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CH02_BIFURCATIONS

CH02_BIFURCATIONS - Chapter 2 Bifurcations 2.1 Types of...

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Chapter 2 Bifurcations 2.1 Types of Bifurcations 2.1.1 Saddle-node bifurcation We remarked above how f ( u ) is in general nonzero when f ( u ) itself vanishes, since two equations in a single unknown is an overdetermined set. However, consider the function F ( x, α ), where α is a control parameter. If we demand F ( x, α ) = 0 and x F ( x, α ) = 0, we have two equations in two unknowns, and in general there will be a zero-dimensional solution set consisting of points ( x c , α c ). The situation is depicted in Fig. 2.1. Let’s expand F ( x, α ) in the vicinity of such a point ( x c , α c ): F ( x, α ) = F ( x c , α c ) + ∂F ∂x vextendsingle vextendsingle vextendsingle vextendsingle ( x c c) ( x x c ) + ∂F ∂α vextendsingle vextendsingle vextendsingle vextendsingle ( x c c) ( α α c ) + 1 2 2 F ∂x 2 vextendsingle vextendsingle vextendsingle vextendsingle ( x c c) ( x x c ) 2 + 2 F ∂x ∂α vextendsingle vextendsingle vextendsingle vextendsingle ( x c c) ( x x c ) ( α α c ) + 1 2 2 F ∂α 2 vextendsingle vextendsingle vextendsingle vextendsingle ( x c c) ( α α c ) 2 + . . . (2.1) = A ( α α c ) + B ( x x c ) 2 + . . . , (2.2) where we keep terms of lowest order in the deviations δx and δα . If we now rescale u radicalbig B/A ( x x c ), r α α c , and τ = At , we have, neglecting the higher order terms, we obtain the ‘normal form’ of the saddle-node bifurcation, du = r + u 2 . (2.3) The evolution of the flow is depicted in Fig. 2.2. For r < 0 there are two fixed points – one stable ( u = r ) and one unstable ( u = + r ). At r = 0 these two nodes coalesce and annihilate each other. (The point u = 0 is half-stable precisely at r = 0.) For r > 0 there are no longer any fixed points in the vicinity of u = 0. In the left panel of Fig. 2.3 we show the flow in the extended ( r, u ) plane. The unstable and stable nodes annihilate at r = 0. 1
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2 CHAPTER 2. BIFURCATIONS Figure 2.1: Evolution of F ( x, α ) as a function of the control parameter α . 2.1.2 Transcritical bifurcation Another situation which arises frequently is the transcritical bifurcation . Consider the equation ˙ x = f ( x ) in the vicinity of a fixed point x . dx dt = f ( x ) ( x x ) + 1 2 f ′′ ( x )( x x ) 2 + . . . . (2.4) We rescale u β ( x x ) with β = 1 2 f ′′ ( x ) and define r f ( x ) as the control parameter, to obtain, to order u 2 , du dt = ru u 2 . (2.5) Note that the sign of the u 2 term can be reversed relative to the others by sending u → − u and r → − r . Consider a crude model of a laser threshold. Let n be the number of photons in the laser cavity, and N the number of excited atoms in the cavity. The dynamics of the laser are approximated by the equations ˙ n = GNn kn (2.6) N = N 0 αn . (2.7) Here G is the gain coefficient and k the photon decay rate. N 0 is the pump strength, and α is a numerical factor. The first equation tells us that the number of photons in the cavity grows with a rate GN k ; gain is proportional to the number of excited atoms, and the loss rate is a constant cavity-dependent quantity (typically through the ends, which are semi-transparent). The second equation says that the number of excited atoms is equal to
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