xn+1 = f1(xn , yn ),
yn+1 = f2(xn , yn ).
xn+1 = F(xn ) + yn ,
yn+1 = bxn .
| F'(x) b | J = | | = -b . | 1 0 |Thus for |b| < 1 , we see that areas will contract by the factor |b| on each iteration of the map. Thus, if the generated sequence of pairs (xn , yn ) remains in a bounded region of the xy plane, then the sequence must asymtotically approach a subset of the region with zero area. This subset is called an attractor. For example, if the sequence becomes attracted to an N-point periodic cycle, then the attractor would be the N points plotted in xy plane, clearly a subset of zero area. Another possible subset of zero area that a sequence might asymptote to is a curve (a 1D subset).
There can be attractors which have noninteger dimension. Such attractors are called strange.
We still have an important question - why 2D dissipative, Hamiltonian and analytic maps are so different?
The dissipative Henon map.
The Hamiltonian standard map.
The quadratic map.
 Edward Ott Strange attractors and chaotic motions of dynamical systems Rev. Mod. Phys. 53, 655-671 (1981)