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Steady-state mode interactions for \(D_3\) and \(D_4\)-symmetric systems. (English) Zbl 0928.37009

The paper deals with the study of generic bifurcations from a \(D_m\)-invariant equilibrium of a \(D_m\)-symmetric dynamical system \((m=3, 4)\) near points of codimension-2 steady-state mode interactions. The approach is based on techniques of normal forms, blowing-up, unfolding and group theory. The main results of the paper are related to symmetry-breaking bifurcations to primary branches, secondary steady-state and Hopf bifurcations, Bogdanov-Takens symmetry-breaking type bifurcations, as well as the study of the structurally stable heteroclinic cycle in a three dimensional representation of \(D_4\).

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
37M20 Computational methods for bifurcation problems in dynamical systems
68W30 Symbolic computation and algebraic computation
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