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Consider the following differential equation
with u Î Rn and a Î R. A periodic solution with period T satisfies the following system
For simplicity the period T is treated as a parameter resulting in the system
If u(t) is its solution then the shifted solution u(t+s) is also a solution to (49) for any value of s. To select one solution, a phase condition is added to the system. The complete BVP (boundary value problem) is
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ì ï ï í
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
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(t) ñdt |
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| | (50) |
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where [u\dot]old is the derivative of a previous solution. A limit cycle is a closed phase orbit corresponding to this periodic solution. This system is discretized using orthogonal collocation [3], the same way as it was done in AUTO [5]. The left hand side of the resulting system is the defining function F(u,T,a) for limit cycles.