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Pseudo-periodic surrogate test to sample time series in stochastic softening Duffing oscillator
Physics Letters A 357 (2006) 204–208 www.elsevier.com/locate/pla
Pseudo-periodic surrogate test to sample time series in stochastic softening Duffing oscillator Chunbiao Gan Department of Mechanics, CMEE, Zhejiang University, Hangzhou 310027, PR China Received 27 September 2005; received in revised form 12 January 2006; accepted 11 April 2006 Available online 25 April 2006 Communicated by A.P. Fordy
Abstract Identification of typical noise-contaminated sample response is a hard task in a nonlinear system under stochastic background since irregularity of the sample response may come from measure noise, dynamical noise, or nonlinear effect, etc., and conventional dynamical methods are generally not useful. Here, the pseudo-periodic surrogate algorithm by Small is employed to test the sample time series in the softening Duffing oscillator under the Gaussian white noise excitation. The correlation dimensions of the noisy periodic and the noise-induced chaotic time series of the system are compared with those of their corresponding surrogate data respectively, the leading Lyapunov exponents by Rosenstein’s algorithm are also presented for comparison. © 2006 Elsevier B.V. All rights reserved. Keywords: Softening Duffing oscillator; Gaussian white noise; Surrogate algorithm; Correlation dimension; Leading Lyapunov exponent
1. Introduction In general, it is not easy to quantify typical dynamical behavior in a nonlinear system under stochastic background. In 1990, Sugihara [1] made tentative distinction between dynamical chaos and measurement error noise due to the basic idea, i.e., if deterministic laws govern the system, then, even if the dynamical behavior is chaotic, the future may to some extent be predicted from the behavior of past values that are similar to those of the present. Recently, the method of surrogate data is used by Small et al. [2–4] to test pseudo-periodic time series data and detect determinism in time series with dynamical noise contamination. As pointed out by Small et al., the method of surrogate data provides a rigorous way to apply statistical hypothesis testing to experimental time series. By the pseudoperiodic surrogate (PPS) algorithm, chaotic and noisy periodic time series are distinguished from the experimental data for the Rössler system and human EGG [2].
E-mail addresses: [email protected], [email protected] (C. Gan). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.040
In [5], Gan has studied the noise-induced dynamics in the softening Duffing oscillator under the Gaussian white noise, where the fractal basin boundary [6,7] and positive leading Lyapunov exponent [8,9] are used to predict the rising of the noise-induced chaos in the softening Duffing oscillator. In this Letter, the PPS algorithm is further employed to test the noisy periodic and the noise-induced chaotic time series in such stochastic oscillatory system. Section 2 briefly describes the stochastic softening Duffing oscillator and the PPS algorithm by Small et al. In Section 3, we present some simulation results of the correlation dimensions for the original sample time series of the system and their corresponding artificial data, where the leading Lyapunov exponents of the original time series are also displayed to support our points. Finally, in Section 4 we conclude. 2. The stochastic softening Duffing oscillator and the PPS algorithm The softening Duffing oscillator under the Gaussian white noise excitation can be written as follows:
C. Gan / Physics Letters A 357 (2006) 204–208
x˙ = y, y˙ = −x + x 3 + ε (−μy + f1 cos ωt + f2 ξ(t)),
(1)
where 0 < ε 1, μ and f1 are the damping coefficient and the amplitude of the harmonic excitation respectively, and ξ(t) is taken as the Gaussian white noise, f2 is the strength of this stochastic excitation. The spectrum density for the Gaussian white noise is assumed to be S0 with the strength D (= 2πS0 ). For system (1), it is well known that the basin surrounded by the stable and unstable manifolds of unperturbed system (ε = 0) will be eroded when ε = 0 and the harmonic excitation is added, and the boundary of the basin can be incursively fractal. Gan in his recent work [5] has shown that fractal basin boundaries can also appear when system (1) is excited only by the Gaussian white noise (ε f1 = 0, ε f2 = 0). From the dynamical theory, fractal basin boundary is generally related to chaotic response, that is, chaos must arise in the system. Based on the necessary condition from the stochastic Melnikov integral and the calculation of the leading Lyapunov exponent from Rosenstein’s algorithm [9], three categories of responses, i.e., almost-harmonic (called noisy periodic here), noise-induced chaotic and thoroughly random responses, are discussed. In the present Letter, the PPS algorithm is further employed to test the first two kinds of noise-contaminated dynamics. Standard surrogate techniques were first suggested by Theiler and his colleagues in 1992 [10]. Using the surrogate data methods, one can test an experimental time series for membership of a specific class of dynamical systems. The algorithm by Theiler et al. tests only for independent noise, linear noise or statically filtered linear noise, while the PPS algorithm presented by Small et al. in [2] can generate surrogates that preserve coarse deterministic features (such as periodic trends) but destroy fine structures (such as deterministic chaos). They showed that PPS method could be applied to differentiate between chaos with dynamic noise and a periodic orbit. The basic algorithm for PPS method