Nonlinear Dynamics and Chaos
Nonlinear Dynamics and Chaos
2nd Edition
ISBN: 9780813349107
Author: Steven H. Strogatz
Publisher: PERSEUS D
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Chapter 3.3, Problem 1E
Interpretation Introduction

Interpretation:

Assuming quasi-static approximation N˙0, the first-order equation for n is derived, and N (t) is expressed in terms of n (t), pc is determined when n*=0 becomes unstable for p>pc, type of bifurcation at the laser threshold pc is determined. The range of parameter which makes quasi-static approximation N˙0 is determined.

Concept Introduction:

The first-order equation for the evolution of N(t) can be derived by substituting quasi-static approximation N˙ 0 in the equation N˙ = -GnN - fN +p 

The pump strength can be evaluated by taking derivative of n˙ with respect number of excited atoms n.

Fixed points are the points where n˙ = 0.

The range of parameter for the approximation can be obtained by considering the rate of change of n˙ is very less than the rate of change of N˙

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Solve the initial value problem: y' = x3 + y³ 3 , y(1) = 2 xy² y(x) = Hint: Notice that the equation on the right is homogeneous and see Homework exercise 23 in section 1.2 of our textbook to review techniques for solving homogeneous equations. Note that we've been given an intial value of the form y(a) = b where a > 0, so this only determines a solution corresponding to the right half of the graph of In(x), i.e., the part of the graph corresponding to positive values of x. Therefore, we should write In(x) instead of ln(|x|), since the left half of the graph is not determined by the initial condition given.
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