Abstract
Bifurcation and stability properties of a nonlinear, autonomous, codimension-3 system are studied. Attention is focused on the vicinity of a compound critical point at which the Jacobian has a double zero eigenvalue of index one and a pair of pure imaginary eigenvalues. A unification technique and harmonic balancing procedure yield simplified differential equations which govern the local dynamics. Thus, equilibrium solutions associated with generic bifurcations, Hopf bifurcations, secondary bifurcations, and bifurcations into quasi-periodic motions on invariant tori are discussed for the first time. Criticality conditions associated with various bifurcations as well as stability conditions are derived in explicit terms. An illustrative example is analyzed.
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