Continuous ``chaotic'' dynamics in two dimensions (Q2367678)

This paper is concerned with a system of equations first introduced by the same authors [Astrophys. J. 386, 215 (1992)] as a model of the dynamical behavior of the magnetic field of a neutron star. It consists of three nonlinear autonomous differential equations and exhibits what appears to be chaotic behavior. A simple coordinate transformation separates the system into a planar autonomous system and a scalar one. However, the planar vector field retains the chaotic appearance of the three-dimensional flow considered. This planar behavior, seemingly a violation of the Poincaré-Bendixson theorem, is seen to be a result of nondifferentiability (with respect to the dependent variables) at a singular point. The authors then introduce intuitively the so-called \(S\)- chaos (not well-defined in mathematics), where the behavior is the result of the sensitivity to initial conditions in the neighborhood of the singularity. Finally, necessary and sufficient conditions are claimed by the authors (without proof) for planar autonomous vector fields to be \(S\)-chaotic.