NOISE-INDUCED CHAOS

November 1, 1999 17:5 WSPC/140-IJMPB

0236

International Journal of Modern Physics B, Vol. 13, No. 28 (1999) 3283–3305 c World Scientific Publishing Company

NOISE-INDUCED CHAOS

J. B. GAO, C. C. CHEN, S. K. HWANG and J. M. LIU Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, CA 90095, USA

Received 10 September 1999

Noise-induced chaos is an interesting phenomenon. However, it is a subtle issue because of the difficulty in distinguishing between true low-dimensional chaos and noise. In this review article, we consider how to define noise-induced chaos and what constitutes a test for noise-induced chaos. The mechanism for noise-induced chaos is studied by considering the long-term growth rate of the logarithmic displacement curves. In particular, we have identified three types of diffusional processes, with the third type, the anomalous diffusion, being the precursor of noise-induced chaos. A number of dynamical systems showing different types of diffusional processes are discussed. We have also pointed out when and how a nonlinear noisy system may be mistaken as a chaotic system in general and noise-induced chaos in particular.

1. Introduction Noise is ubiquitous in nature and in man-made systems, such as in nonlinear solid state devices, in physiological systems, and in fluid flows. In nonlinear dynamical systems, it can induce a number of interesting phenomena, such as stochastic resonance1 (for a recent review, see Ref. 2), noise-induced instability,3,4 noise-induced order,5 and noise-induced chaos.6 – 9 Noise-induced chaos was first observed in a driven nonlinear oscillator by Crutchfield et al.6 and later carefully studied via the noisy logistic map.7 The main idea of Crutchfield et al. is that intrinsic noise truncates the period-doubling cascade. That is, the periodic motions with high periods of a clean system is replaced by chaos-like motions when there is noise. Indeed, when a period-doubling cascade is observed in experimental situations, such as in fluid flows,10 in semiconductor lasers,11 in chemical reactions,12 in plasma reactors,13 and in biological systems,14 only the first few period-doubling bifurcations have been observed. A different scenario for noise-induced chaos to occur involves noise-induced hopping between two periodic states8,9 or noise-inhibited hopping between a periodic state and a metastable chaotic state.8 The former is irregular in time and resembles that occurring in a deterministic system caused by changes in a parameter. The 3283

November 1, 1999 17:5 WSPC/140-IJMPB

3284

0236

J. B. Gao et al.

latter refers to that the system virtually spends all of its time in the metastable chaotic state. One may expect that the above two scenarios for noise-induced chaos are both generic. We thus would expect that noise-induced chaos should be readily observed experimentally. However, to date, not many true low-dimensional chaotic systems, either intrinsically deterministic or induced by noise, have been identified experimentally. This discrepancy simply reflects the difficulty in objectively defining lowdimensional deterministic chaos in general and noise-induced chaos in particular. This controversy on the deterministic or stochastic character of an experimentally observed irregular phenomenon can be glimpsed from an argument made by Argoul et al.,15 referring to the work on Belousov–Zhabotinskii (BZ) reaction by Roux et al.16 and Hudson et al.17 : “after measuring the largest Lyapunov exponent, whose positivity confirmed sensitive dependence upon initial conditions, the demonstration of the determinism was complete, despite objections by certain experts in the kinetics of the BZ reaction”. This discrepancy has stimulated us to re-examine the noisy logistic map.18 We find that noise can indeed induce chaos. However, this is not associated with the main period-doubling cascade. We have also identified three basic conditions for noise to induce chaos and shown that when noise induces chaos, the complete period-doubling cascade is inhibited; otherwise, the cascade is simply masked by noise. In this review article, we shall first consider how to define noise-induced chaos and what criteria a test for noise-induced chaos must satisfy. We then consider the mechanism for noise-induced chaos in great detail. Whenever appropriate, we shall point out how a nonlinear noisy system may be mistaken as a chaotic system in general and noise-induced chaos in particular. In order for the paper to be directly relevant to experimental data analysis, throughout the paper, we shall work with a scalar time series, x(1), x(2), . . . , x(N ). For convenience, we shall first normalize the scalar time series into the unit interval [0, 1]. We then use the time delay embedding procedure19 to construct vectors of the form: Xi = (x(i), x(i + L), . . . , x(i + (m − 1)L)), with m being the embedding dimension and L the delay time. For the analysis of purely chaotic signals, m and L has to be chosen properly. This is the issue of optimal embedding (see Refs. 20 and 21 and references therein). Mathematically speaking, optimal embedding is not defined for the analysis of noisy data, since noisy data is infinite-dimensional. We have found, however, that optimal embedding parameters for a chaotic signal are usually also optimal for the analysis of the corresponding noisy chaotic signal. Note that when a piece of the trajectory O1 = {Xi , Xi+1 , Xi+2 , . . .} is close to another piece of the trajectory O2 = {Xj , Xj+1 , Xj+2 , . . .}, where |i − j|  1, then O1 and O2 can be viewed as two trajectories. The dynamic noise associated with O1 is typically uncorrelated to that associated with O2. From time series analysis point of view, this point is obvious and trivial. However, if one knows the equations for the system and wants to study the effects of noise on the dynamics by iterating or integrating the equations, this issue becomes subtle. One needs to consider two

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3285

noise realizations when generating two nearby trajectories, as was suggested by Paladin et al.22 This paper is organized as follows. In Sec. 2, we first discuss how to define noise-induced chaos, we then consider the criteria that a test for noise-induced chaos should satisfy. We argue that the time-dependent exponent curves defined by Gao and Zheng21 can serve as the basis for the study of noise-induced chaos. When reviewing the definition of the time-dependent exponent curves, 21 we shall also point out how a simple stochastic process may be misinterpreted as chaos by other methods. We also review the logarithmic displacement curves defined by Gao23 in Sec. 2, which is very useful for a quantitative study of the effects of noise on chaotic systems and diffusional processes in noisy nonlinear oscillators. To make some concepts in Sec. 2 more concrete, in Sec. 3, we shall briefly review the results obtained with the noisy logistic map.18 In Sec. 4, we characterize the behavior of noisy nonlinear oscillators by the logarithmic displacement curves. We shall identify three types of diffusional processes and argue that type (iii) anomalous diffusion is the very intrinsic mechanism for noise to induce chaos. Nonlinear dynamics in a semiconductor laser is of much recent interest. Very interestingly, all three types of diffusional processes discussed in the paper have been observed in an optically injected semiconductor laser model.24,25 These studies are reviewed in Sec. 5. To further deepen our understanding of when a noisy chaotic system should be regarded as simply noisy rather than still chaotic and why simple diffusional processes are pseudo-chaos, we deviate in Sec. 6 to consider desynchronization in chaotic systems due to noise and failure of secure communication using pseudo-chaos. Conclusions and discussions are given in Sec. 7.

