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In continuous-time systems there are eight generic codim 2 bifurcations that can be detected along the torus curve:
1:1 resonance. We will denote this bifurcation by R1
2:1 resonance point, denoted by R2
3:1 resonance point, denoted by R3
4:1 resonance point, denoted by R4
Fold-Neimarksacker point, denoted by LPNS
Chenciner point, denoted by CH.
Flip-Neimarksacker point, denoted by PDNS
Double Neimarksacker bifurcation point, denoted by NSNS
To detect these singularities, we first define 6 test functions:
y1=k-1
y2=k+1
y3=k-1/2
y4=k
y5=(v1*)TW1LC×Mv1M
y6=Re(d)
y7=det( M + In )
y8=det(( M2 \odot M2 ) - In )
where v1M is computed by solving
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(80) |
The normalization of v1M is done by requiring åi=1Nåj=0msjá(v1M)ij,(v1M)ijñti=1 where sj is the Gauss-Lagrange quadrature coefficient.
By discretization we obtain
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To normalize (v1*)W1 we require åi=1Nåj=1m|((v1*)W1)ij|1=1. Then ò01 áv1*(t),v1(t)ñdt is approximated by (v1*)TW1LC×Mv1M and if this quantity is nonzero, v*1W is rescaled so that ò01 áv1*(t),v1(t)ñdt=1.
We compute j1W* by solving
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and normalize j1W1* by requiring åi=1Nåj=1m|((j1*)W1)ij|1=1. Then ò01 áj1*(t),F(u0,1(t))ñdt is approximated by (j1*)TW1(F(u0,1(t)))C and if this quantity is nonzero, j*1W is rescaled so that ò01 áj1*(t),F(u0,1(t))ñdt=1.
We compute h20,1M by solving
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a1 can be computed as (j*W1)T(B(t,v1M,[`v]1M))C.
The computation of (h11,1)M is done by solving
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The expression for the normal form coefficient d becomes
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In the 7th test function, M is the monodromy matrix.
In the 8th test function, M2 = (M - In)2, restricted to the subspace without the two eigenvalues with smallest norm.
The singularity matrix is:
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(81) |