How Do You Solve Nonlinear Oscillations Using Poincaré Mapping?

[kidsmoker]

Homework Statement

http://img51.imageshack.us/img51/853/39983853.jpg


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...

Please help! :-( Thanks.

 

[mighty2000]

Hello there,

Your solution for Q3.1 looks correct to me. For Q3.2, I think you have the right idea but your solution is a bit off. The correct solution should be:

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

For Q3.3, your solution is also correct. To find the fixed points, you need to set x(2pi)=x(0) and y(2pi)=y(0). This will give you two equations with two unknowns (A and B). Solving these equations will give you the fixed points. However, I would recommend using a numerical method such as Newton's method to solve for the fixed points, as they can be quite complicated to solve algebraically.

I hope this helps! Let me know if you have any other questions.