presented in [2,3] can be performed according to the following steps: (i) construct the w from the scalar time series vector delay embedding {zt }N−d t=1 N {yt }t=1 according to zt = [yt , yt+τ , yt+2τ , . . . , yt+de τ ], where the embedding dimension de and embedding lag τ can be selected according to Ref. [11], and the embedding window dw = de τ − 1; (ii) choose an initial condition s1 ∈ A at random and let i = 1, where A = {zt | t = 1, 2, . . . , N − dw } is called the reconstructed attractor; (iii) choose a near neighbor zj ∈ A of si according to the probability distribution Prob(zj = zt ) ∝ exp{(−zt − si )/ρ}, where the parameter ρ is the noise radius; (iv) let si+1 = zj +1 be the successor to si and make increment i; (v) if i < N go to step (iii); (vi) the surrogate time series is recorded as {(st )1 } ≡ {(s1 )1 , (s2 )1 , (s3 )1 , . . . , (sN )1 }, i.e., the scalar first components of {st }t . In the above algorithm, the noise radius ρ can be selected according to the suggestions by Small et al. [2] such that the expected number of sequences of length two or more that are identical for data and surrogates is maximized. This selection criteria of noise radius can provide a balance between: (a) too
205
much randomization (few identical sequences of length); and (b) too little (data and surrogate near identical). 3. Simulation results on the noise-contaminated time series In this section, the PPS algorithm is employed to test the noise-contaminated dynamics in system (1). As studied by Small et al., the PPS algorithm is a robust method to test noisy periodic time series data against the null hypothesis of a periodic orbit with uncorrelated noise. In Figs. 1 and 2, we all set ε = 0.1, ω = 1.0, D = 1.0 for simulating the sample time series from the system, unless otherwise indicated. The x-axis in Figs. 1(a)–(d) and 2(a)–(d) means dynamical evolving time t in seconds, while the y-axis in Figs. 1(a)–(b) and 2(a)–(b) and Figs. 1(c)–(d) and 2(c)–(d) represents the displacement x in meter in system (1) and mean divergence of the system’s sample response respectively. Here, the leading Lyapunov exponent of the system’s response can be evaluated by the least-square fit from the curve in Figs. 1(c)–(d) and 2(c)–(d) (see [9]). An obvious linear slope in Fig. 2(c) or (d) means the response is a chaotic one. It deserves to notify that, the values of the leading Lyapunov exponents (the slopes of the obvious linear regions in the mean divergence curves) are not accurately calculated, the main goal in current work is just to show nonchaotic and chaotic responses in stochastic system (1). Due to the particularity of the system discussed in this Letter, the initial state is hard to be chosen for performing the simulation of the system’s response since the basin is readily eroded by the external excitation. For all the original sample time series presented in Figs. 1 and 2, the initial states are picked up from the results of safe basins given in [5]. For each sample time series of system (1) and 30 PPS data sets, we estimated the correlation dimension according to the algorithm by Judd in [12]. The algorithm described in [12] estimates correlation dimension dc as a function of viewing scale ε0 , hence dc (ε0 ). Therefore, the correlation dimension for each time series is not a single number, but a curve. For the periodic signals contaminated with the Gaussian white noise (see Fig. 1(a), (b)), the data and surrogates are indistinguishable and in this case the null hypothesis of a periodic orbit with uncorrelated noise cannot be rejected, see Fig. 1(e), (f). This means that periodic nature still dominates the dynamical behavior though it is contaminated by the dynamical noise and seems irregular. These results are consistent with what the leading Lyapunov exponents predicted (see Fig. 1(c), (d)), from which one learns that the slope of the curve is almost flat and the leading Lyapunov exponent is nearly zero. In Fig. 2, we also tested this algorithm on noise-induced chaotic time series. For these chaotic ones, the original signals and their corresponding surrogates are clearly distinct when ε0 tends to zero, and in this case the null hypothesis of a periodic orbit with uncorrelated noise should be rejected. From Fig. 2(c), (d), apparent incline appears in the curve which tells one that the leading Lyapunov exponent is positive and finite and the time series is a chaotic one. From Figs. 1 and 2, the noisy periodic time series can be easily picked out from
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C. Gan / Physics Letters A 357 (2006) 204–208
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1. PPS test for the noisy periodic time series. The top panels show the original time series from system (1) for (a) non-zero damping force case, μ = 5.0, f1 = 3.8, f2 = 0.02 and (b) zero damping force case, μ = 0.0, f1 = 0.0, f2 = 0.02. Also shown in (c) and (d) are the leading Lyapunov exponents by Rosenstein’s algorithm for each (see [4]). Comparisons of the correlation dimensions for pseudoperiodic time series and surrogate data are presented in (e) ρ = 0.002 and (f) ρ = 0.001, where the thick real line and the thin scatter lines represent the results from the original and the surrogate data respectively. In these noisy periodic cases, there is no distinction between the data and surrogates.
other noise-contaminated dynamics of the system, while for the noise-induced chaotic ones, other measures, such as the time history and the leading Lyapunov exponents, etc., should be complemented for a proper description on the data’s dynamical nature. All the results shown in Figs. 1 and 2 coincide with what Small et al. predicted about noisy data in [2].
