Henon map bifurcations

Saddle-node bifurcation

To the left above you see that for a = -0.1, b = 0.5 the red unstable separatrix of the saddle point x1 goes to attractor x2 . As it follows from the right picture the stable blue separatrix confines basin of attraction of this point (in the basin colors show how fast points go to attractor). Points are attracted faster along the yellow line corresponding to unstable separatrix.

Two points x1 , x2 are born in a saddle-node bifurcation under decreasing a. This bifurcation corresponds to tangent bifurcation in 1D. The left image with a = 0.07, b = 0.5 shows how the stable manifold bud appears before the bifurcation. Note that in this image colors show how fast points go to infinity (and purple points are delayed longer).

Period doubling bifurcations

For a = -0.5 the left fixed point turns into a saddle and period-2 cycle appears (two yellow regions to the right). The red unstable separatrix of the saddle x1 goes to the saddle x1 and the black stable one goes in reverse direction. Near x1 the red separatrix crosses the red one and makes oscillating heteroclinic structure. You can see it to the right (and for large N > 15 to the left). But all points leave this repeller and are attracted to the stable cycle.
This period-2 cycle correspond to the stable fixed points of the map f o2. Under decreasing a the next period doubling takes place and it turns into a saddle too. The last two pictures correspond to attracting period-4 cycle. You see heteroclinic structures made by crossing of the red and black separatrises. It would be interesting toe see separatrises of this new saddle :)

Henon strange attractor and homoclinic structures

Henon strange attractor a = -1.1 and b = 0.46 are plotted to the right. Compare it with the unstable manifolds Wu (red and green curves) of the hyperbolic points x1,2 to the left. To "refine" stable manifolds Ws (inside the blue and black regions), increase N. Then Wu become thinner and "disappear". Ws become very complicated.

You see that stable and unstable manifolds have many homo- and heteroclinic intersection points. Therefore in the region of these intersections there are countable set of unstable periodic orbits and continuum of chaotic orbits.

And something attracts all neighbouring points to this very strange set :)

Fractal repellers

At last for a = -1.4 all orbits go to infinity. To the right you see "trace" of stable manifolds.
One more funny repeller for b > 1. You can see unstable period-2 cycle "trace" here.

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updated 6 July 07