To the left above you see that for a = -0.1, b = 0.5 the red unstable
separatrix of the saddle point x_{1} goes to attractor
x_{2 }. 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 x_{1 }, x_{2} 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). |
And something attracts all neighbouring points to this very strange set :)
One more funny repeller for b > 1. You can see unstable period-2 cycle "trace" here. |