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(1-x_n).

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 x_F 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  (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<r₁=3 there is a trivial unstable fixed point at 0 and a stable fixed point at x_F =(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  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 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  z_n+1 = z_n² + (1-r/2)r/2 . Here a relation to the mandelbrot map z_n+1 = z_n² + 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.