TAKENS-BOGDANOV RANDOM-WALKS

Takens-Bogdanov bifurcations have Linearized dynamics that are nondiag onalizable. The nonlinear response of such systems to perturbations is quite distinct from systems with diagonalizable linearizations. While Takens-Bogdanov systems are of central importance in the theory of lo cal codimension-two bifurcations, their physical relevance is unclear because they are topologically fragile. Only those properties that are robust under the breaking of the exact degeneracy required for nondia gonalizability are likely to have any physical significance. In this p aper we consider the steady-state noise response of degenerate and nea rly degenerate nodes in N dimensions. Escape times are computed for th e subcritical case and shown to obey scaling relations that are differ ent from those of normal systems. The scaling behavior at high noise l evels can be extracted from the related Fokker-Planck equation and is robust to weak breaking-of the exact degeneracy.