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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

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