# Lyapunov exponents of a damped, driven harmonic oscillator

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1. May 6, 2016

### Ananthan9470

The problem statement, all variables and given/known data
I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by $\ddot{x} + 2\beta \dot{x} + \omega_0^2 x = fcos(\omega t)$
Lyapunov exponent is $\lambda$ in the equation $\delta x(t) = \delta x_0 e^{\lambda t}$

The attempt at a solution
The gerenal solution of the sytem is given by $Acos(\omega t - \delta) + Ce^{r_1 t} + De^{r_2 t}$

Consider two initial points 1 and 2. The solutions evolve to give $Acos(\omega t - \delta) + C_1e^{r_1 t} + D_1e^{r_2 t}$ and $Acos(\omega t - \delta) + C_2e^{r_1 t} + D_2e^{r_2 t}$ having initial points $Acos(\delta) + C_{1,2} + D_{1,2}$.

Hence we have, $\delta x(t) = C e^{r_1 t} + D e^{r_2 t}$ and $\delta x(0) = C + D$ where $C = C_1 - C_2$ and $D = D_1 - D_2$

So now my problem now comes down to being able to write $Ae^{x} + Be^{-x}$ in the form of $e^{y}(A+B)$ and figuring out y. And I don't know how I can do that. Am I doing this right? Or am I completely off track?

Ps. A and $r_1$ and $r_2$ have complex form depending on $\beta$ and $\omega$ etc.
$r_1$ and $r_2$ in the equation can be changed into the form x and -x

2. May 11, 2016

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 12, 2016

### Fred Wright

Here's my suggestion and I'm no expert so you can take it or leave it. First, solve the differential equation explicitly. I found
r1= -β+√(β202)
r2= -β-√(β202)
δ=atan((ω202)/2ωβ).
You have δx(t) = e-βt((C1-C2)e+√(β202)t + (D1-D2)e-√(β202)t)
I assert that when β202 << 1, λ≅-ω0.

Last edited: May 12, 2016