Past exam of the topics in applied mathematics course of the University of Oxford / 2011 / B / 5 / a

A periodic orbit of least period $k$ is a list of distinct points $x_0,\dots ,x_{k-1}$ satisfying $f(x_i)=x_{i+1}$ for $i<k-1$ and $f(x_{k-1})=x_0$. Equivalently $f^k(x_0)=x_0$, with no smaller positive iterate fixing $x_0$. For $k=1$ this is a fixed point.

Follow a perturbation through one complete circuit. The chain rule givesAfter $m$ circuits the linear perturbation is $M^m\delta x_0$. With distinct eigenvalues, the eigenvectors span over the complex numbers, and each component is multiplied by ${\lambda }_j^m$. Thus the periodic orbit is linearly asymptotically stable exactly whenThese are the orbit's discrete Floquet multipliers. Any modulus greater than one causes instability; a unit modulus prevents decay in the linear system, and nonlinear stability then needs further analysis. The finitely many intermediate derivative factors do not affect convergence to zero between complete circuits.

A period-$k$ orbit of a Poincaré map corresponds to a closed trajectory, hence a periodic orbit of the underlying flow, meeting the chosen section at the $k$ distinct successive return points. Its time period is the sum of the $k$ return times, not necessarily $k$ times a single common time. Stability of the return-map orbit describes transverse stability of that closed trajectory; the flow itself also has the neutral time-translation direction.