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Mathematical definition

Consider the following differential equation

du

dt

= f(u,a)
(47)

with u Î Rn and a Î R. A periodic solution with period T satisfies the following system

ì
ï
í
ï
î
du

dt

= f(u,a)
u(0) = u(T)
 .
(48)

For simplicity the period T is treated as a parameter resulting in the system

ì
ï
í
ï
î
du

dt

= T0f(u,a)
u(0) = u(1)
 .
(49)

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

ì
ï
ï
í
ï
ï
î
du

dt

- Tf(u,a)
= 0
u(0) - u(1)
= 0
ó
õ
1

0 
áu(t),
×
u
 

old 
(t) ñdt
= 0
(50)

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.