Mathematical definition

Consider the following differential equation

$\displaystyle \frac{du}{dt} = f(u,\alpha)$     (44)

with $u \in \ensuremath{\mathbf{R}}^n$ and $\alpha \in \ensuremath{\mathbf{R}}$. A periodic solution with period $T$ satisfies the following system
$\displaystyle \left\{ \begin{array}{l}
\frac{du}{dt} = f(u,\alpha) \\
u(0) = u(T)
\end{array}\right.\ .$     (45)

For simplicity the period $T$ is treated as a parameter resulting in the system
$\displaystyle \left\{ \begin{array}{l}
\frac{du}{d\tau} = T\:f(u,\alpha) \\
u(0) = u(1)
\end{array} \right.\ .$     (46)

If $u(\tau)$ is its solution then the shifted solution $u(\tau+s)$ is also a solution to (46) for any value of $s$. To select one solution, a phase condition is added to the system. The complete BVP (boundary value problem) is
$\displaystyle \left\{ \begin{array}{rl}
\frac{du}{d\tau} - Tf(u,\alpha) & = 0 \...
... \\
\int_0^1 \langle u(t),\dot u_{old}(t) \rangle dt & = 0
\end{array} \right.$     (47)

where $\dot u_{old}$ is the derivative of a previous solution. A limit cycle is a closed phase orbit corresponding to this periodic solution.