This Demonstration shows a model of the magnetic field of a neutron star. It consists of two concentric spherical bodies with opposite charges and differing angular velocities. Assume there is viscous damping between them, as well as a uniform charge distribution on the outer body, such that its rotation yields a uniform magnetic field.

where and are the Cartesian components of the magnetization of the inner sphere, and is related to damping parameters. The reduced 2D system evidently violates the Poincaré–Bendixson theorem.

References

[1] F. W. Cummings, D. D. Dixon, and P. E. Kaus, "Dynamical Model of the Magnetic Field of Neutron Stars," Astrophysical Journal, 386(1), 1992 pp. 215–221. adsabs.harvard.edu/full/1992ApJ...386..215C.

[2] D. D. Dixon, F. W. Cummings, and P. E. Kaus, "Continuous 'Chaotic' Dynamics in Two Dimensions," Physica D65(1–2), 1993 pp. 109–116. doi:10.1016/0167-2789(93)90008-O.