# Homework Help: Nonlinear Oscillations

1. Jan 19, 2010

### kidsmoker

1. The problem statement, all variables and given/known data

http://img51.imageshack.us/img51/853/39983853.jpg [Broken]

2. The attempt at a solution

Q3.1

I get the general solution as

$$x(t) = Ae^{3t}+Be^{-t} + cost - 2sint$$ .

Q3.2

Letting

$$y=\dot{x}$$

and using the general solution, we get

$$y=3Ae^{3t}-Be^{-t}-sint-2cost$$ .

Therefore the solution in the form they ask for is

$$(Ae^{3t}+Be^{-t} + cost - 2sint, 3Ae^{3t}-Be^{-t}-sint-2cost, t)$$ .

Or am I misunderstanding?

Q3.3

$$x_{0}=x(0)=A+b-1 , y_{0}=y(0)=3A-B-2$$ .

Solving these simultaneously gives

$$A=0.25(x_{0}+y_{0}+1) , B=0.25(3x_{0}-y_{0}-5)$$

so the Poincaré mapping is

$$(0.25(x_{0}+y_{0}+1)e^{3t}+0.25(3x_{0}-y_{0}-5)e^{-t}+cost-2sint, 0.75(x_{0}+y_{0}+1)e^{3t} - 0.25(3x_{0}-y_{0}-5)e^{-t}-sint - 2cost, t)$$

Is that correct so far?

To find the fixed points, do I let x(2pi)=x(0), y(2pi)=y(0) and solve for x(0) and y(0)? I tried this but I get something ridiculously complicated, so I'm worried i'm not understanding the question correctly at all...