2. Test for Noise-Induced Chaos Noise-induced chaos is a subtle issue. Its subtlety lies in the fact that it is very hard to distinguish true low-dimensional chaos from noise. For example, Osborne and Provenzale observed that 1/f α noise generates time series with a finite correlation dimension26 and converging K2 entropy estimates.27 Defining chaos by calculating the positive Lyapunov exponents is also controversial, as can be seen from our citation of Argoul et al.’s argument15 in Sec. 1. Indeed, Dammig and Mitschke28 found that a finite positive value for the largest positive Lyapunov exponent is obtained from some stochastic data by the algorithm of Wolf et al.29 Hence, in order to have a deep understanding of noise-induced chaos, we need first to define chaos carefully. Mathematically speaking, a noisy system, no matter how weak the noise is, has infinite dimensions. Experimentally speaking, one would be more interested in a certain range of finite scales. If the noise is very weak, then its influence on the dynamics may be limited to very small scales, leaving the dynamics on finite scales deterministic-like. Here we will adopt this experimentalist’s point of view and define chaos by the exponential divergence between nearby trajectories on certain finite

November 1, 1999 17:5 WSPC/140-IJMPB

3286

0236

J. B. Gao et al.

scales. Note that this definition is different from the one based on simply calculating the positive Lyapunov exponent, in which the concept of scale is absent. Noise-induced chaos can be defined in a similar manner. That is, a certain amount of noise induces exponential divergence between nearby trajectories on certain finite scales in a dynamical system. Since exponential divergence between nearby trajectories is involved, a test for noise-induced chaos has to be able to give an estimate of the largest positive Lyapunov exponent. On the other hand, this suggests that one may actually be able to find a suitable test for noise-induced chaos from the proposed algorithms for estimating the positive Lyapunov exponents. These algorithms can be roughly grouped into three types: (1) the well-known algorithm of Wolf et al.29 for estimating the largest positive Lyapunov exponent, (2) a complexity measure proposed by Paladin et al.,22 which is a weighted version of the largest positive Lyapunov exponent, and (3) a direct dynamical test for deterministic chaos proposed by Gao and Zheng.21 Before we proceed, we need to carefully examine which algorithm can serve as a basis for the study of noise-induced chaos. A test for noise-induced chaos has to satisfy a set of minimal criteria. First, we require that different researchers should interpret the obtained results consistently when they apply the test to the same problem. This can be ensured if there is no free parameters to select, or if there is a simple, strict procedure for choosing the parameters when using the algorithm. The second criterion, which is as important as the first, is that it will not fool us by interpreting a simple noisy process, such as white noise or a linear Gaussian process, as chaos. A particular type of linear Gaussian process is the so-called surrogate data.30 It is obtained by taking the Fourier transform of the original signal, randomizing the phase and taking the inverse Fourier transform. Hence, a surrogate data shares the same mean, variance, and the power spectrum of the original signal. Note that the above two criteria ensure in some degree the notion of determinism. We will show below that among the three algorithms mentioned above, only the method of Gao and Zheng21 satisfies these two criteria. The test of Gao and Zheng21 has an additional appealing feature that it has incorporated the notion of scale into the algorithm. Hence, it can be used as the basis for this study without any modification. Below we shall first review this method. At the same time, we shall point out why the algorithms of Wolf et al.29 and Paladin et al.22 fail to satisfy the two criteria discussed above. 2.1. Time-dependent exponent Λ(k) curves The time-dependent exponent Λ(k) curves are defined as [21]    kXi+k − Xj+k k Λ(k) = ln kXi − Xj k

(1)

with r ≤ kXi − Xj k ≤ r + ∆r, where r and ∆r are prescribed small distances. The angle brackets denote ensemble averages of all possible pairs of (Xi , Xj ). The integer k, called the evolution time, corresponds to time kδt, where δt is the sampling time.

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3287

Note that geometrically (r, r + ∆r) defines a shell, and a shell captures the notion of scale. Figure 1(a) shows the Λ(k) curves for the chaotic Lorenz system: dx = −16(x − y) + Dη1 (t) , dt dy = −xz + 45.92x − y + Dη2 (t) , dt dz = xy − 4z + Dη3 (t) . dt

(2)

with hηi (t)i = 0, hηi (t)ηj (t0 )i = δij δ(t − t0 ), i, j = 1, 2, 3. Note that D2 is the variance of the Gaussian noise terms and D = 0 describes the clean Lorenz system. We observe from Fig. 1(a) that the Λ(k) curves are composed of three parts. These curves are linearly increasing for 0 ≤ k ≤ ka . They are still linearly increasing for ka ≤ k ≤ kp , but with a slightly different slope. They are flat for k ≥ kp . Note that the slope of the second linearly increasing part gives an estimate of the largest positive Lyapunov exponent.21 ka is related to the time scale for a pair of nearby points (Xi , Xj ) to evolve to the unstable manifold of Xi or Xj . It is on the order of the embedding window length, (m − 1)L. kp is the prediction time scale. It is longer for the Λ(k) curves that correspond to smaller shells. The difference between the slopes of the first and second linearly increasing parts is caused by the discrepancy between the direction defined by the pair of points (Xi , Xj ) and the unstable manifold of Xi or Xj . This feature was first observed by Sato et al.31 and was used by them to improve the estimation of the Lyapunov exponent. The first linearly increasing part can be made smaller or can even be eliminated by adjusting the embedding parameters such as by using a larger value for m. Note that the second linearly increasing parts of the Λ(k) curves collapse together to form an envelope. It is this very feature that forms the direct dynamical test for deterministic chaos.21 This is because the Λ(k) curves for noisy data, such as white noise or the surrogate data of a chaotic signal, are only composed of two parts, an increasing (but not linear) part for k ≤ (m − 1)L and a flat part.21 Furthermore, different Λ(k) curves for noisy data separate from each other; hence an envelope is not defined. One can expect that the behavior of the Λ(k) curves for a noisy chaotic system lies in between that of the Λ(k) curves for a clean chaotic system and that of the Λ(k) curves for white noise or for the surrogate data of a chaotic signal. This is indeed so. An example is shown in Fig. 1(b) for the noisy Lorenz system with D = 4. We observe from Fig. 1(b) that the Λ(k) curves corresponding to different shells now separate. Therefore, an envelope is no longer defined. This separation is larger between the Λ(k) curves corresponding to smaller shells, indicating that the effect of noise on the small-scale dynamics is stronger than that on the large-scale dynamics. Also note that ka + kp is now on the order of the embedding window length and is almost the same for all the Λ(k) curves. With stronger noise (D > 4), the Λ(k) curves will be more like those for white noise.21

