Abstract
Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2
(codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except
the weak resonances) with no more than three Floquet multipliers on the unit circle are covered.
The resulting formulas are independent of the dimension of the phase space and involve solutions of
certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle,
as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the
cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2
bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundaryvalue
algorithms based on orthogonal collocation, rather than shooting techniques, which greatly
expands its applicability. The implementation is described in detail with numerical examples, where
numerous codim 2 bifurcations of limit cycles are analyzed for the first time.
Original language | English |
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Pages (from-to) | 722-788 |
Number of pages | 67 |
Journal | SIAM Journal on Applied Dynamical Systems |
Volume | 12 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2013 |
Keywords
- normal forms
- limit cycles
- bifurcations
- codimension 2