Prototypical Hamiltonian Mapping — Standard Map

5See (1) G.M.Zaslavsky and B.V.Chirikov, Stochast ic instabilities in nonlinear oscillations , Sov. Phys, Usp. 14 (1972) 549, (2) B.V.Chirikov, A universal instability of many oscillator systems , Physics Reports 52 (1979) 265, and (3) G.Casati, B.V.Chirikov, D.L.Shepelyansky and 

Two-dimensional Area-Preserving Maps Consider a Hamiltonian of the form 

Letting Qn and p,i be the values of q and p just prior to the delta-function impulse, equations 4.42 may be integrated to yield the mapping 

Since the absolute value of the Jacobian J = a qn+i,Pn+i)/d qn,Pn) = 1, we see that this discrete-time map is indeed area-preserving. 

It is customary to put both variables on the unit torus that is, q and p are taken to be periodic variables with period equal to one. The sole control parameter, Q, determines the extent of the nonlinearity. As a increases, there is a dramatic transition between regular smooth orbits and trajectories that are almost completely chaotic. 

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