# Linearization around a limit cycle versus stability of Poincare map

Thank you in advance for answering the following silly question. I am currently studying 2D limit cycles, say of the system

$\dot{x} = f(x)$.

Assume that the periodic solution $\gamma(t)$ is a stable limit cycle. There is plenty of literature available to study stability of limit cycles, e.g., stability of fixed points of the Poincare map.

However, my first intuition to study stability was to linearize the problem around the limit cycle trajectory $\gamma(t)$ and to study the stability of the linearized problem (with time dependent coefficients):

$\dot{\tilde{x}} = \mathcal{D}(f)(\gamma(t)) \tilde{x}$,

where $\tilde{x}$ is a small perturbation. My expectation would be that $\tilde{x}\to 0$ as $t\to \infty$. However if I numerically simulate this linear system for some sample problem, rather than observing $\tilde{x}\to 0$, I see that $\tilde{x}$ is non-zero and periodic (with same period as $\gamma(t)$).

I am sure that this indicates a fundamental misunderstanding, but can someone help me understand why this might occur?

• Floquet theory. – nonlinearism Oct 16 '15 at 21:10
• Something something can't believe they let Matt out of prison something something. – oxeimon Oct 16 '15 at 23:52