logistic map
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The first and most simple system that we will consider is the logistic map having only one degree of freedom x and one system parameter r:

xn+1 = rxn(1-xn).

Its time scale is discrete: n=1, 2, ... . Starting at an arbitrary point x0 in the interval [0..1] and applying this map again and again yields a series of values which either converges to a fixed point xF or alternates between several fixed points or even exhibits completely irregular deterministic-chaotic behaviour, depending on the value of the system parameter r.

Remark: systems having a discrete time scale can behave chaotically even if they only have one degree of freedom. However, systems having a continuous time scale need at least 3 degrees of freedom for chaotic motions [Guckenheimer83]. Why? Obviously trajectories do never cross since each point in the state space defines a unique state of the system with a unique future development. Therefore a non-diverging trajectory in a two-dimensional space only can converge to a fixed point or a periodic orbit.

The logistic map is a non-linear system (quadratic in x ) whose output is fed back  (xn+1 depends on xn) and whose domain is mapped to itself (the interval [0..1]). These are necessary preconditions for the occurence of chaotic motion.

How to use the applet: in the left window you can adjust the parameter r using the arrow buttons or by mouse-clicking into the diagram. At the same time the parabola on the right side is recalculated. If you press the draw button a series of iterations starting from a randomly chosen value x0 is calculated and drawn. The color is switched from yellow to black after few iterations. Depending on the value of r the black iterations will display a fixed point or a periodic orbit or irregular behaviour.

At the logistic map one of the routes to chaos can be shown, i.e. the period doubling route. For r<r1=3 there is a trivial unstable fixed point at 0 and a stable fixed point at xF =(r-1)/r . For r1=3 the slope of the parabola at the stable fixed point (i.e. the first derivation) becomes larger than 1 and so the fixed point becomes unstable. The sequence of  xn now alternates between two values. For further incresing values of r further bifurcations occur. The distances between successive values rm where these bifurcations occur  get smaller and the equation  holds. The constant = 4.6692016... is called Feigenbaum constant. For values larger than r inf = 3.57... the sequence xn becomes irregular - the system behaves chaoticly. However, you can find periodic gaps within the chaotic range above rinf  , for example there is a 3-cycle at r=3.83.

Using the substitution x = -z/r + 1/2 with z in [-r/2 .. r/2] the logistic map becomes  zn+1 = zn2 + (1-r/2)r/2 . Here a relation to the mandelbrot map zn+1 = zn2 + c becomes obvious.

 

Literature

Although the logistic map has a very simple structure it is exhibiting essential characteristics of chaotic systems. Therefore it is frequently used as an introducing example. So you will find comprehensive descriptions in  [Guckenheimer83], [Schuster88], [Leven89] or [Ott93], e.g.

 

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