November 1, 1999 17:5 WSPC/140-IJMPB

3288

0236

J. B. Gao et al.

Fig. 1. Time dependent exponent Λ(k) versus evolution time k curves for (a) clean and (b) noisy Lorenz system. Six curves, from bottom to top, correspond to shells (2−(i+1)/2 , 2−i/2 ) with i = 8, 9, 10, 11, 12, and 13. The sampling time for the system is 0.03 and embedding parameters are m = 4, L = 3. 5000 points are used in the computation.

We can now fully appreciate why in the presence of noise the algorithms of Wolf et al.29 and Paladin et al.22 cannot be used as a test for chaos. Their methods are somewhat equivalent to estimating the Lyapunov exponent (or a complexity measure) by Λ(k)/k, where Λ(k) is defined as in Eq. (1) but with a modification: it is computed for kXi − Xj k < r and kXi+k − Xj+k k < R, where r and R are prescribed distance scales. The condition kXi − Xj k < r amounts to grouping some of our small shells together to form a small ball. The condition kXi+k − Xj+k k < R

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3289

is presumably to set the time scale k smaller than kp , the prediction time scale. For clean chaotic signals, if the embedding parameters are chosen such that the adjusting time scale ka is very small compared to kp and the linearly increasing parts of the Λ(k) curves form a tight envelope, then an estimation of the exponent will not depend on the specific choice of r and R. However, one is usually not so lucky as to choose those specific optimal embedding parameters. Hence, different researchers may obtain different estimates of the exponent by choosing different sets of values for r and R. For example, an inexperienced researcher might unfortunately set k so much larger than kp that a value of almost zero is obtained for the exponent. The problem associated with estimating an exponent from stochastic data using the algorithms of Wolf et al.29 and Paladin et al.22 is much more serious than that associated with estimating an exponent from a clean chaotic signal. Since now the Λ(k) curves do not form an envelope, different researchers typically will obtain different values for the exponent by choosing different sets of r and R (and embedding parameters). Also one may instinctly choose k  (m − 1)L. Then the estimated exponent may quite often be very close to zero. For a stochastic data set, since the Λ(k) curves always increase for k < (m − 1)L due to the correlations introduced by the embedding procedure,21 a positive finite value for the exponent can always be obtained, thus resulting in a false interpretation of the stochastic data being chaotic. The method of Gao and Zheng21 has avoided all these problems because all the important information, including the time scales ka and kp , and whether the envelope exists or not, is automatically shown by the Λ(k) curves. For a quantitative understanding of how noise affects chaos, it is more convenient to work with the logarithmic displacement curves than with the time dependent exponent Λ(k) curves. The former is even more useful for the study of diffusional processes caused by noise. Hence, we review its definition in the next subsection. 2.2. Logarithmic displacement curves The logarithmic displacement curves is obtained by rewriting Eq. (1) as23 D(k) = hlnkXi+k − Xj+k ki = hlnkXi − Xj ki + Λ(k) ,

(3)

and plotting hlnkXi+k − Xj+k ki as a function of the evolution time k. Figures 2(a) and (b) show the corresponding displacement curves for Figs. 1(a) and (b), respectively. We see that for large values of k where Λ(k) is constant, the curves collapse nicely. This is because all hlnkXi+k − Xj+k ki curves are estimating the same quantity, the logarithm of the diameter of the attractor. For small values of k where the Λ(k) curves coincide to form a linearly increasing envelope for the clean Lorenz system, we see the displacement curves separate. Thus, within time period kp , the pairs (Xi , Xj ) on an average are keeping memory as which shell they originate from. By adding noise, this separation shrinks, reflecting loss of memory. Note that this shrinkage of separation is more severe for the displacement curves corresponding to smaller shells, reflecting again that the effect of noise on small-scale dynamics

November 1, 1999 17:5 WSPC/140-IJMPB

3290

0236

J. B. Gao et al.

Fig. 2. The corresponding logarithmic displacement curves for Fig. 1. Six curves, now from top to bottom, correspond to shells (2−(i+1)/2 , 2−i/2 ) with i = 8, 9, 10, 11, 12 and 13.

is stronger than that on large-scale dynamics. The stronger the noise is, the more the separation shrinks. This can be quantified by taking the ratio of the separation between the displacement curves for the noisy and the clean systems. More concretely, we take two logarithmic displacement curves corresponding to different shells, denote them as D1 and D2 , calculate the area between them, do this both for the noisy and the clean systems, and take their ratio. Since this ratio is a normalized area, we denote it by NA. This procedure can be well approximated by the following formula23 : P (D1 (ki ) − D2 (ki )) with NA ≈ Pi (D1 (ki ) − D2 (ki )) i

noise

without noise

(4)

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3291

with (m−1)L < ki ≤ kp , i = 1, 2, 3, . . . . NA typically decreases from 1 to 0 with the strength of the noise. A NA versus noise strength curve for the noisy Lorenz system described by Eq. (4) can be found in Ref. 23. For a chaotic system in the presence of noise to remain mostly chaotic rather than noisy, the value of NA computed based on certain finite scale shells should be around 1. To make some concepts in this Section more concrete, we briefly review in the next Section some results obtained on the noisy logistic map.18 3. Noise-Induced Chaos in the Noisy Logistic Map The noisy logistic map is defined by xn+1 = µxn (1 − xn ) + Pn ,

