Poincaré map

A local transverse section to a flow is mapped back to itself by the next intersection of an orbit. Fixed points represent periodic orbits, and the derivative of this return map determines their transverse stability.

For $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, use entry $(x,h)$ and exit $(h,Y)$. If $\mu =k^2>0$, $x>k$, integration gives $Y=h[(h+k)(x-k)/((h-k)(x+k)){]}^{\lambda /(2k)}$. A smooth global return has $P(x)=\nu +cY+O(Y^2)$. If $\lambda >2k$, its derivative vanishes as $x\downarrow k$, and a small $\nu -k>0$ yields an attracting fixed point. If $\mu =-k^2<0$, the passage is $Y=h\exp [-(\lambda /k)(\arctan (h/k)-\arctan (x/k))]$. For small parameters the resulting global return is strongly contracting, with an attracting fixed point near $\nu$ on either side of zero.

If a planar equilibrium with purely imaginary linear eigenvalues has radial return map $r\mapsto r+\kappa r^3+O(r^4)$, then $\kappa <0$ gives nonlinear attraction and $\kappa >0$ gives repulsion. Vanishing linear decay therefore does not imply a centre.