Stochastic webs with Fourfold Rotation Symmetry

Abstract

Weak chaos in nonlinear Hamilton systems has been studied extensively over the last two decades. A lot of attention was paid to the two-dimensional Chirikov-Taylor symplectic (i.e. area-preserving) standard map, modelling (part of) the motion of a charged particle in an inhomogeneous kicking electric field. This map displays weak chaos if it is perturbed from integrability (Chirikov, 1979; Lichtenberg and Lieberman, 1983). In the standard map, the chaotic motion is trapped between KAM-curves if the perturbation from integrability (here the strength of the electric field) is smaller than some threshold value. This causes a separation of chaotic regions in so-called stochastic layers. Only if the perturbation exceeds the threshold value, the chaotic motion is not trapped and the stochastic layers connect to form a two-dimensional infinite net of chaotic saddle connections, a so-called stochastic web, along which chaotic motion proceeds. The latter process is called stochastic diffusion. In the standard map, the threshold for stochastic diffusion is related to the break-up of the last KAM-curve.