0 < xn < 1 ,

(5)

where µ is the bifurcation parameter and Pn is a Gaussian random variable with zero mean and standard deviation σ. We will refer to σ as the noise level. The bifurcation diagrams for the clean and the noisy systems are plotted in Figs. 3(a) and (b), respectively. The noisy system behavior at parameter values µ = 3.55, 3.63, 3.74, and 3.83 has been specifically studied in Ref. 18. The clean system at these parameter values is periodic with periods 8, 6, 5 and 3, respectively, as indicated in Fig. 3(a). Note that µ = 3.55 belongs to the main 2n cascade, while µ = 3.83 belongs to the period (3)-doubling cascade. Based on the difference between Figs. 3(a) and (b), one is tempted to conclude that the clean periodic motions have indeed become chaotic in the presence of noise. However, the actual behavior of the noisy logistic map is more complex and more interesting than this scenario. We have found18 that the behavior of the noisy system corresponding to the main period-doubling sequence is simply diffusive. That is, the noisy time series follows closely the periodic pattern of the clean periodic signals. Hence, the main period-doubling sequence is only masked by noise. Noise-induced chaos is associated with P3, P5 and P6 periodic windows. The chaotic states adjacent to these periodic windows are much less susceptible to noise than those adjacent to the main period-doubling sequence. These behaviors are shown by the Λ(k) curves plotted in Figs. 4(a)–(d), where for each figure, six curves, from bottom to top, correspond to shells (r, r + ∆r)i = (2−(i+1)/2 , 2−i/2 ) with i = 7, 8, . . . , 12. Clearly, the linearly increasing segments of the Λ(k) curves for µ = 3.74 and σ = 0.002 form a very tight envelope, while for µ = 3.55, the Λ(k) curves only show a noise-like feature. Noise-induced chaos for µ = 3.63 and 3.83 is, however, only defined for very specific scales, since the linearly increasing segments of the Λ(k) curves collapse together only for Λ(k) curves corresponding to two shells: shells (2−(i+1)/2 , 2−i/2 ) with i = 9, 10 for µ = 3.63 and i = 7, 8 for µ = 3.83. Note that the noisy system behavior at µ = 3.55 would be taken as chaotic were one to estimate the largest positive Lyapunov exponent or the complexity measure by using the algorithms of Wolf et al.29 or Paladin et al.22 Indeed, it

November 1, 1999 17:5 WSPC/140-IJMPB

3292

0236

J. B. Gao et al.

Fig. 3. The attractor versus bifurcation parameter µ for the logistic map of Eq. (5), (a) clean system and (b) noisy system. 100 iterations were plotted after an initial 10000 iterations for each increment in the bifurcation parameter. The parameter was incremented 1000 times in the interval [2.95, 3.95].

was based on the calculation of the Lyapunov exponent that Crutchfield et al.7 concluded that the noisy logistic map has become chaotic. Echoing our citation to the argument of Argoul et al.15 in Sec. 1, we surmise that those experts in the kinetics of the BZ reaction might have observed simple diffusional processes or completely noisy dynamics, thus have questioned if those processes can be classified as deterministically chaotic.

4. Mechanism for Noise-Induced Chaos To find the mechanism for noise-induced chaos, we need to carefully study the behavior of a stochastically driven nonlinear oscillator. In terms of the long term

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3293

Fig. 4. Time dependent exponent Λ(k) versus evolution time k curves for (a) µ = 3.55 and σ = 0.01; (b) µ = 3.63 and σ = 0.005; (c) µ = 3.74 and σ = 0.002; and (d) µ = 3.83 and σ = 0.005. Six curves, from bottom to top (in terms of the Λ(k) values for large k), correspond to shells (2−(i+1)/2 , 2−i/2 ) with i = 7, 8, 9, 10, 11 and 12.

growth rate of the logarithmic displacement curves, we have identified three types of different behavior.18,23 – 25 They are: (i) hlnkXi+k −Xj+k ki ∼ ln k α , with α = 1/2. This type of scaling law is caused by Brownian-like motions executed by the phase points around the deterministic limit cycle.23 To observe such Brownian-like motions, the noise should not be too strong and the convergent flow toward the deterministic limit cycle should not be too weak. (ii) hlnkXi+k − Xj+k ki ∼ ln k α , with α < 1/2. This type of scaling law can be observed when the noise is very strong or the convergent flow toward the deterministic limit cycle is very weak. When either of these two conditions holds, noise will instantly kick phase points away from the attractor to a region where nonlinearity is very strong. Since the motion is bounded, the diffusional processes for large time scales will then be weaker than the standard Brownian motion. A weak convergent flow toward an attractor is typically associated with a parameter near a bifurcation point. We term the behavior of the above two types of noisy oscillators “simple diffusional processes”. By simple we mean that the noisy time series typically follows the periodic pattern of the clean periodic signal closely18 ; hence its long-term behavior can be readily predicted. Such diffusional processes might, however, be interpreted

November 1, 1999 17:5 WSPC/140-IJMPB

3294

0236

J. B. Gao et al.

as chaotic if one were to compute the Lyapunov exponent using the algorithm of Wolf et al.29 Hence, we also give it a name “pseudo-noise–induced chaos”. (iii) hlnkXi+k − Xj+k ki ∼ ln k α , with α > 1/2. This type of scaling law is observed when the noise strength is very weak. We call it “anomalous diffusion”. When the noise strength is increased, chaos-like feature can be observed. Hence, we may also call it “pre-noise-induced chaos”, in contrast to the above “pseudonoise–induced chaos”. We have argued18 that this type of diffusion is the very condition for the periodic states themselves to be susceptible to noise-induced chaos. This is because deterministic chaos is characterized by short-term exponential divergence between nearby orbits. To induce chaos by adjusting the noise level, we are trying to make the displacement curves of the noisy system grow exponentially for a short period of time, then level off. This is possible only if a noisy oscillator can execute anomalous diffusion with very weak noise. Type (i) diffusion can be observed from stochastically driven nonlinear oscillators, such as a stochastically driven Van der Pol’s oscillator: dx = y + D1 η1 (t) , dt dy = −(x2 − 1)y − x + D2 η2 (t) , dt

(6)

and a stochastically driven Rayleigh’s oscillator: dx = y + D1 η1 (t) , dt dy = −(y 2 − 1)y − x + D2 η2 (t) , dt

