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Let us now come back to the Belousov–Zhabotinsky reaction. The variable which signals chaotic behavior in this system is the concentration c of the Ce4+ ions. It is measured by the selective light absorption of these ions. The mean residence time of the substances in the open reactor (i. , r−1 ) acts as an external control parameter corresponding to ∆T in the previous experiment. 8) T →∞ T 0 T →∞ T 0 This function measures the correlation between subsequent signals. , 1981). 9) that is, both quantities contain the same information.

To make the analogy to critical phenomena even closer, we observe that λ = log 2r scales with a power law near the “critical point” rc = 1/2. λ ∝ (r − rc ) . 20) This shows that even the simple transition to chaos in the triangular map displays some features that are reminiscent of an equilibrium phase transition. As we have already mentioned before, we will investigate this aspect more generally in Chapter 4. It should also be noted here that the definition of the Liapunov exponent can be extended to higher dimensional maps.

What happens then? Figure 25: The fixed points of f , for a) r < 1 and b) 1 < r < 3. 38 4 Universal Behavior of Quadratic Maps Figure 26: a) f (x) and f 2 (x) = f [ f (x)] for r > r1 . b) Generation of two new stable fixed points in f 2 via a pitchfork bifurcation. (The bifurcation diagram looks like a pitchfork, see p. 159) Figure 26 shows fr (x) together with its second iterate fr2 (x) for r > r1 . We note four properties of f 2 (the index r is dropped for convenience): 1. It has three extrema with f 2 = f [ f (x)] f (x) = 0 at x0 = 1/2, because f (1/2) = 0, and at x1, 2 = f −1 (1/2), because f [ f [ f −1 (1/2)]] = f (1/2) = 0.