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Unformatted text preview: 16 Period doubling route to chaos We now study the routes or scenarios towards chaos. We ask: How does the transition from periodic to strange attractor occur? The question is analogous to the study of phase transitions: How does a solid become a melt; or a liquid become a gas? We shall see that, just as in the study of phase transitions, there are universal ways in which systems become chaotic. There are three universal routes: Period doubling Intermittency Quasiperiodicity We shall focus the majority of our attention on period doubling. 16.1 Instability of a limit cycle To analyze how a periodic regime may lose its stability, consider the Poincar e section: x 0 x 1 x 2 The periodic regime is linearly unstable if  x 1 x  <  x 2 x 1  < . . . 150 or  x 1  <  x 2  < . . . Recall that, to first order, a Poincar e map T in the neighborhood of x is described by the Floquet matrix T i M ij = . X j In a periodic regime, x ( t + ) = x ( t ) . But the mapping T sends x + x x + Mx. Thus stability depends on the 2 (possibly complex) eigenvalues i of M . If i > 1, the fixed point is unstable.   There are three ways in which i > 1:   i i Re Im +1 1 1. = 1 + , real, > 0. x is amplified is in the same direction: x 1 x 2 x 3 x 4 This transition is associated with Type 1 intermittency. 2. = (1 + ). x is amplified in alternating directions: x 3 x 1 x 0 x 2 This transition is associated with period doubling. 151 3. = i = (1 + ) e i .  x  is amplified, x is rotated: x 0 1 2 3 4 This transition is associated with quasiperiodicity. In each of these cases, nonlinear effects eventually cause the instability to saturate. Lets look more closely at the second case, 1. Just before the transition, = (1 ), > 0. Assume the Poincar e section goes through x = 0. Then an initial pertur bation x is damped with alternating sign: x 1 x 3 0 x 2 x 0 Now vary the control parameter such that = 1. The iterations no longer converge: x 1 0 x 0 x 3 x 2 We see that a new cycle has appeared with period twice that of the original cycle through x = 0. This is a period doubling bifurcation. 16.2 Logistic map We now focus on the simplest possible system that exhibits period doubling. 152 In essence, we set aside ndimensional ( n 3) trajectories and focus only on the Poincar e section and the eigenvector whose eigenvalue crosses ( 1). Thus we look at discrete intervals T, 2 T, 3 T, . . . and study the iterates of a transformation on an axis. We therefore study first return maps x k +1 = f ( x k ) and shall argue that these maps are highly relevant to ndimensional ows....
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 Fall '06
 DanielRothman
 The Land

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