(7)

with hηi (t)i = 0, hηi (t)ηj (t0 )i = δij δ(t − t0 ), i, j = 1, 2. The phase diagrams and the logarithmic displacement curves for these two systems are shown in Figs. 5 and 6, respectively. Clearly we observe that hlnkXi+k − Xj+k ki ∼ ln k 1/2 , for large values of k. Type (i) diffusion can also be observed from experimental data. An example is the fluctuating velocity signals measured at a fixed point in the near wake behind a circular cylinder.23,32 The phase diagram for such a signal corresponding to a Reynolds number Re = 136.5 is shown in Fig. 5(c). Its displacement curves are shown in Fig. 6 as group “c”. Again we observe hln kXi+k − Xj+kki ∼ ln k 1/2 , for large values of k. Note that time series with phase diagrams similar to those depicted in Fig. 5 frequently appear in diverse dynamical systems, for example, in chemical reactions,33 in parametrically forced surface waves,34 and in plasma physics.35 When the main purpose of an experiment is to look for chaos-like motions, it is very likely for these diffused limit cycles to be interpreted as strange attractors. Indeed, conventional

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3295

Fig. 5. Phase-diagrams for (a) the Van der Pol’s oscillator, (b) the Rayleigh’s oscillator and (c) a fluctuating velocity signal measured in a near wake behind of a circular cylinder. For all three systems, L = 6. The sampling time is 0.2 both for (a) and (b) and the sampling frequency for (c) is 300 Hz.

analysis (such as power spectra, dimension calculation, etc.) of our wake velocity signal suggested that the signal exhibited chaotic features.32 Type (ii) diffusion is as common as type (i) diffusion. For example, it can be observed in the stochastically driven Van der Pol’s and Rayleigh’s oscillators when the stochastic forcing is much stronger than the one we have studied above. It can also be observed in the fluctuating velocity signals measured in the near wake

November 1, 1999 17:5 WSPC/140-IJMPB

3296

0236

J. B. Gao et al.

Fig. 6. Three logarithmic displacement hlnkXi+k − Xj+k ki curves (from top to bottom) corresponding to shells (2−(i+1)/2 , 2−i/2 ) with i = 5, 6 and 7 for (a) the Van der Pol’s oscillator, (b) the Rayleigh’s oscillator and (c) a fluctuating velocity signal measured in a near wake behind of a circular cylinder. Also shown in (a–c) (as triangles) is a curve generated from ln k α with α = 0.5. The embedding parameters are m = 6 and L = 3 for all three systems.

behind a cylinder in two different scenarios: (a) for Re around 136, the signal measured at the center of the vortex is characterized by an exponent α < 1/2, while the signal measured around and outside of the vortex is characterized by α = 1/2; and (b) for flows of larger Reynolds numbers, typically α < 1/2. The first scenario indicates that nonlinearity is nonuniform inside the vortex, while the second scenario suggests that a sequence of bifurcations may have occurred with increasing Reynolds number. The absence of type (iii) anomalous diffusion for the wake flows of low Reynolds numbers (Re ≤ 200) suggests that low-dimensional chaos is unlikely to be observed in cylinder wakes. This argument is consistent with that made by Noack and Eckelmann36 based on numerical simulations. The simplest system in which to observe type (ii) diffusion is perhaps the main period (2)-doubling cascade of the noisy logistic map. An example of the displacement curves is shown in Fig. 7 for µ = 3.55 as group “d” curves, where we find α = 0.25. Type (iii) anomalous diffusion is associated with very week noise. It is the precursor for noise-induced chaos. They are found to be associated with the noisy logistic map at µ = 3.63, 3.74 and 3.83. The logarithmic displacement curves for these states are shown in Fig. 7 as group “a-c” curves, where we find α = 1.0, 1.0, and 1.5 for µ = 3.63, 3.83 and 3.74, respectively. Type (iii) diffusion and noised-induced chaos has also been observed in a semiconductor laser model.25 Hence, we surmise that type (iii) anomalous diffusion is a generic mechanism for noised-induced chaos.

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3297

Fig. 7. Three logarithmic displacement hlnkXi+k − Xj+k ki curves (from top to bottom) corresponding to shells (2−(i+1)/2 , 2−i/2 ) with i = 12, 13 and 14 for (a) µ = 3.74 and σ = 0.0003; (b) µ = 3.83 and σ = 0.001; (c) µ = 3.63 and σ = 0.0003; and (d) µ = 3.55 and σ = 0.0005. To separate these different groups of curves from each other, groups (a) and (b) curves are shifted upward by 2 and 1 units, while groups (c) and (d) curves are shifted downward by −0.5 and −0.2 units, respectively. Also shown in (a) (as diamonds), (b) (as triangles), (c) (as circles) and (d) (as squares) are curves generated from ln k α with α = 1.5, 1.0, 1.0 and 0.25, respectively.

Very interestingly, all three types of diffusional processes have been observed in an optically injected semiconductor laser model.24,25 Since the nonlinear dynamics of a semiconductor laser is of much recent research interest, we review the relevant results in the following Section separately. 5. Diffusional Processes in an Optically Injected Semiconductor Laser Model Like the logistic map, an optically injected semiconductor laser follows a perioddoubling route to chaos. This was first numerically predicted by Sacher et al.37 and later experimentally confirmed by Simpson et al.38,39 Remarkably, all the observed phenomena were reproduced by a simple single-mode model.38,39 When the physical quantities are properly normalized, the single-mode model takes the following form40 : da 1 ˆ + φ) + µfa , = [ˆ γn n ˆ − γˆp (2a + a2 )](1 + a) + ξ cos(Ωτ dτ 2 dφ b ξ ˆ + φ) + µfφ , ˆ − γˆp (2a + a2 )] − = − [ˆ γn n sin(Ωτ dτ 2 1+a 1+a

(8)

dˆ n = −ˆ γs n ˆ − (2a + a2 ) − [ˆ γn n ˆ − γˆp (2a + a2 )](1 + a)2 . dτ Here τ is the time normalized to the photon lifetime. a is the normalized field amplitude, φ is the phase difference between the amplitude of the injected and

