Next: Continuation of LPC
Up: Period Doubling
Previous: Mathematical definition
  Contents
In MatCont / CL_MatCont there are four generic codim 2 bifurcations that can be detected along
the flip curve:
- Strong 1:2 resonance. We will denote this bifurcation with R2
- Fold - flip, Limit Point - Period Doublingor We will denote this bifurcation with LPPD
- Flip - Neimark-Sacker, denoted as PDNS
- Generalized period doubling point, denoted as GPD
To detect these singularities, we define 4 test functions:
- f1=(v1*)TW1LC×Mv1M
-
f2=(y1*)TW1(F(u0,1(t)))C
-
f3=det((M \odot M ) - In )
-
f4=c
where c is the coefficient defined in (55), M is the monodromy matrix and \odot is the bialternate product.
The v1's and y1's are obtained as follows. For a given z Î C1([0,1],\mathbb Rn) we consider three different discretizations:
Formally we further introduce LC×M which is a structured sparse matrix that converts a vector zM of values in the mesh points into a vector zC of values in the collocation points by zC=LC×MzM.
We compute v1M by solving
The normalization of v1M is done by requiring åi=1Nåj=0msjá(v1M)(i-1)m+j,(v1M)(i-1)m+jñDi=1 where sj is the Gauss-Lagrange quadrature coefficient and Di is the length of the i-th interval.
By discretization we obtain
(v1W*)T |
é ê
ê ë
|
|
ù ú
ú û
|
disc
|
=0. |
|
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=[1/2]. We compute y1W* by solving
(y1W*)T |
é ê
ê ë
|
|
ù ú
ú û
|
disc
|
=0 |
|
and normalize y1W1* by requiring åi=1Nåj=1m|((y1*)W1)ij|1=1. Then ò01 áy1*(t),F(u0,1(t))ñdt is approximated by (y1*)TW1(F(u0,1(t)))C and if this quantity is nonzero, y*1W is rescaled so that ò01 áy1*(t),F(u0,1(t))ñdt=1.
a1 can be computed as (y*W1)T(B(t,v1M,v1M))C.
The computation of (h2,1)M is done by solving
|
ì ï í
ï î
|
| |
(B(t;v1M,v1M))C + 2a1(F(u0,1(t)))C, t Î [0,1] |
|
| | |
| | |
|
|
|
The expression for the normal form coefficient c becomes
|
c= |
1
3
|
((v*1W1)T(C(t;v1M, |
1
T
|
v1M,v1M)C+3(B(t;v1M,(h2,1)M)C |
|
-6 |
a1
T
|
(v*1W1)T(A(t)v1(t))C). |
|
|
|
|
The singularity matrix is: