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A Hopf bifurcation theorem for singular differential–algebraic equations

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  • Beardmore, R.
  • Webster, K.

Abstract

We prove a Hopf bifurcation result for singular differential–algebraic equations (DAE) under the assumption that a trivial locus of equilibria is situated on the singularity as the bifurcation occurs. The structure that we need to obtain this result is that the linearisation of the DAE has a particular index-2 Kronecker normal form, which is said to be simple index-2. This is so-named because the nilpotent mapping used to define the Kronecker index of the pencil has the smallest possible non-trivial rank, namely one. This allows us to recast the equation in terms of a singular normal form to which a local centre-manifold reduction and, subsequently, the Hopf bifurcation theorem applies.

Suggested Citation

  • Beardmore, R. & Webster, K., 2008. "A Hopf bifurcation theorem for singular differential–algebraic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 1383-1395.
    • Handle: RePEc:eee:matcom:v:79:y:2008:i:4:p:1383-1395
    DOI: 10.1016/j.matcom.2008.03.009
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