November 1, 1999 17:5 WSPC/140-IJMPB

3298

0236

J. B. Gao et al.

injection field, and n ˆ is the normalized carrier density of the injected laser. γˆn , γˆp , γˆs , and b are intrinsic laser parameters determined experimentally,41 where the first three are normalized to the photon decay rate, which is the inverse of the photon lifetime. ξ is a dimensionless parameter characterizing the strength of the injection ˆ is the frequency detuning (also normalfield received by the injected laser, and Ω ized to the photon decay rate) of the injection field from the free-running frequency of the injected laser. The latter two parameters together with the injection current density are the three controlling parameters in the experiments. The injection current density is, however, kept fixed for this study. Finally fa and fφ are the normalized Langevin noise-source parameters that characterize the spontaneous emission in the laser. Since the strength of noise can differ by orders of magnitude in different semiconductor lasers, to infer how different noise levels affect the laser dynamics, a coefficient µ is introduced before fa and fφ . Hence, µ = 0 corresponds to an idealized, but unrealistic, clean system in the absence of any noise, while µ = 1 corresponds to the experimentally determined noise level41 for the laser under consideration. Note that the dynamic noise in Eq. (8) is multiplicative, since fφ is multiplied by 1/(1 + a), where a is an unknown state variable. In contrast, the noise in Eq. (2) is simply additive. When the noise brought in by the injection field is also taken into account, additional multiplicative noise terms will appear in Eq. (8). It is thus clear that mathematically speaking, understanding of the effects of spontaneous emission noise on the laser dynamics is also useful for the understanding of the effects of noise from the injection field on the laser dynamics. The dynamics of the above system has been experimentally mapped as a funcˆ 39 where two separate chaotic regions are identified. In particular, tion of ξ and Ω, ˆ is two scenarios for period-doubling route to chaos have been found: (1) when Ω 38 fixed, the system period-doubles to chaos when ξ is varied ; and (2) when ξ is ˆ is varied.39 The effects of fixed, the system also period-doubles to chaos when Ω noise on the laser dynamics corresponding to a specific period-doubling route to ˆ = 0 and ξ varied is studied in Ref. 24. It is found that this specific chaos with Ω route to chaos is very similar to the main period-doubling cascade to chaos of the logistic map, where noise-induced chaos is not observed. With the experimentally determined noise level, i.e. µ = 1, type (ii) diffusion is observed. When µ is reduced, type (i) diffusion is observed. These results are shown in Fig. 8. Noise-induced chaos has recently been carefully studied by considering the second scenario for period-doubling route to chaos.25 With ξ = 0.04, we have found ˆ around that the specific period-doubling route following a path of increasing Ω ˆ Ω = 0.04732 (corresponding to a frequency detuning of 1.67 GHz) is still like the main period-doubling route to chaos of the logistic map, where type (ii) diffusion has been observed, while noise-induced chaos does not occur. The specific route ˆ around Ω ˆ = 0.07959 (corresponding to a frefollowing a path of decreasing Ω quency detuning of 3.04 GHz) is, on the other hand, quite different. Both type (iii) anomalous diffusion and noise-induced chaos have been observed. Figure 9 shows

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3299

Fig. 8. Four logarithmic displacement hlnkXi+k − Xj+k ki curves (from top to bottom) of the dynamics of the optically injected semiconductor laser corresponding to shells (2−(i+1)/2 , 2−i/2 ) ˆ = 0 and ξ = 0.015. (a) µ = 1 and (b) µ = 0.051/2 . Also shown with i = 7, 8, 9 and 10 for Ω in (a) (as diamonds) and (b) (as squares) are curves generated from ln k α with α = 0.3 and 0.5, respectively.

< ln || Xi+k - X j+k || >

2

(a) 0

(b)

2

4 0

500

1000

1500

2000

Evolution time k

Fig. 9. Four logarithmic displacement hlnkXi+k − Xj+k ki curves, from top to bottom, corˆ = 0.04732 and responding to shells (2−(i+1)/2 , 2−i/2 ) with i = 9, 10, 11 and 12 for (a) Ω ˆ = 0.07959 and µ = 0.00071/2 . Curves generated from ln k α with α = 0.4 µ = 0.0071/2 and (b) Ω and 0.9 are also depicted in (a) as blank squares and (b) as circles, respectively.

ˆ = 0.04732 and 0.07959, where we find the logarithmic displacement curves for Ω α = 0.4 and 0.9, respectively. Type (iii) diffusion and noise-induced chaos have also been observed for other periodic states embedded inside the chaotic region, such as ˆ = 0.07409. at Ω

November 1, 1999 17:5 WSPC/140-IJMPB

3300

0236

J. B. Gao et al.

µ=0

µ=1

15 Ω = 0.04372

Ω = 0.04372

Ω = 0.07933

Ω = 0.07933

Ω = 0.07959

Ω = 0.07959

10

5

0

Magnitude (A.U.)

15

10

5

0 15

10

5

0 10

5

0

5

10

10

5

0

5

10

Frequency (GHz) ˆ = 0.04732, 0.07933, Fig. 10. Optical spectrum of the optically injected semiconductor laser for Ω and 0.07959, for the cases of µ = 0 and µ = 1.

Researchers in optics may feel more at home with the optical spectra. We have indeed observed that different types of diffusional processes have different manifestations on the optical spectra. While the spectra of type (i) and type (ii) processes are similar, they are quite different from those of type (iii) processes or ˆ = 0.04732, those of noise-induced chaos. Figure 10 shows the optical spectra at Ω 0.07933, and 0.07959, both for the clean (µ = 0) and the noisy (µ = 1) systems. ˆ = 0.04732 and 0.07959 is periodic; hence, the spectra consist The clean system at Ω ˆ = 0.07933 is chaotic and the spectra of only discrete peaks. The clean system at Ω is dominated by a broad pedestal with some strong secondary peaks. The spectra ˆ = 0.07933 does not vary much when noise is present, indicating that this parat Ω ˆ = 0.07959 ticular chaotic state is quite insensitive to noise. The noisy spectra at Ω

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3301

ˆ = 0.07933, indicating noise-induced chaos has occurred. is very similar to that at Ω ˆ The noisy spectra at Ω = 0.04732, however, is still dominated by two very strong peaks, indicating that the system is mostly just masked by noise. 6. Desynchronization in Chaotic Systems and Failure of Secure Communication Using Pseudo-Chaos Since Pecora and Carroll42 have theoretically proven and experimentally demonstrated that two identical chaotic systems can be synchronized (for a recent review, see Ref. 43), there has been much interest in using the broadband and unpredictable features of a chaotic waveform to achieve secure communication. Here we resort to the concept of chaotic synchronization and secure communication to deepen our understanding of (1) when a noisy chaotic system ought to be regarded as simply noisy rather than still mostly deterministically chaotic, and (2) why simple diffusional processes are pseudo-chaos. The understanding on how noise affects chaotic systems and nonlinear oscillators sheds light on the desynchronization in chaotic systems due to noise and failure of secure communication using pseudo-chaos. First we consider a drive-response synchronization scheme, which is schematically shown in Fig. 11. The drive system is a modified chaotic Lorenz system44 : dx = σ(y(t) − x(t)) , dτ dy = rx(t) − y(t) − 20x(t)z(t) , dτ dz = 5x(t)y(t) − bz(t) , dτ with σ = 16, r = 45.6 and b = 4. The response system is

(9)

dxr = σ(yr (t) − xr (t)) , dτ dyr (10) = rv(t) − yr (t) − 20v(t)zr (t) , dτ dzr = 5v(t)yr (t) − bzr (t) , dτ where v(t) = x(t)+n(t) and n(t) is white Gaussian noise with zero mean and power spectral density σn2 . Note that the channel noise n(t) is simply a measurement noise Drive X

Response x(t)

+

Y Z Fig. 11.

n(t)

x r(t)

v(t) Yr

Xr

Zr

A schematic for drive-response synchronization.

