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Chaos and Nonlinear DynamicsAn Introduction for Scientists and Engineers$

Robert C. Hilborn

Print publication date: 2000

Print ISBN-13: 9780198507239

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198507239.001.0001

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(p.584) Appendix H Other Universal Features for One-Dimensional Iterated Maps

(p.584) Appendix H Other Universal Features for One-Dimensional Iterated Maps

Chaos and Nonlinear Dynamics
Oxford University Press

H.1 Introduction

In this appendix we describe, without any derivations or proofs, some additional universal quantitative features that appear for one-dimensional iterated maps, beyond those introduced in Chapter 5. The numerical values for these features can be derived by methods similar to those used in Chapter 5 and Appendix F. However, the details of the derivations would take us too far afield; therefore, the treatment will be purely descriptive.

H.2 Power Spectrum

We have seen that as the period-doubling bifurcations proceed, new subharmonics of the fundamental period of the system appear with each new bifurcation. Thus, if we calculate a power spectrum of the trajectories of the iterated map function, we would expect to see new components appear at the 2n subharmonic when period 2n is “born” at the nth bifurcation. However, because the system is nonlinear, we also expect frequency components at the frequencies corresponding to all the possible sum and difference frequencies for all the harmonics present. These notions are perhaps best understood through a simple example.

For a parameter value below the first period-doubling bifurcation for an iterated map function, the trajectory has a period that we shall call T = 1. After transients die away, every iteration value is the same. Thus, the power spectrum would have just a single frequency component, namely at the frequency v = 1/T = 1. After the first period-doubling bifurcation occurs, we now have period T =2 behavior. Thus, the power spectrum has a component at v = 1/2. Because the system is nonlinear, however, there will also be a second harmonic component at v = 2×(1/2) = 1. (We will ignore any higher frequency harmonics.) After the second period-doubling bifurcation, the system exhibits period T = 4 behavior, and hence has a component of the power spectrum at v = 1/4. Again, however, the nonlinearities produce power spectrum components at v = 1/2, 3/4, and 1. This evolution of the power spectrum is shown in Fig. H. 1.

Feigenbaum (FEI80) showed that the total “intensity” (the sum of the Fourier transform amplitudes) associated with the new frequency components (e.g. 1/4 and 3/4 after the second period-doubling bifurcation at the supercycle parameter value after a period-doubling bifurcation) are smaller than those associated with the previous bifurcation by a universal constant factor, whose value is approximately (p.585)

Appendix H Other Universal Features for One-Dimensional Iterated Maps

Fig. H.1. The evolution of the power spectrum for several period-doubling bifurcations. The logarithm of the Fourier amplitude is plotted as a function of frequency. The frequency scale is chosen such that period 1 behavior has a frequency of 1 associated with it. All frequencies higher than v = 1 are ignored here. Top left: period-2 supercycle. Top right: period-4 supercycle. Note the new components at v = 1/4 and v = 3/4. Bottom: period-8 supercycle. New components appear at v = 1/8, 3/8, 5/8, and 7/8. According to theory the sum of the amplitudes of the new frequency components at each period-doubling should be 8.17 dB smaller than the components associated with the previously existing frequency. On these diagrams, 8.17 dB corresponds to about 0.8 of a vertical division.

0.1525. [Since power spectra are often plotted on logarithmic scales, this factor corresponds to a logarithmic difference of 10 log10 (0.1525) = − 8.17 dB (dB = decibels).] This power spectrum ratio has been observed in a few experiments (GMP81, TPJ82). The observations seem to agree reasonably well with this universality prediction, though it is often difficult to determine these average values from the experimental data because the different odd-number harmonics have different strengths. (See the references cited for a discussion of various averaging techniques.)

If we find the Fourier power spectrum for a chaotic signal, we find that the power is distributed continuously as a function of frequency. Figure H.2 shows the power spectrum for a signal from the logistic map function with A = 3.609. This parameter value results in chaotic trajectories that alternate between two bands as shown in Fig. 5.9. The power spectrum is continuous (like that of a “noisy” signal) (p.586)

Appendix H Other Universal Features for One-Dimensional Iterated Maps

Fig. H.2. The logarithm (base 10) of the power spectrum for the logistic map with A = 3.609 corresponding to two-band chaotic behavior. Note the continuous range of frequencies present for chaotic behavior. The alternation between the two bands results in a broad maximum in the spectrum near v = 0.5.

but shows a broad maximum near v = 0.5 corresponding to the alternation between the two chaotic bands.

H.3 Effects Due to Noise

In our discussion of nonlinear systems, we have so far completely avoided the question of noise, that is, of uncontrollable outside influences, usually of a random nature, that limit the level of precision possible in any real scientific measurement. For nonlinear systems, this noise has some obvious effects. For example, if the experimental quantity corresponding to the control parameter for a system is “noisy,” then any effect that occurs (theoretically) at some well-defined value of the control parameter will be smeared out by this noise. To be concrete, let us think about the electrical voltage used as a control parameter in the diode circuit of Chapter 1. Even though we try to control this voltage rather precisely, there is always some small amount of electrical “noise” present. In this case, the noise manifests itself as small fluctuations in the control parameter voltage. Thus, when we say that the first period-doubling bifurcation occurs at V = 1.3345 volts, we really mean that the bifurcation occurs at V = 1.3345 ±0.0002 volts if the noise (p.587) level is about 0.0002 volts. Clearly, when we proceed to higher-order bifurcations, the voltage difference between successive bifurcations will eventually become smaller than the fluctuating noise level and the bifurcations, including the transition to chaos, will be smeared together.

