# Floquet theory and Poincaré theorem on the continuation of periodic orbit

I read about the Floquet theory and a theorem that it named Poincaré's theorem of the continuation of periodic orbit.

Poincaré's Theorem: Consider a dynamical system depending on the parameter $$\varepsilon$$ $$\dot x = f_\varepsilon (x)$$ Suppose that there exists for every $$|\varepsilon|< \varepsilon_0$$ a costant of motion $$E_\varepsilon(x)$$ for the system. Let $$p_0(t)$$ be a periodic orbit for the "unperturbed" system ( that is for $$\varepsilon=0$$ ) with "energy" $$E_0(p_0)=\bar E$$. Suppose that $$p_0(t)$$ has $$1$$ as Floquet multiplier with multiplicity 2. Then there exists $$\varepsilon_1$$ such that if $$|\varepsilon|<\varepsilon_1$$ the system $$\dot x = f_\varepsilon (x)$$ has a periodic orbit with energy equals to the energy of $$p_0(t)$$.

Floquet multipliers: Denoting the flow with initial data $$x$$ at time $$t$$ of the dynamical system $$\dot x = f_\varepsilon (x)$$ as $$\phi_\varepsilon^t (x)$$. Let $$T$$ the period of the periodic orbit of the unperturbed system and $$x_0$$ a point of the periodic orbit. Consider the matrix $$d\phi_0^T(x_0)$$. The eigenvalue of $$d\phi_0^T(x_0)$$ are called Floquet multiplier.

Now comes my question:

Consider the dynamic system

$$\begin{cases} \dot x = y - \varepsilon (x^2+y^2)x \\\dot y= -x - \varepsilon (x^2+y^2)y \end{cases}$$

Through the Lyapunov function $$V(x,y)=(x^2+y^2)/2$$ one can show that there is no periodic orbits for the system ( the origin is an asymptotically stable equilibrium ). The problem is that I think the hypothesis of the Poincaré theorem are verified:

-$$(x(t),y(t))=(cost,sint)$$ is a periodic solution with period $$T=2\pi$$;

-the matrix $$d\phi_0^T(x_0)$$ is the identity, so two eigenvalue $$1$$ (multiplicity $$2$$);

So the only thing to miss the Poincaré theorem is that there is no costant of motion $$E_\epsilon$$ for the considered system. Wherevere, I am quite sure that the system has an integral of motion.

Another thing that "I feel" is: there is a problem of "dimension"? Maybe the dimension 2 is too low and that's why the Poincarè theorem doesn't work in this case

So my first question is:

-Has the system I considered an integral of motion?

-If yes, why the Poincaré theorem doesn't work?

N.B. Another thing that "I feel" is: there is a problem of "dimension"? Maybe the dimension 2 is too low and that's why the Poincarè theorem doesn't work in this case

Thank you