In CL_MatCont there are five generic codim 2 bifurcations that can be detected along the fold curve:
A Generalized Hopf (GH) marks the end (or start) of an LPC curve.
To detect the generic singularities, we first define bp+5 test functions, where bp is the number of branch parameters:
In the yi expressions w is the vector computed in (60) and bi (branch parameter) is a component of a.
In the second expression ybp+1, we compute v1M by solving
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By discretization we obtain
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To normalize (j1*)W1 we require åi=1Nåj=1m|((v1*)W1)(i-1)m+j|1=1. Then ò01 áj1*(t),v1(t)ñdt is approximated by (j1*)TW1LC×Mv1M and if this quantity is nonzero, j*1W is rescaled so that ò01 áj1*(t),v1(t)ñdt=1.
So the third expression for the normal form coefficient b becomes
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In the fourth expression, M is the monodromy matrix.
In the fifth expression, M2 = (M - In)2, restricted to the subspace without the two eigenvalues with smallest norm.
The number of branch parameters is not fixed. If the number of branch parameters is 3 then this matrix has three more rows and columns. This singularity matrix is automatically extended:
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