# Differential Equations Problem

• Nov 5th 2008, 02:44 PM
flaming
Differential Equations Problem
A two-mode laser produces two di erent kinds of photons. The number of photons in the laser eld of each type are n1 and n2:
These vary in time and are governed by the DE system
n*1(n with dot on top) = G1*N*n1 - k1*n1

n*2(n with dot on top) = G2*N*n2 - k2*n2
where N(t) = N0 - alpha1*n1 - alpha2*n2

For a given experiment the parameters are given by G1 = 2; G2 = 1; alpha1 =
1 ;alpha2 = 2;k1 = 2;k2 = 2 and N0 = 4

(a) Find and classify all the equilibrium points and sketch the solutions in
the phase plane.
(b) What is the long term behaviour of the laser?
• Nov 6th 2008, 05:05 AM
shawsend
flaming . . . you know how to get started with this? Let me give you some tips:

(1) Write the system as pretty as possible else some will not want to bother interpreting your writing. Here's what I did after a lot of processing:

\$\displaystyle n'_1=6n_1-2n_1^2-4n_1n_2\$

\$\displaystyle n'_2=2n_2-2n_2^2-n_1n_2\$

See. That makes a lot of difference. Now shake and bake:

(a) Find the equilibrium points by setting the right side to zero. Flat out, I'm not going to do that by hand (just use Solve in Mathematica). Maybe you want to. Anyway they are: \$\displaystyle e_1=(0,0),\; e_2=(0,1),\; e_3=(3,0)\$

(b) Linearize it by calculating the Jacobian matrix for each equilibrium point.

(c) Find the eigenvalues for each matrix.

(d) Based on the sign of the eigenvalues, determine the stability of the equilibrium points.

(e) Finally, rely heavily on Mathematica to plot the results. You can use VectorFieldPlot to draw the vector field easily. Use NDSolve to solve the system for some select initial conditions and superimpose these solutions in the vector field to illustrate how the solutions follow the field. Use Point to show where the eq. points are and by plotting a lot of solutions, show how the solutions are affected by the stability of the eq. points.

See. Poke-a-poke :)