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A torus bifurcation of limit cycles (Neimark-Sacker, NS) generically corresponds to a bifurcation to an invariant torus, on which the flow contains periodic or quasi-periodic motion. It can be characterized by adding an extra constraint G=0 to (50) where G is the torus test function which has four components from which two are selected. The complete BVP defining a NS point using a minimally extended system is
| |
ì ï ï ï í
ï ï ï î
|
| |
| |
|
ó õ
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1
0
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áx(t), |
×
x
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old
|
(t) ñdt |
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|
|
| | (77) |
|
where
is defined by requiring
N3 |
æ ç ç
ç è
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ö ÷ ÷
÷ ø
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= |
æ ç ç ç
ç ç è
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ö ÷ ÷ ÷
÷ ÷ ø
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. |
| (78) |
Here v1 and v2 are functions and G11,G12,G21 and G22 are scalars and
where the bordering functions v01,v02,w11,w12, vectors w21and w22 are chosen so that N3 is nonsingular [].
This method (using system (78) and (79)) is implemented in the curve definition file neimarksacker. The discretization is done using orthogonal collocation over the interval [0 2].