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fall_lectureset_03

fall_lectureset_03 - Example Laser threshold pump partially...

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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 self-organization out of cooperative interaction of atoms (Haken 1983, Strogatz’s 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 = - α o N(t) N n τ = = 1 k = - - α = - α - ° ° 2 o o n (GN k)n Gn n Gn(N n) kn or
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2 ² The corresponding bifurcation diagram is: No physical meaning n * N 0 x * =r n * =0 N 0 =k/G lamp laser ³ 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
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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 µ Linear stability analysis Consider is stable when r < 0 is unstable r > 0 ° what about when r = 0? The linear analysis fails!! For the non-zero 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 0 ° = * x 0 ° - ± = - = ± 2 * df r 3( r) 2r dx x r
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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 non-linear 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 saddle-node, r = r S
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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. ° No bifurcations arise so long as = 0 0 0 0 f(x ,r ) 0 i.e.,(x ,r ) = ° Consider the system x f(x,r) = f(x,r) 0 0 0 (x ,r ). 0 0 df 0 (x ,r ) dx 0 0 (x ,r ), 0 0 df 0 (x ,r ) dx °¹ The figure below illustrates the idea through two points along a solution curve. At (x 1 ,r 1 ), the derivative df/dx does not vanish , where as at (x 2 ,r 2 ), the derivative df/dx vanishes .
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