Prove that no periodic orbits exist

Question

Please help!

I need to prove for the system below that no periodic orbits exist when $V_0=0$: $\frac{d^2x}{dt^2}+\zeta_1 \frac{dx}{dt}+(k_{11}+k_{12}x^2)x-2\gamma\frac{dy}{dt}=V_0 cos(\omega t),$ $\frac{d^2y}{dt^2}+\zeta_2 \frac{dy}{dt}+(k_{21}+k_{22}y^2)y+2\gamma\frac{dx}{dt}=0$.

x and y are displacements, and for mechanical understanding, $\zeta_1$ and $\zeta_2$ are damping parameters, $k_{11}, k_{12}, k_{21}, k_{22}$ are stiffness parameters, $V_0$ is a prescribed input voltage and $\gamma$ is the rotation rate. Can someone help me out here?

Answer 1

You start with the standard trick to get from dynamic equations to energy equations, which is multiply first equation by $\dot x$, second by $\dot y$ and sum them:

$$ \ddot x\dot x+\zeta_1\dot x^2+(k_{11}+k_{12}x^2)x\dot x -2\gamma \dot y\dot x= \frac{d}{dt}\left(\frac12\dot x^2+\frac{k_{11}}2x^2+\frac{k_{12}}4x^4\right)+\zeta_1\dot x^2-2\gamma\dot x\dot y=0,\\ \ddot y\dot y+\zeta_2\dot y^2+(k_{21}+k_{22}y^2)y\dot y +2\gamma \dot x\dot y= \frac{d}{dt}\left(\frac12\dot y^2+\frac{k_{21}}2y^2+\frac{k_{22}}4y^4\right)+\zeta_2\dot y^2+2\gamma\dot x\dot y=0,\\ \frac{dE}{dt}=-\zeta_1\dot x^2-\zeta_2\dot y^2, \qquad\text{where}\qquad E=\frac14\left(2\dot x^2+2\dot y^2+2k_{11}x^2+2k_{21}y^2+k_{12}x^4+k_{22}y^4\right). $$

So now we know for sure, that if both $\zeta_1,\zeta_2>0$ (or both $<0$), then energy is strictly decreasing (or increasing) and therefore there are no periodic orbits.

However, if $\zeta_1=\zeta_2=0$ or $\zeta_1\zeta_2<0$, then with some parameters there can be periodic orbits.

For example, if $k_{11}=k_{21}=1$ and everything else is zero, it's just two harmonic oscillator and every orbit is periodic. Or if $\zeta_2=-\zeta_1=1$, all $k=0$ and $\gamma=1$, then for $z=(\dot x, \dot y)$ it is linear equation $\dot z=Az$ with pure imaginary eigenvalues. Again, every orbit is periodic.