Noise can also be present in the dependent variable being monitored. (This would correspond to noise in the value of x used in a one-dimensional iterated map function.) This kind of noise can be studied numerically for iterated maps by adding to the value of xn+1 calculated from the map function a (usually) small amount of noise. This is done in practice by using a “random number generator” available in most computer languages. Formally, we write

(H.3-1) Appendix H Other Universal Features for One-Dimensional Iterated Maps
where σn is a random number, usually chosen so that the average value of the σn s is 0 (we choose the average value to be 0 so that positive and negative values are equally likely). The average of the squares of the random numbers is some fixed value, whose square root is denoted by σ. Crutchfield, Farmer and Huberman (CFH82) studied the effects of such “additive noise” on the iterates of the logistic map. They found, as we might anticipate, that in the presence of noise, chaotic behavior begins apparently at lower values of the control parameter A. In fact, they showed that the difference between the parameter value at which chaos begins in the absence of noise A and the value at which it begins in the presence of noise A* obeys a universal power law expression
(H.3-2) Appendix H Other Universal Features for One-Dimensional Iterated Maps
where Appendix H Other Universal Features for One-Dimensional Iterated Maps and Appendix H Other Universal Features for One-Dimensional Iterated Maps, the reciprocal of the power spectrum scaling number. (δ is, of course, the Feigenvalue 4.669 ….) This result can be derived by recognizing that if noise of average size σm is large enough to “hide” all subharmonics whose index m is greater than some value, then chaos is apparently present for A = Am. To hide the next lower subharmonic, that is, to push chaotic behavior to lower values of A, we need to increase σ by an amount proportional to μ, since it is μ that gives the relative size of successive components in the power spectrum as discussed in the previous section. Thus, we can write
(H.3-3) Appendix H Other Universal Features for One-Dimensional Iterated Maps
where σ1 is the amount of noise needed to push chaos all the way to the parameter value A 1. We can take the logarithm of Eq. (H.3-3) to solve for m and then use that value in the result derived in Exercise 2.4-1 to obtain
(H.3-4) Appendix H Other Universal Features for One-Dimensional Iterated Maps
(p.588) which is the same as Eq. (H.3-2). This behavior has been verified numerically for the logistic map (CFH82) and in a few experiments on electronic oscillators (TPJ82 and YEK82).

In practice, the situation is a bit more complicated. In some cases, noise can simply mask the period-doubling cascade without actually inducing chaotic behavior (characterized, as usual, by a positive average Lyapunov exponent) while in other cases, the noise can indeed induced chaotic behavior. See GHL99 for a nice study of these two cases in the context of the logistic map model.

Since systems with chaotic behavior are sensitive to small changes in initial conditions, those systems can serve as “noise amplifiers” (FOE93). That is, the sensitive dependence can amplify small, microscope noise up to macroscopic levels.

H.4 Further Reading

M. J. Feigenbaum, “Universal Behavior in Nonlinear Systems,” Los Alamos Science 1, 4–27 (1980) (reprinted in [Cvitanovic, 1984]). Provides a quite readable introduction to the universal features of one-dimensional iterated maps.

M. Giglio, S. Musazzi, and U. Perini, “Transition to Chaotic Behavior Via a Reproducible Sequence of Period-Doubling Bifurcations,” Phys. Rev. Lett. 47, 243–46(1981).

J. Testa, J. Perez, and C. Jeffries, “Evidence for Universal Chaotic Behavior of a Driven Nonlinear Oscillator,” Phys. Rev. Lett. 48, 714–17 (1982).

J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, “Fluctuations and Simple Chaotic Dynamics,” Phys. Reports 92, 45–82 (1982).

The following three papers are reprinted in both [Hao, 1984] and [Cvitanovic, 1984]:

J. P. Crutchfield and B. A. Huberman, “Fluctuations and the onset of chaos,” Phys. Lett. A 77, 407–10 (1980).

J. Crutchfield, M. Nauenberg, and J. Rudnick, “Scaling for external noise at the onset of chaos,” Phys. Rev. Lett. 46, 933–35 (1981).

B. Shraiman, C. E. Wayne, and P. C. Martin, “Scaling theory for noisy period-doubling transitions to chaos,” Phys. Rev. Lett. 46, 935–9 (1981).

R. F. Fox and T. C. Elston, “Amplification of intrinsic fluctuations by the Lorenz equations,” Chaos 3, 313–23 (1993).

J. B. Gao, S. K. Hwang, and J. M. Liu, “When Can Noise Induce Chaos?” Phys. Rev. Lett. 82, 1132–35 (1999).