lecnotes16 - 16 Period doubling route to chaos We now study...

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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 | ψx 1 ψx 0 | < | ψx 2 ψx 1 | < . . . 150
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or | νψx 1 | < | νψx 2 | < . . . Recall that, to first order, a Poincar´ e map T in the neighborhood of ψx 0 is described by the Floquet matrix ωT i M ij = . ωX j In a periodic regime, ψx ( t + φ ) = ψx ( t ) . But the mapping T sends ψx 0 + νψx ψx 0 + Mνψx. 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: | | λ i λ i Re Im +1 −1 1. = 1 + π , π real, π > 0. νψx 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 + π ). νψx is amplified in alternating directions: x 3 x 1 x 0 x 2 This transition is associated with period doubling. 151
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3. = ± = (1 + π ) e ± . | νψx | is amplified, νψx is rotated: x 0 1 2 3 4 γ γ γ This transition is associated with quasiperiodicity. In each of these cases, nonlinear effects eventually cause the instability to saturate. Let’s look more closely at the second case, ◦ − 1. Just before the transition, = (1 π ), π > 0. Assume the Poincar´ e section goes through x = 0. Then an initial pertur- bation x 0 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
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In essence, we set aside n -dimensional ( 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 n -dimensional flows. For clarity, we adopt a biological interpretation. Imagine an island with an insect population that breeds in summer and leaves egges that hatch the following summer.
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