View source for Lin's method

===Bifurcation to a simple periodic orbit===

We look for a <math>2\omega</math> periodic solution <math>x(t)</math> that stays near the orbit of <math>q\ .</math>   If <math>x=q+z</math> and  <math>x(-\omega)=x(\omega)\ ,</math> then 
:<math>
\dot z(t)= D_x f(q(t),0)z(t) + g(z(t),\mu,t),\quad
z(\omega) - z(-\omega)= q(-\omega)-q(\omega).
</math>

Let <math>\rho =\min\{\text{Re} \lambda| \lambda\in \sigma D_x f(p,0), \text{Re}\lambda>0\},\quad
\nu =\min\{-\text{Re} \lambda| \lambda\in \sigma D_x f(p,0), \text{Re}\lambda<0\}\ .</math>
We assume  <math>0< \rho <\nu</math> and we make the generic assumptions:

(H1) <math>\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt \neq 0\ .</math>

(H2) <math>|q(t)| \sim C_1 e^{\rho t}, \; t\to-\infty\ ,</math> and <math>|\psi(t)| \sim C_2 e^{-\rho t}, \; t\to\infty\ .</math>

There is one bifurcation equation  <math>G(\omega,\mu)=0\ .</math> 
Since <math>|q(t)| \sim C_3 e^{-\nu t}, \; t\to \infty\ ,</math> and <math>|\psi(t)| \sim C_4 e^{\nu t}, \; t\to -\infty\ ,</math> we drop the smaller term <math>\psi(-\omega)q(\omega)</math> in the estimate:
:<math> G(\omega,\mu) =\psi(\omega)q(-\omega) + \mu \int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt + o(e^{-2\rho \omega}+|\mu|).
</math>

There are two important  cases:

(I) The dominant eigenvalue is real and simple. In this case <math>\psi(\omega)q(-\omega) \sim C_5 e^{-2\rho\omega}\ ,</math> and 
:<math>\mu\sim C_5 e^{-2\rho\omega}\left(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt\right)^{-1}.
</math>
The bifurcation to periodic orbits can happen only on one side of <math>\mu\ ,</math> and <math>\mu \to 0</math> as <math>\omega \to\infty\ .</math>

(II) The dominant eigenvalues are a pair of simple complex eigenvalues
<math>\rho \pm i \theta\ .</math> In this case <math>\psi(\omega)q(-\omega)\sim 
 C_6 e^{-2\rho\omega}\sin(2\theta\omega+\beta)\ ,</math> and
:<math>\mu\sim C_5 e^{-2\rho\omega}\sin(2\theta\omega + \beta)\left(\int_{-\infty}^\infty \psi(t)D_\mu f(q(t),0)dt\right)^{-1}.
</math>
There exist infinitely many periodic orbits when <math>\mu=0\ .</math>  The values of <math>\omega</math> are almost equally spaced. The periodic orbits can occur on both sides of <math>\mu=0\ .</math> The number of periodic orbits decreases as <math>|\mu|</math> increases.