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Unformatted text preview: 1 & Example: Laser threshold At low energy levels each atom oscillates acting as a little antenna, but all atoms oscillate independently and emit randomly phased photons. At a threshold pumping level, all the atoms oscillate in phase producing laser! This is due to selforganization out of cooperative interaction of atoms (Haken 1983, Strogatzs book) Active material pump & laser light partially reflecting mirror Let n(t)  no. of photons Then, gain  loss (escape or leakage thru endface) & gain coeff > 0 no. of excited atoms Note that k > 0, a rate constant Here typical life time of a photon in the laser Note however that (because atoms after radiation of a photon, are not in an excited state), i.e., = & n = GnN kn = o N(t) N n = = 1 k = =  & & 2 o o n (GN k)n Gn n Gn(N n) kn or 2 & The corresponding bifurcation diagram is: No physical meaning n * N x * =r n * =0 N =k/G lamp laser Pitchfork bifurcation Examples: We have already seen the example of buckling of a column as a function of the axial load: Another example is that of the onset of convection in a toroidal thermosyphan mg g g fluid heating coil 3 & The normal form for pitchfork bifurcation is: The behavior can be understood in terms of the velocity functions as follows: The bifurcation diagram is then: Supercritical pitchfork = = & 3 x r x x f(x) x f(x) r < 0 x f(x) r = 0 x f(x) r > 0 X * r stable stable r =0 unstable stable Linear stability analysis Consider is stable when r < 0 is unstable r > 0 & what about when r = 0? The linear analysis fails!! For the nonzero equilibria: & eigenvalue is negative if r > 0 i.e., these bifurcating equilbiria are asymp. stable. = = & 3 x r x x f(x) = * The equilibria are at x 0, r & = = 2 * df r 3(0) r dx x & = * x & =  = 2 * df r 3( r) 2r dx x r 4 & Subcritical pitchfork The resulting bifurcation diagram is: = + =  & 3 * The normal form is x r x x , with equilibria x 0, r X * r unstable unstable r =0 unstable stable Usually, the unstable behavior is stabilized by higher order nonlinear terms, e.g., The resulting bifurcation diagram can be shown to be: = + & 3 5 x r x x x X * r unstable unstable r =0 unstable stable subcritical pitchfork, r = r P supercritical saddlenode, r = r S 5 & Connection between simple bifurcations and the implicit function theorem Let be an equilibrium. Let f be continuously differentiable w.r.t. x and r in some open region in the (x, r) plane containing Then if in a small neighborhood of we must have: & has a unique solution x=x(r) such that f(x(r),r)=0 & furthermore, x(r) is also continuously differentiable....
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 Fall '10
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