### 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.