# Plotting Invariant Manifolds of the Henon Map

Given the following map:

\begin{align} & x_{n+1}=-y_n+2x_n^2 \\ & y_{n+1}=\beta x_n \end{align}

for $$β \in (0,1)$$, $$x_n \in \mathbb{R}, y_n \in \mathbb{R}$$ (which is a one parameter version of the Henon Map), I have done my analysis from which I have deducted the existence of 2 fixed points:

$$$$(\bar{x_1},\bar{y_1})=(0,0) \quad (\bar{x_2},\bar{y_2})=\left( \frac{β+1}{2},β\frac{(β+1)}{2}\right)$$$$ The origin appears to be stable spirals (since the eigenvalues of the corresponding Jacobian are $$(λ_1,λ_2) \in \mathbb{C}$$ and $$|λ_1|<1, |λ_2|<1)$$ while the $$(\bar{x_2},\bar{y_2})$$ is an unstable saddle point.

That's about it, for the math involved. What I need to do now is first plot the phase space of this map around the fixed points with Mathematica for the various values of $$β$$. Perhaps it is a naive question but how can one do that since it is an iterating map? Could I use Stream Plot or a similar command?

Second, I would like to plot both the stable and unstable manifolds of the map. These are defined by the eigenvalues of the saddle point. I do know that all I have to do is get the correspoding eigenvectors $$(\vec{u}, \vec{v})$$ for the eigenvalues of the saddle point $$(\bar{x_2},\bar{y_2})$$.

By taking the slope of $$σ_1=u_2/u_1=-\sqrt{β^2+β+1}$$ I can then start with perhaps $$x_1=10^3$$ I.C. close to the unstable manifold and use $$(x_1,σ_1 x_1)$$ for the map to plot the points. Then I do the same for the stable manifold but now with the slope $$v_2/v_1=\sqrt{β^2+β+1}$$ and the inverse map. It is a simple procedure.

Problem is that I am not able to put it down to Mathematica and plot it. I just dont know how to do that. I have found something similar here: Basins of Attraction, but that would be for continuous phase space.

I would really appreciate your help.

Thank you.

One possibility is to plot the trajectories using StreamPlot:

b = 0.8; StreamPlot[{-y + 2 x^2, b x}, {x, -3, 3}, {y, -1, 3}]


And here you can scan through plots for different values of b:

Manipulate[StreamPlot[{-y + 2 x^2, b x}, {x, -10, 10}, {y, -10, 30}], {b, 0, 1}]

• Thank you for your answer. That is not however what I am looking for here. I am interested in invariant manifolds which define the motion in the phase space. Apr 20, 2016 at 1:22
• What this does is to "plot the phase space of this map around the fixed points" which is exactly what you asked for (in bold). You can adjust the x and y ranges if you want to see more closely about the fixed points. Apr 20, 2016 at 1:45
• Oh you are referring to my first question, sorry I rushed into answering. Thank you for helping! The hard part though is the second one. :/ Apr 20, 2016 at 1:48
• I think you need to be clearer about what you want. The eigenvectors are vectors (straight lines). The region of attraction of the {0,0} equilibrium is the parabolic region you can see in the image above, or plot for {y, -1, 20} and it's clearer. Everything else is the unstable region. Apr 20, 2016 at 1:57
• There is a homoclinic "curve" which covers the attractor and is defined by both the stable and unstable eigenvectors that give rise to the sadle. I need to plot that homoclinic orbit. Apr 20, 2016 at 2:00