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Problem: Write in explicit form Poincare's map for $\ddot x+\delta\dot x+\omega_0^2x=\gamma\cos\omega t$.

Find the stationary points and examine their stability.

An attempt at a solution:

The characteristic equation of the homogenous equation is $z^2+\delta z+\omega_0^2=0$

Its roots are $$z_{1,2}=\frac{-\delta\pm\sqrt{\delta^2-4\omega_0^2}}{2}$$

The solution to the homogenous differential equation is thus $$x(t)=Ae^{z_1t}+Be^{z_2t}$$

Let us now use the method of undetermined coefficients to find a particular solution as follows:

We will be looking for a solution in the form $$x_p(t)=a\gamma\cos\omega t + b\gamma\sin\omega t$$

Then $$\dot x_p(t)=-a\omega\gamma\sin\omega t + b\omega\gamma\cos\omega t,$$ $$\ddot x_p(t)=-a\omega^2\gamma\cos\omega t - b\omega^2\gamma\sin\omega t$$

Substituting we get $$-a\omega^2\gamma\cos\omega t - b\omega^2\gamma\sin\omega t + \delta(-a\omega\gamma\sin\omega t + b\omega\gamma\cos\omega t)+\omega_0^2(a\gamma\cos\omega t + b\gamma\sin\omega t)=\gamma\cos\omega t$$

Equating the coefficients, we obtain the following system: $$-a\omega^2+b\delta\omega+\omega_0^2a=1$$ $$-b\omega^2-a\delta\omega+\omega_0^2b=0$$

Therefore, $$a=\frac{\omega_0^2-\omega^2}{(\omega_0^2-\omega^2)^2+(\delta\omega)^2}\land b=\frac{(\delta\omega)^2}{(\omega_0^2-\omega^2)^2+(\delta\omega)^2}$$

Now the general solution is $$Ae^{z_1t}+Be^{z_2t}+a\gamma\cos\omega t + b\gamma\sin\omega t$$

Let us consider the initial conditions $x(0)=x_0, \dot x(0)=y_0$.

Using the initial conditions, we get $$A=x_0-a\gamma-B, \quad B=\frac{y_0-x_0z_1+a\gamma z_1-b\gamma\omega}{z_2-z_1}$$

Replace $\theta$ with $\omega t$, now lets write the equation as an autonomous system as such: $$\dot x=y$$ $$\dot y=-\delta y-\omega^2_0x+\gamma\cos\theta$$ $$\dot\theta=\omega (\text{mod}\quad 2\pi)$$

WLOG we may consider $\theta(0)=0$. Then the flow is $\phi^t(x_0,y_0,0)=(x(t),y(t),\omega t)$.

The section is $\Gamma^0=\{(x,y,\theta):\theta=0\}$.

Finally Poincare's map is $P(x_0,y_0)=(x(\frac{2\pi}{\omega}),y(\frac{2\pi}{\omega}))$.

I want to examine the stability of all periodic solutions corresponding to fixed points of Poincare's map. Finding the fixed points seems beyond me now. I need help with that. Furthermore, how could I go about finding fundamental matrixes for the respctive periodical solutions and the respective monodromy matrices? Any help would be appreciated.

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The equation you have has an analytical solution, since it is a mass-spring-damper with periodic excitation (forced linear 2D ODE). Use Laplace transform to find the exact solution for x(t), y(t), and you get your Poincare map for free.

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  • $\begingroup$ Yes, thank you, sir. I know I need to find the exact solution, but I don't know how. I can't say that I am familiar with the Laplace transform. $\endgroup$ – superAnnoyingUser May 10 '13 at 3:55

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