
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: x_{n+1 }= rx_{n}(1x_{n}). Its time scale is discrete: n=1, 2, ... . Starting at an arbitrary point x_{0 }in the interval [0..1] and applying this map again and again yields a series of values which either converges to a fixed point x_{F} or alternates between several fixed points or even exhibits completely irregular deterministicchaotic 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 nondiverging trajectory in a twodimensional space only can converge to a fixed point or a periodic orbit. The logistic map is a nonlinear system (quadratic in x ) whose output is fed back (x_{n+1 }depends on x_{n}) and whose domain is mapped to itself (the interval [0..1]). These are necessary preconditions for the occurence of chaotic motion. At the logistic map one of the routes to chaos can be shown, i.e. the period doubling route. For r<r_{1}=3 there is a trivial unstable fixed point at 0 and a stable fixed point at x_{F }=(r1)/r . For r_{1}=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 x_{n } now alternates between two values. For further incresing values of r further bifurcations occur. The distances between successive values r_{m} 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 x_{n} becomes irregular  the system behaves chaoticly. However, you can find periodic gaps within the chaotic range above r_{inf } , for example there is a 3cycle at r=3.83. Using the substitution x = z/r + 1/2 with z in [r/2 .. r/2] the logistic map becomes z_{n+1} = z_{n}^{2 }+ (1r/2)r/2 . Here a relation to the mandelbrot map z_{n+1} = z_{n}^{2 }+ 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.




