Nonlinear Oscillations

**Showcase_22** · December 29th 2009, 06:09 AM

Hi guys! I think I posted a question similar to this before. I think i've made some progress in solving questions of this type, but i'm still getting stuck on bits of it.

Find the general solution to $\ddot{x}-2\dot{x}-3x=10 \sin t$ .

The method for this is long so i'll just put the answer.

This comes out to be $x(t)=Ae^{3t}+Be^{-t}-2 \sin t + \cos t$ where $A,B$ are constants.

Write this equation in the three variables $x,y=\dot{x}$ and $\dot{s}=1$ so that the equations are autonomous and first order.

$\dot{x}=y$
$\dot{y}=10 \sin s +3x+2y$
$\dot{s}=1$

We know that $\frac{ds}{dt}=1$ . This gives $\frac{dx}{dt}=\frac{ds}{dt}. \frac{dx}{ds}=\frac{dx}{ds}$ .

Applying a similar method gives:

$\frac{dy}{dt}=\frac{ds}{dt}.\frac{dy}{ds}=\frac{dy }{ds}$

We also have:

$ \frac{d^2x}{dt^2} = \frac{d}{dt} \left(\frac{dx}{dt}\right) = \frac{d}{ds} \left( \frac{dx}{ds}\right) \cdot \frac{ds}{dt} = \frac{d^2x}{ds^2} = \frac{dy}{ds} $ since $y=\frac{dx}{ds}$ .

This gives my 3 equations as:

$\frac{dt}{ds}=1$
$\frac{dx}{ds}=y$
$\frac{dy}{ds}-2y-3x=10 \sin t$

(Kudos to Danny for this!)

Write down the corresponding solution $(x(t),y(t),s(t))$ .

$x(p) \in \mathbb{R}^3$
$x(p)=(\dot{x}, \dot{y},\dot{s})=(y,10 \sin s+3x+2y,1)$

But we know the general solution.

$(x(t),y(t),s(t))=(Ae^{3t}-2 \sin t+ \cos t, 3Ae^{3t}-2 \cos t - \sin t, t+C)$

The vector field in $\mathbb{R}^3$ is now periodic in $s$ with period $2 \pi$ .

Write down the Poincare map in the plane $s= 2 \pi$ for a set of initial conditions $(x_0,y_0)$ lying in the plane $s=0$ .

So we're going from $s=0$ to $s= 2 \pi$

$(x(0),y(0),0)=(x_0,y_0,0)$

This gives:

$A+1=x_0$
$3A-2=y_0$
$C=0$

$\Rightarrow 4A-1=x_0+y_0$
$\Rightarrow A=\frac{1}{4}(x_0+y_0+1)$

So this gives:

$(x(t),y(t),s(t))=$ $\left( \frac{1}{4}(x_0+y_0+1)e^{3t}-2 \sin t+ \cos t, \frac{3}{4}(x_0+y_0+1)e^{3t}-2 \cos t- \sin t, t \right)$

Find it's fixed point and discuss it's stability in terms of eigenvalues.

This is where I get stuck. I only have one example in my notes that does not have an $e^{3t}$ in it. As for the part on eigenvalues, I have no idea how to even get a matrix from this!!

I'd appreciate some help on this, there are still more bits of the question that I have to try and do!!

**scienceismylady** · January 9th 2010, 04:21 PM

Are you solving the equation $\ddot{x}-\ddot{x}-3x=10\sin t$ or $\ddot{x}-2\ddot{x}-3x=10\sin t$ ? I found complex eigenvalues $\lambda=\frac{1}{2}\pm\frac{1}{2}\sqrt{13}$ instead of the $\lambda=3,-1$ that you did.

Also, for getting matrices, assume that $x=x$ and $y=x'$ . Differentiating these gives $x'=y$ and $y'=x''$ , which you can then use to turn any second order equation into a system of first order equations.

**Showcase_22** · January 10th 2010, 02:46 AM

I'm so sorry! I did mean to write $ \ddot{x}-2\ddot{x}-3x=10\sin t $ .

I also think i've tried $x=x$ and $y=\dot{x}$ . Haven't I just written it as $ \dot{y}=10 \sin s +3x+2y $ ?

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