This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2 Bifurcations 2.1 Types of Bifurcations 2.1.1 Saddlenode 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, ), where is a control parameter. If we demand F ( x, ) = 0 and x F ( x, ) = 0, we have two equations in two unknowns, and in general there will be a zerodimensional solution set consisting of points ( x c , c ). The situation is depicted in Fig. 2.1. Lets expand F ( x, ) in the vicinity of such a point ( x c , c ): F ( x, ) = F ( x c , c ) + F x vextendsingle vextendsingle vextendsingle vextendsingle ( x c , c) ( x x c ) + F vextendsingle vextendsingle vextendsingle vextendsingle ( x c , c) ( c ) + 1 2 2 F x 2 vextendsingle vextendsingle vextendsingle vextendsingle ( x c , c) ( x x c ) 2 + 2 F x vextendsingle vextendsingle vextendsingle vextendsingle ( x c , c) ( x x c )( c ) + 1 2 2 F 2 vextendsingle vextendsingle vextendsingle vextendsingle ( x c , c) ( c ) 2 + ... (2.1) = A ( c ) + B ( x x c ) 2 + ... , (2.2) where we keep terms of lowest order in the deviations x and . If we now rescale u radicalbig B/A ( x x c ), r c , and = At , we have, neglecting the higher order terms, we obtain the normal form of the saddlenode bifurcation, du d = 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 halfstable 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 2 CHAPTER 2. BIFURCATIONS Figure 2.1: Evolution of F ( x, ) as a function of the control parameter . 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 ( x x ) with = 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 n . (2.7) Here G is the gain coefficient and k the photon decay rate.the photon decay rate....
View
Full
Document
This note was uploaded on 09/30/2011 for the course PHYS 221a taught by Professor Staff during the Fall '11 term at UCSD.
 Fall '11
 staff

Click to edit the document details