Chaos and order in 4D symplectic maps and 2 1/2 degree of freedom Hamiltonian systems

Supervisor

Prof. Robert Dewar Phone: 6125 2949

The symplectic map problem can be regarded as a model for a Poincaré plot of nonadiabatic particle motion in a static 3-D magnetic field, or as a stroboscopic plot of adiabatic motion in a 3-D field when there is a periodic pumping term. Even when there are many invariant tori (KAM surfaces), which act as barriers to diffusion, diffusion along channels criss-crossing the entire phase space can still occur in 4-D phase space because the tori are only 2-dimensional, allowing sufficient topological freedom for the particle to "go around" the torus. However this topological (Arnol’d) diffusion is difficult to visualize. Also, the efficient determination of the critical nonlinearity for breakup of the last KAM surface in a 4-D map is still an unsolved problem, so the transition from Arnol’d to Chirikov diffusion is difficult to determine.

This can be studied using the 4-D symplectic map introduced by Froeschlé to model the time evolution of elliptic galaxies, which is actually two 2-D standard maps with nonlinearity parameters a and b, respectively, and a coupling term proportional to the parameter c.

y'1 = y1 + (a/2p)sin 2px1 + (c/2p)sin 2p(x1 + x2) y'2 = y2 + (b/2p)sin 2px2 + (c/2p)sin 2p(x1 + x2) x'1 = x1 + y'1 x'2 = x2 + y'2

This area of research has already produced two medal-winning Honours theses, and there is much more that can be done, either at the mathematical end, e.g. by generalizing the flux-minimization concept to 4D, by extending the work in to designing an experiment for the H-1NF heliac, or by developing new visualization methods using the WEDGE (or a combination of these).