# Hamiltonian Chaos - Demo 7

## Standard Map: Arnold Diffusion

At *a=1.2* all the original tori spanning the range of *x* have disappeared and chaotic orbits are quite evident. There also, of course, still remain periodic orbits (discrete repeating points in the map) and quasiperiodic orbits (the elliptical orbits surrounding then). It is interesting to enlarge different regions of the plot and to start orbits from various initial points to study the resulting structures, substructures.....

The disappearance of the last of the original tori means that a chaotic orbit starting from some initial condition near the origin can eventually "diffuse" to large values of *y*. This is known as Arnold diffusion. In the following demonstration (the same as above but over a larger *y* scale) start an initial condition near the origin and watch over many iterations. Some orbits starting within the subsidiary tori will remain bounded, but others will lead to trajectories that eventually diverge to large values. The motion in the *y* direction is diffusive, with <y^{2}> growing proportionate to the iteration number.

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Last modified Thursday, March 2, 2000

Michael Cross