2D invertible maps

A general 2D map can be written as (the page is based on [1])
    xn+1 = f1(xn , yn ),
yn+1 = f2(xn , yn ).
    (1)
The map is invertible if (1) can be solved uniquely for xn and yn as functions of xn+1 and yn+1, xn = g1(xn , yn ) and yn = g2(xn , yn ).   That is, it is possible to go either forwards or backwards in time. A 2D invertible map can easily be constructed from a 1D noninvertible map F(x) as follows:
    xn+1 = F(xn ) + yn ,
yn+1 = bxn
.
    (2)
For b = 0,   xn+1 = F(xn ), and the noninvertible 1D map is recovered. However, as long as b ≠ 0, no matter how small it is, the map (2) is invertible:   xn = yn+1/b ,   yn = xn+1 - F(yn+1/b) . On the other hand, if b is sufficiently small, the variation of x is well described by the 1D map F(x). Furthermore, for small b, the range of of yn will be small compared to that for xn [see (2)].
The Jacobian of the map is
      | 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.

[1] Edward Ott Strange attractors and chaotic motions of dynamical systems Rev. Mod. Phys. 53, 655-671 (1981)


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updated 27 June 04