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A Fold bifurcation of limit cycles (Limit Point of Cycles, LPC) generically corresponds to a turning point of a curve
of limit cycles. It can be characterized by adding an extra constraint G=0 to (50) where G is the Fold test function. The complete BVP defining a LPC point using the minimal extended system is
| |
ì ï ï ï í
ï ï ï î
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| |
| |
|
ó õ
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1
0
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áu(t), |
×
u
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old
|
(t) ñdt |
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|
|
| | (72) |
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where G is defined by requiring
N1 |
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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. |
| (73) |
Here v is a function, S and G are scalars and
where the bordering functions v01,w01, vector w02 and scalars v02 and w03 are chosen so that N1 is nonsingular [].
This method (using system (73) and (74)) is implemented in the curve definition file limitpointcycle. The discretization is done using orthogonal collocation in the same way as it was done for limit cycles.