Physica D 50 (1991) 155-i76 North-Holland Heteroclinic orbits in a spherically invariant system Dieter Armbruster Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA and Pascal Chossat Uniuersit~ de Nice, Laboratoire de Math~matiques (UA CNRS 168), Parc Valrose, 06034 Nice Cede.x, France Received 14 June 1990 Revised manuscript received 3 December 1990 Accepted 4 December 1990 Communicated by J. Guckenheimer The existence and stability of structurally stable heteroclinic cycles are discussed in a codimension-2 bifurcation problem with O(3)-symmetry,when the critical spherical modes 1= 1 and l = 2 occur simultaneously. Several types of heteroclinic cycles are found which may explain aperiodic attractors found in numerical simulations for the onset of convection in a self-gravitating fluid in a spherical shell. 1. Introduction Steady state mode interactions for a convection problem in a spherical shell have been analyzed by Friedrich and Haken [7]. Their numerical study of the resulting dynamical system revealed apparently chaotic trajectories which connected different equilibria of the system. These trajectories are suggestive of heteroclinic connections: They start from a neighborhood of an equilibrium and successively explore regions of phase space close to other equilibria before eventually coming back to the first one. We prove the existence of these heteroclinic cycles by applying the ideas developed recently for systems with symmetry by Guckenheimer and Holmes [11], Armbruster, Guckenheimer and Holmes [2], and Melbourne, Chossat and Golubitsky [15] and study their asymptotic stability. We find three other types of heteroclinic cycles which were not noticed in Friedrich and Haken's paper. All these objects share the following two properties: (i) they are robust under small perturbations of the equations, as long as these perturbations do not break the spherical symmetry of the system; (ii) for initial conditions close to the heteroclinic cycle, the asymptotic behavior exhibits intermittency, with long periods of time spent close to each of the equilibria (or limit cycles, if any) of the cycle, and sudden "jumps" from one stateto another. The physical interpretation of these solutions, when applied to planetary convection, is that "sudden" changes in the pattern of convection can exist without the need of an externally triggering event. Moreover, one feature of some of these heteroclinic cycles is that states with reversed polarities are successively explored which is very reminiscent of the aperiodic reversal of the earth's magnetic dipole field in geological times. We conjecture that the geodynamo follows a heteroclinic cycle of the same kind as those presented in this work. 0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)