x_{n+1} = f_{1}(x_{n} , y_{n} ),
y_{n+1} = f_{2}(x_{n} , y_{n} ). | (1) |
x_{n+1} = F(x_{n} ) + y_{n} ,
y_{n+1} = bx_{n} . | (2) |
| 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 (x_{n} , y_{n} ) 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.
[1] Edward Ott Strange attractors and chaotic motions of dynamical systems Rev. Mod. Phys. 53, 655-671 (1981)