The loss of stability of symmetrical equilibrium positions

Systems of differential equations possessing a finite (or compact) symmetry group and depending on one parameter are considered. The nature of the los s of stability of equilibrium positions is investigated in cases when, owin g to symmetry, the linearized problem has multiple eigenvalues. Conditions are presented that determine whether the loss of stability when the paramet er is varied is soft or hard, for double eigenvalues lambda - zero or pure imaginary. Cases of triple zero eigenvalues lambda, corresponding to tetrah edral (or cubic) symmetry, are considered. (C) 1999 Elsevier Science Ltd. A ll rights reserved.