November 1, 1999 17:5 WSPC/140-IJMPB

3302

0236

J. B. Gao et al.

when viewed by the drive system. However, it is a dynamic noise from the point of view of the response system. This situation should remind us of the noise brought in by an injection field in terms of the operation of a driven semiconductor laser. The effect of weak channel noise on chaotic synchronization was studied by Tsimiring and Suschik.45 They found that synchronization was severely degraded by the bursting caused by the noise. The bursting is caused by local positive Lyapunov exponents. Here we shall study the effect of noise on desynchronization in chaotic systems more quantitatively. We define the input and output SNR to be Input SNR = 10 log10 (σx2 /σn2 ) , Output SNR = 10 log10 (σx2 /σe2 ) , where σx2 is the power (i.e. variance) of the transmitted signal x(t) and σe2 is the power of the synchronization error e(t) = x(t) − xr (t). (By this definition, D = 4 of Eq. (2) corresponds to 10 dB.) Figure 12(a) shows the variation of Output SNR versus Input SNR. We observe that when the Input SNR is around 20 dB, the Output SNR starts to drop almost vertically. It actually happens that the two systems are desynchronized before the Input SNR is reduced to 20 dB. Next we compute the Λ(k) (and displacement) curves from xr (t) and then compute NA according to Eq. (4). Note that the Λ(k) curves corresponding to Input SNR = 30 dB are very similar to those of Fig. 1(b). (It goes without saying that the Λ(k) curves from xr (t) for Input SNR > 30 dB are in between those of Figs. 1(a) and (b), while those for Input SNR < 30 dB look noisier than Fig. 1(b).) For the computation of NA, we choose to work with two logarithmic displacement curves corresponding to shells (2−(i+1)/2 , 2−i/2 ) with i = 10 and 11. Figure 12(b) shows the variation of NA versus Output SNR. We observe that when Output SNR ≥ 35 dB, NA is close to 1. This means that the dynamics on the specific scale chosen by us has not been affected much by the small amount of noise. However, for Output SNR ≤ 25 dB, NA 1, indicating that the dynamics has been more like noise than chaos. Hence, desynchronization is basically because of the fact that the response

Fig. 12.

(a) Output SNR versus Input SNR and (b) NA versus Output SNR curves.

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3303

signal xr (t) is no longer chaotic. When a system is not chaotic, then chaotic synchronization is not defined. Synchronization can occur not only in chaotic systems, but also in nonlinear oscillators. A simple example is the following drive-response system using the logistic map: xn+1 = f (xn ) + Pn(1) ,

0 < xn < 1 ,

yn+1 = f (yn ) + [f (xn ) − f (yn )] + Pn(2) ,

(11) 0 < yn < 1 ,

(12) (1)

where f (x) = µx(1 − x) is the logistic map with µ = 3.55,  = 0.5, and Pn (2) and Pn are Gaussian random variables with zero mean and standard deviation σ. (1) When the x system is clean (i.e. Pn = 0), we know from Sec. 3 that the system (2) is periodic with period 8. When the y system is also clean (i.e. Pn = 0), then after several tens of iterations, y synchronizes to x perfectly. When both x and y systems are noisy but the noise level is small (such as σ = 0.0005), the x and y systems can still be very well synchronized. This is because in this situation, the values for the synchronization error, y(n) − x(n), are on the order of 10−4 , while the values for the signals x(n) and y(n) are on the order of 1. Recalling from the discussion in Sec. 4 that the x system is a diffusional process of type (ii) and might be interpreted as “chaos”, there is a tendency for the Eqs. (11) and (12) to be viewed as an example for chaotic synchronization. However, secure communication using Eqs. (11) and (12) will simply fail: because the noisy signals x(n) and y(n) follow closely the period 8 pattern of the clean signal,18 the values of x(n) and y(n) can be readily estimated. 7. Discussions In this paper, we have carefully considered what constitutes a test for noise-induced chaos. In particular, we have pointed out when and how a simple noisy signal may be mistaken as deterministic chaos. The mechanism for noise-induced chaos has been studied in great detail. By considering the long-term growth rate of the logarithmic displacement curves, we have identified three types of diffusional processes and have identified the third type, the anomalous diffusion, to be the intrinsic mechanism for noise-induced chaos. Numerous examples for the three different diffusional processes have also been given. As an application of these understandings, a specific section on the desynchronization in chaotic systems due to noise and on the failure of secure communication using pseudo-chaos is also included. Can noise-induced chaos be observed? In an experimental setting, if the equations for the evolution of the system are unknown, then one is at most able to determine if the motion is chaotic or not. However, in a deliberately designed experiment, when the equations for the motion are known and the noise source can be freely turned on and off, one can certainly study if noise-induced chaos has occurred and what noise level can induce chaos.

