Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to apply Floquet theory to analyse the stability of the periodic solutions for a system of differential equations. I understand the theoretical portion but how can I actually find the Floquet exponents. I searched a lot but i couldn't find any simple example. Would someone help me with an example for a two by two system of ode. I will be grateful.

PS: For example my differential equation is: $\dot{x}=\mu-d x$ $\mu$ and $d$ are periodic functions of t. I am adding more work that i did on this problem. I made it homogeneous equation by using a transformation $y=x-\frac{\mu}{d}$ and the equation becomes: $$\dot{y}=-d y$$ Solving this gives me: $$y=c_0 e^{\int{-d(t)dt}}$$ Now the $$\phi(t)=e^{\int{-d(t)dt}}$$ and $C=1$. Here $\phi(t)$ is the fundamental matrix. This means that the Floquet multiplier is $1$. Is it so?

share|cite|improve this question
I saw another post on floquet multipliers which was answered by Anon but the in that question, the differential equation was very simple because it was it was homogeneous, i guess. – math Jan 10 '13 at 2:28
Do you have access to the book "An introduction to Dynamical Systems" by Arrowsmith and Place? See section 2.6 and the corresponding problems, especially 2.6.5, which has an excellent example. Regards – Amzoti Jan 10 '13 at 5:29
Thank you very much @Amzoti I will look into this book. – math Jan 10 '13 at 5:50
Also, do these notes Complex Eigenvalues, Floquet help, Periodic Linear Systems? Regards – Amzoti Jan 10 '13 at 15:28
@Amzoti I have another question: If i have an equation like this: $$\dot{x}=\mu g(x)-d x$$. Where $\mu$ and $d$ are periodic in time. The function $g(x)$ is a Sigmoidal Hill function ($g(x)=\frac{x^n}{x^n+K^n}$) where $n$ is anything like 1,2,3... and K is a constant. How can I prove that there exists a periodic solution to this equation? I could not find anything relevant to my problem and I am really struggling with it. Sorry if I am being repetitive. – math Jan 10 '13 at 15:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.