Bifurcations and chaos in the ϕ4theory on a lattice

Abstract

The authors have studied numerically and analytically the discrete phi ⁴ model defined by Bak and Pokrovsky (see Phys. Rev. Lett., vol.47, no.13, p.958-61, 1981). The model may describe Peierls systems, structural instabilities, metal-insulator transitions, etc., in solid state physics. The phi ⁴ theory can be formulated as an area-preserving two-dimensional mapping. This mapping exhibits infinite series of period-doubling bifurcation leading to chaos. The bifurcations are characterised by universal numbers delta =8.72109 ... and alpha =4.0180 ..., which appear to be identical to those found by Bountis (1981) for the Henon mapping, but different from the Feigenbaum numbers for dissipative systems. In addition, novel features arise because of marginally stable fixed points and the splitting of one 2-cycle orbit into two 2-cycle orbits.