# Necessary and sufficient condition for a periodic system to have a non-trivial periodic solution.

Consider the linear periodic system in $\mathbb{R}^n$ $$\begin{cases} \dot{x}(t) & = A(t)x(t),\\ x(0) & = x_0, \end{cases} \label{eq:Floquet} \tag{1}$$ where $A(t)$ is a real $n\times n$ matrix function which is smooth in $t$ and periodic of period $T>0$. Floquet theory states that there exists at least 1 non-trivial solution $\chi(t)$ satisfying $$\chi(t+T) = \mu\chi(t), \ \ t\in(-\infty, \infty), \label{eq:Floquet2} \tag{2}$$ where $\mu$ is an eigenvalue of the Floquet matrix. $\mu$ is more well-known as a Floquet multiplier of the system. What is the necessary and sufficient conditions so that \eqref{eq:Floquet} has a non-trivial $T$-periodic solution? By non-trivial I meant a periodic solution with minimal period $T$.

It seems like according to \eqref{eq:Floquet2}, one would want to impose condition on the Floquet matrix such that it has eigenvalue $\mu=1$, but I know nothing about this, not to mention that this sounds like a very strong condition. Spefically, I am looking for conditions that stem from Floquet theory.

• So, if you know that for any $x(0)$ there exists some special matrix $A$ that describes you what happens after time $T$, namely $x(T) = A x(0)$, and you want to find non-trivial periodic solutions with period that is commensurable to $T$, what kind of eigenvalues this matrix should have? – Evgeny Jan 4 '17 at 8:55
• @Evgeny real and 1? Oh wait, complex I think....... ? – Chee Han Jan 4 '17 at 18:30
• Close enough. If you know that there is a periodic solution with period $\frac{p}{q}T$, the trajectory which started from point $x(0)$ must return to itself after time $pT$. But at the same time it passes through the point $x(pT)$ which relationship with $x(0)$ is known. Do you see how you can answer your question now? – Evgeny Jan 4 '17 at 22:12
• @Evgeny Apologise for the late reply, but from what I understand, your comment leads to the same conclusion as mine (see above). I can surely say that $x(t)=\Phi(t)x(0)$, where $\Phi(t)$ is a fundamental matrix of the system such that $\Phi(T)=I$. Now $x(T)=\Phi(T)x(0)$, where $\Phi(T)$ is the Floquet matrix (as far as how I defined it). So in order to have a non-trivial periodic solution, one requires $\Phi(T)$ to have unit eigenvalue? – Chee Han Jan 5 '17 at 19:59
• @Evgeny I had a thought about this yesterday again, and I think it suffices to quote Floquet theorem, which is statement (1) above and mention that there exists a non-trivial $T$-periodic solution if $\Phi(T)$ has at least one eigenvalue $\mu=1$. If you can say anything more than this, I would be happy to listen about it. – Chee Han Jan 5 '17 at 20:01