifolds for Henon map. The parameter values used are a=1.4 and b=0.3. For these parameter values the Henon map is nonhyperbolic, i.e. the stable and unstable manifolds are tangent and angle between them is zero for a nonzero fraction of points on the attractor. I found that there is no noticeable correlation between the invariant density and angle between manifolds, even if one computes the invariant density at the image points. There was found definite scaling of the finite time Lyapunov exponents measured by going 13 iterations into the past with angle between manifolds.
It is well known that many real physical systems have nonhyperbolicities for a certain common parameter values. Forced damped pendulum is one of the examples of systems possessing such behavior. Henon map is another, although purely mathematical, example of such system. Many mathematical results for dynamical systems are restricted to hyperbolic systems. For example, the shadowing lemma which tells us whether there is a true trajectory in the neighborhood of a computer-generated trajectory is proven only for hyperbolic systems. The nonhyperbolicities produce extremely complicated types of behavior and understanding properties of nonhyperbolic systems has become recently a subject of interest in nonlinear science.
In this paper I studied the Henon map
with parameter values a=1.4 and b=0.3 . It was shown  that at this values of parameters there exist points on the attractor with arbitrarily small angle between stable and unstable manifolds. Figure 1 shows the angle distribution for the above parameter values for 50000 points on the attractor.
Figure 1. Probability vs. angle for Henon attractor.
As one can readily see, the angle is not bounded away from zero for the parameter values used. In the vicinity of the nonhyperbolic fixed points we have a neutral equilibrium, because there is no stable or unstable directions. Therefore it is interesting to see what happens to the invariant density and finite time Lyapunov exponents at the points on the attractor. In the following part of the paper I will present the results of numerical simulations and analysis of the results.
II. ANGLE COMPUTATION
In order to determine whether the invariant set is hyperbolic or nonhyperbolic, we first need to calculate the angles between stable and unstable manifolds for points x belonging to the set. In this section I present the numerical method that I used to compute the angle between stable and unstable manifolds on the Henon attractor .
In order to find the stable direction at the point x we first iterate this point forward under the Henon map N times (I used N=8) and we get the point MN(x). Then we generate 2 points in the vicinity of MN(x) by adding and subtracting a small vector e=(e,e), where e was taken to be 10-8 , to the coordinates of the point MN(x). After that we iterate both created points under the inverse map M-1(x) N times and the resulting vector s=M-N(MN(x)+e )-M-N(MN(x)-e ) points along the stable direction at point x. In the similar way the unstable direction was found: u=MN(M-N(x)+e )-MN(M-N(x)-e ) and the angle between them defined as the minimum of two angles of intersection was computed using
This method was used to generate Figure 1. The result on Figure 1 looks very similar to the analogous result obtained in .
III. INVARIANT DENSITY
The invariant density was computed by iterating the Henon map starting at a point inside the basin of attraction for 100 iterates to get a point on the attractor. Then I iterated the resulting point for another 50000 iterates and recorded the values. Next the attractor was covered by 200 by 200 equally spaced grid and number of points inside every grid cell was counted. Figure 2 shows the invariant density on the Henon attractor for my parameter values. The region shown on the plot is for x from [-1.4,1.4] and y from [-0.4,0.4]. The plot is done in grayscale and darker points indicate higher values of invariant density.
Figure 2. The invariant density on 200x200 grid on Henon attractor for x from [-1.4,1.4] and y from [-0.4,0.4] computed for 50000 iterates.
IV. FINITE TIME LYAPUNOV EXPONENTS
I estimated finite time Lyapunov exponents at given point (x1, y1) by linearizing the Henon map and then evaluating its Jacobian at the point and then at the image (or pre-image, if I wanted to go into the past) of the point and then repeating this procedure for several time steps. Then I took the Jacobians, multiplied them all together and computed the eigenvalues of the resulting matrix. The natural logarithm of the smallest magnitude eigenvalue divided by the number of time steps gave the negative exponent and the natural logarithm of the other eigenvalue gave the other exponent:
This method is described in greater detail in . I noticed that for small number if iterations (t=3…5), both exponents sometimes came out negative, but as I increased the number of iterations past 10, the positive exponents always came out positive.
V. ANALYSIS OF THE RESULTS
As Dr. Daniel Lathrop suggested, I was looking for correlation between the invariant density and angle between stable and unstable manifolds for points on the attractor. To verify this hypothesis, I covered the region of the attractor for x from [-1.4,1.4] and y from [-0.4,0.4] with 200x200 grid and calculated the average angle inside every grid point. Figure 3 is the scatter plot of invariant density vs. the angle where angle is plotted on the x axis and density is on the y axis.
Figure 3. Angle vs. invariant density on the Henon attractor.
There seems to be no correlation between the invariant density and the angle between manifolds. I plotted the density at the image point vs. the angle at the point on the attractor and there was no correlation either.
Let us now perform the same kind of analysis with the finite time Lyapunov exponents. Figure 4 shows the scatter plot of the negative Lyapunov exponent computed by going 3 steps into the future. One can see definite structure there, buy there is no direct scaling of the exponent with angle.
Figure 4. Scaling of the negative Lyapunov exponent computed by going 3 steps into the future with the angle.
I found interesting that as angle goes to either zero or p there are only certain values of the Lyapunov exponent that are present. There are 6 values as angle goes to zero and 3 values as angle goes to p. There is also some kind of discontinuity present at the angles between 1.0 rad and 1.2 rad.
Figure 5. Scaling of the positive Lyapunov exponent measured by going 13 steps into the past with angle between stable and unstable manifolds. Green line is a fit
My final attempt to find some scaling involved computing the finite time Lyapunov exponents by into the past. I went 13 steps into the past for every point on the attractor and computed the positive Lyapunov exponent. The result is shown on Figure 5. In this case there is a definite trend for positive Lyapunov exponent to increase with increasing angle. I also measured the negative Lyapunov exponent for this case. The result is presented on Figure 6. The negative Lyapunov exponent increases in magnitude with increasing angle.
Figure 6. Scaling of the negative Lyapunov exponent measured by going 13 steps into the past with angle between stable and unstable manifolds.
From the first look these scalings look like a power law. I tried to fit the data on Figure 5 to a power law and obtained the following scaling:
where x is the angle and y is the positive Lyapunov exponent. This result is quite approximate and more work will be required to obtain the correct scaling law.
In this paper I presented the numerical techniques and results of the experiments on the Henon map with the parameter values a=1.4 and b=0.3 which give rise to nonhyperbolicities at a nonzero fraction of points on the attractor. There was found no correlation between the angle and both the invariant density and image invariant density. There was found that finite time Lyapunov exponents measured by going into the past increase in magnitude with increasing angle between stable and unstable manifolds.
 Ying-Cheng Lai, Celso Grebogi, James A Yorke and Ittai Kan, Nonlinearity 6 (1993) 779-797.
 Edward Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993)