Chaos and order in 4D symplectic maps and 2 1/2 degree of freedom Hamiltonian systems
A possible Honours project: Supervisor: Prof Robert L. Dewar+61 (0)2 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 [1,2]. 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 [3], 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/2π)sin 2πx1 + (c/2π)sin 2π(x1 + x2)
y'2 = y2 + (b/2π)sin 2πx2 + (c/2π)sin 2π(x1 + x2)
x'1 = x1 + y'1
x'2 = x2 + y'2
A visualization of periodic and chaotic orbits and KAM tori made using Mathematica 3.0 3-D graphics and colour coding for the fourth dimension is presented below (annotations added with Adobe Illustrator).
Figure: Puncture plots for the Froeschlé map with parameters a = 0.2, b = 0.1 and c = 0.05.
This area of research has already produced two medal-winning Honours theses, and there is much more [2,3] that can be done, either at the mathematical end, e.g. by generalizing the flux-minimization concept [4,5] to 4D, by extending the work in [2] to designing an experiment for the H-1NF heliac, or by developing new visualization methods using the WEDGE (or a combination of these).
References
[1] A.J. Lichtenberg, Arnol'd diffusion in a torus with time-varying fields, Phys. Fluidss B 4, 3132 (1992).
[2] S. Hardy, Arnol'd diffusion in magnetic confinement geometries, Honours Thesis, Department of Physics & Theoretical Physics, The Australian National University, 1993.
[3] V. Robins,Periodic orbits and invariant tori in symplectic maps, Honours Thesis, Department of Mathematics, The Australian National University, 1994.
[4] R.L. Dewar and A.B. Khorev. Rational Quadratic-Flux Minimizing Circles for Area-Preserving Twist Maps . Physica D 85, 66-78 (1995).
[5] S.R. Hudson and R.L. Dewar. Analysis of perturbed magnetic fields via construction of nearby integrable fields . Phys. Plasmas 6, 1532-1538 (1999).
©Robert L. Dewar, 1999-2004
Last modified on September 22, 2004. Please send comments or enquiries to Robert Dewar
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