November 1, 1999 17:5 WSPC/140-IJMPB

3304

0236

J. B. Gao et al.

A comment is worth making on why only a few cases of low-dimensional deterministic chaos have been convincingly identified experimentally. It has been thought that low-dimensional chaos would be most likely observed at the end of a bifurcation sequence, such as a period-doubling cascade. We surmise that in an experimental situation, an observed period-doubling cascade is more like the main 2n cascade of the logistic map. Were this true, then neither could noise induce chaos, nor could the associated chaos be readily identified as low-dimensional, since those chaotic states are very sensitive to noise. We can only hope that sometimes an observed period-doubling sequence is more like the period(3)-doubling sequence of the logistic map so that noise-induced chaos or the associated chaos at the end of the bifurcation sequence could be unambiguously identified as low-dimensional. Hence, we encourage future experimentalists not to stop at somewhere of a bifurcation sequence, but to fully explore it. Acknowledgment J. B. Gao would like to thank Professors X. Q. Er, Z. B. Lin, and Z. M. Zheng of the Lab for Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences for providing him with the fluctuating velocity signals measured in the near wake behind a cylinder and for many useful discussions. This work is partly supported by the U.S. Army Research office under the contract No. DAAG55-98-1-0269. References 1. R. Benzi, A. Sutera and A. Vulpiani, J. Phys. A14, L453 (1981). 2. L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998). 3. A. R. Bulsara, E. W. Jacobs and W. C. Schieve, Phys. Rev. A42, 4614 (1990). 4. Z. Y. Chen, Phys. Rev. A42, 5837 (1990). 5. Matsumoto and I. Tsuda, J. Stat. Phys. 31, 87 (1983). 6. J. P. Crutchfield and B. A. Huberman, Phys. Lett. A74, 407 (1980). 7. J. P. Crutchfield, J. D. Farmer and B. A. Huberman, Phys. Rep. 92, 46 (1982). 8. R. L. Kautz, J. Appl. Phys. 58, 424 (1985). 9. F. Arecchi, R. Badii and A. Politi, Phys. Rev. A32, 403 (1985). 10. R. J. Wiener, G. L. Snyder, M. P. Prange and D. Frediani, Phys. Rev. E55, 5489 (1997); J. Vonstamm, U. Gerdts, T. Buzug and G. Pfister, Phys. Rev. E54, 4938 (1996); T. Buzug, J. Vonstamm and G. Pfister, Phys. Rev. E47, 1054 (1993); D. Rockwell, F. Nuzzi and C. Magness, Phys. Fluids 3, 1477 (1991). 11. L. Chusseau, E. Hemery and J. M. Lourtioz, Appl. Phys. Lett. 55, 822 (1989); T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis and P. M. Alsing, Appl. Phys. Lett. 64, 3539 (1994); Phys. Rev. A51, 4181 (1995); Appl. Phys. Lett. 67, 2780 (1995). 12. T. Geest, C. G. Steinmetz, R. Larter and L. F. Olsen, J. Phys. Chem. 96, 5678 (1992); G. Rabai, J. Phys. Chem. A101, 7085 (1997). 13. P. A. Miller and K. E. Greeberg, Appl. Phys. Lett. 60, 2859 (1992). 14. G. A. Ruiz, Chaos Solitons & Fractals 6, 487 (1995). 15. F. Argoul, A. Arneodo and P. Richetti, in A Chaotic Hierarchy, eds. G. Baier and M. Klein (World Scientific, Singapore, 1991), pp. 80–83.

November 1, 1999 17:5 WSPC/140-IJMPB

0236

Noise-Induced Chaos

3305

16. J. C. Roux, J. S. Turner, W. D. McCormick and H. L. Swinney, in Nonlinear Problems: Present and Future, eds. A. R. Bishop, D. K. Campbell and B. Nicolaenko (NorthHolland, Amsterdam, 1982), p. 409. 17. J. L. Hudson, J. C. Mankin, J. McCullough and P. Lamba, in Nonlinear Phenomena in Chemical Dynamics, eds. C. Vidal and A. Pacault (Springer, Berlin, 1982), p. 44. 18. J. B. Gao, S. K. Hwang and J. M. Liu, Phys. Rev. Lett. 82, 1132 (1999). 19. N. H. Packard, J. P. Crutchfield, J. D. Farmer and R. S. Shaw, Phys. Rev. Lett. 45, 712 (1980); F. Takens, in Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 898, eds. D. A. Rand and L. S. Young (Springer-Verlag, Berlin, 1981), p. 366. 20. J. B. Gao and Z. M. Zheng, Phys. Lett. A181, 153 (1993). 21. J. B. Gao and Z. M. Zheng,, Europhys. Lett. (25), 485 (1994); Phys. Rev. E49, 3807 (1994). 22. G. Paladin, M. Serva and A. Vulpiani, Phys. Rev. Lett. 74, 66 (1995); V. Loreto, G. Paladin and A. Vulpiani, Phys. Rev. E53, 2087 (1996). 23. J. B. Gao, Physica D106, 49 (1997). 24. J. B. Gao, S. K. Hwang and J. M. Liu, Phys. Rev. A59, 1582 (1999). 25. S. K. Hwang, J. B. Gao and J. M. Liu, submitted to Phys. Rev. E. 26. A. R. Osborne and A. Provenzale, Physica D35, 357 (1989). 27. A. Provenzale, A. R. Osborne and R. Soj, Physica D47, 361 (1991). 28. M. Dammig and F. Mitschke, Phys. Lett. A178, 385 (1993). 29. A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Physica D16, 285 (1985). 30. J. Theiler, S. Eubank, A. Longtin and B. Galdrikian, Physica D58, 77 (1992). 31. S. Sato, M. Sano and Y. Sawada, Prog. Theor. Phys. 77, 1 (1987). 32. Z. B. Lin, X. Q. Er and J. B. Gao, Sci. in China (Ser. A) 23, 277 (1993) (in Chinese). 33. X. G. Wu and R. Kapral, Phys. Rev. Lett. 70, 1940 (1994). 34. S. Ciliberto and J. Gollub, J. Fluid Mech. 158, 381 (1985). 35. S. A. Stafford, M. Kot and J. R. Roth, J. Appl. Phys. 68, 488 (1990). 36. B. R. Noack and H. Eckelmann, Physica D56, 151 (1992). 37. J. Sacher, D. Baums, P. Panknin, W. Elsasser and E. O. Gobel, Phys. Rev. A45, 1893 (1992). 38. T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis and P. M. Alsing, Appl. Phys. Lett. 64, 3539 (1994); Phys. Rev. A51, 4181 (1995). 39. V. Kovanis, A. Gavrielides, T. B. Simpson and J. M. Liu, Appl. Phys. Lett. 67, 2780 (1995). 40. S. K. Hwang and J. M. Liu, to be published. 41. T. B. Simpson and J. M. Liu, J. Appl. Phys. 73, 2587 (1993); J. M. Liu and T. B. Simpson, IEEE Photon. Technol. Lett. 4, 380 (1993); IEEE J. Quantum Electron. 30, 957 (1994). 42. L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990). 43. L. M. Pecora, T. L. Carroll, G. A. Johnson and D. J. Mar, Chaos 7, 520 (1997). 44. K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, IEEE Trans. Cir. & Syst.-II 40, 626 (1993). 45. L. S. Tsimiring and M. M. Suschik, Phys. Lett. A213, 155 (1996).