Shigehiro Ushiki = Henon Julia Page


Movies of Julia sets of complex Henon maps in C2

Movies are stereo graphic. Look at the right picture with your left eye and the left picture with your right eye. Fractal objects, "Henon Julia set" lives in complex two dimensional space, that is, in real four dimensional space. You are looking at the projection of the object to a real three dimensional space. The object is either rotating in the four dimensional space or changing their form according to the variation of the parameter. Rotation in the four-dimensional space is quite different from the rotation in the three dimensional space.

The Henon mapping considered in this page is given by the following formula.
X = x^2 + c + b y
Y = x


See "Second Julia sets of complex dynamical systems in C^2 -- computer visualization -- " for more explanation.


Movies of Julia sets of complex Henon maps near saddle node bifurcation parameter.

Here we have some movies of Julia sets of complex Henon maps for b = -0.5. The fixed point of the Henon map has a saddle node bifurcation for parameters satisfying the equation c=0.25 * (b-1)^2. For such parameters, the fixed point has 1 as an eigenvalue. So, for b= -0.5, the saddle node bifurcation ( parabolic bifurcation ) occurs for parameter c = 0.5625. In our movies, when parameter c is below 0.5625, the system has an attractive fixed point. And when c is above this value, the fixed point becomes unstable and the Julia set becomes complicated.


Julia sets for volume preserving complex Henon maps

Here we have some movies of Julia sets of complex Henon maps which preserves the volume. In the henon map with our parameters, -b is the determinant of the map. More precisely, the algebraic determinant of the Jacobian matrix is equal to -b. And the determinant of the map regarded as a mapping of the real four dimensional euclidean space R^4, the determinant is equal to abs(b)^2.