# Lyapunov exponents of a damped, driven harmonic oscillator

• Ananthan9470

#### Ananthan9470

Homework Statement
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 system 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

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.

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