Universität Hamburg - Fachbereiche - Fachbereich Mathematik

Explanation of the Applet Invariant Curves

We deal with a plane discrete dynamical system, defined by the delayed logistic map

(x,y)-> (y,a*y*(1-x)) which exhibits the appearance of invariant curves.

We are interested in the influence of the parameter a on the dynamics. This parameter together with others (see below) can be directly changed by the user, especially by using the scrollbar.

After having started the applet, an orbit of length nIt is shown, initiateted in (x,y). The points of the orbit have pixel-thickness d, the orbits are drawn with some delay, given by v to visualize the dynamics.

[xMin,xMax] x [yMin,yMax] represents the window in phase space.

The user can either change the parameters in the textfields and click on the button draw again or he can start a bifurcation scenario with different parameters a before and after a certain Hopf bifurcation at a=2. Every further bifurcation scenario button click changes a. For a<2 all orbits oscillates into the hyperbolic fixed point (1-1/a,1-1/a). For a=2 the fixed point is non-hyperbolic with two eigenvalues of modulus one (sixth root of unity). For a>2 invariant curves are borne increasing in size with a.

In any situation one can choose the starting point (x,y) just by a mouse click in the plane immediately followed by a simulation. The invariant curve undergoes a phase locking at a=2.17640 with rotation number 1/7. The phase locking interval ends at 2.20066. Further clicks on the bifurcation scenario button shows the attractor by zooming into it.