Based on the results presented in [5] and the above correlation dimension comparisons, one knows that some quantitative methods, such as the fractal basin boundary, the leading Lyapunov exponent and the surrogate test, etc., can be improved to identify the noise-contaminated dynamical responses.
C. Gan / Physics Letters A 357 (2006) 204–208
207
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. PPS test for the noise-induced chaotic time series. The top panels show the original time series from system (1) for (a) non-zero damping force case, μ = 5.0, f1 = 3.88, f2 = 0.01 and (b) zero damping force case, μ = 0.0, f1 = 0.0, f2 = 0.08. Also shown in (c) and (d) are the leading Lyapunov exponents by Rosenstein’s algorithm for each (see [4]). Comparisons of the correlation dimensions for noise-induced chaotic time series and surrogate data are presented in (e) ρ = 0.002 and (f) ρ = 0.0008, where the thick real line and the thin scatter lines represent the results from the original and the surrogate data respectively. In these cases, the data and surrogates are clearly distinguishable.
4. Summary and discussion To date, surrogate techniques have largely been applied within the nonlinear time series community to screen data prior to analysis with nonlinear methods. The present Letter further demonstrates applications of PPS algorithm to the noise-
induced dynamics in the softening Duffing oscillator under the Gaussian white noise excitation. From the numerical results presented in Section 3, it is interesting to find that the noisy periodic time series (zero leading Lyapunov exponent) and the noise-induced chaotic ones (finite positive leading Lyapunov exponent) can be easily identified
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C. Gan / Physics Letters A 357 (2006) 204–208
by the PPS algorithm. The surrogate test is still in its infancy, it needs to be further improved to test the random-predominant (infinite leading Lyapunov exponent, not given in the present Letter) time series arising from a nonlinear stochastic system. Acknowledgement The author acknowledges the financial support from the National Natural Science Foundation of China under Grant No. 10302025. References [1] G. Sugihara, R.M. May, Nature 344 (19) (1990) 734. [2] M. Small, D. Xu, R.G. Harrison, Phys. Rev. Lett. 87 (18) (2001) 188101.
[3] M. Small, C.K. Tse, IEEE Trans. Circuit Systems I: Fund. Theory Appl. 50 (5) (2003) 663. [4] M. Small, C.K. Tse, Physica D 164 (2002) 187. [5] C. Gan, Chaos Solitons Fractals 25 (2005) 1069. [6] J. Xu, Q.S. Lu, K.L. Huang, Acta Mech. Sinica 12 (1996) 281. [7] C.B. Gan, Q.S. Lu, K.L. Huang, Acta Mech. Solida Sinica 11 (3) (1998) 253. [8] H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, Cambridge Univ. Press, Cambridge, 1997. [9] M.T. Rosenstein, J.J. Collins, C.J. Luca, Physica D 65 (1993) 117. [10] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Physica D 58 (1992) 77. [11] H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York, 1996. [12] K. Judd, Physica D 56 (1992) 216.
Pseudo-periodic surrogate test to sample time series in stochastic softening Duffing oscillator Chunbiao Gan Department of Mechanics, CMEE, Zhejiang University, Hangzhou 310027, PR China Received 27 September 2005; received in revised form 12 January 2006; accepted 11 April 2006 Available online 25 April 2006 Communicated by A.P. Fordy
Abstract Identification of typical noise-contaminated sample response is a hard task in a nonlinear system under stochastic background since irregularity of the sample response may come from measure noise, dynamical noise, or nonlinear effect, etc., and conventional dynamical methods are generally not useful. Here, the pseudo-periodic surrogate algorithm by Small is employed to test the sample time series in the softening Duffing oscillator under the Gaussian white noise excitation. The correlation dimensions of the noisy periodic and the noise-induced chaotic time series of the system are compared with those of their corresponding surrogate data respectively, the leading Lyapunov exponents by Rosenstein’s algorithm are also presented for comparison. © 2006 Elsevier B.V. All rights reserved. Keywords: Softening Duffing oscillator; Gaussian white noise; Surrogate algorithm; Correlation dimension; Leading Lyapunov exponent
1. Introduction In general, it is not easy to quantify typical dynamical behavior in a nonlinear system under stochastic background. In 1990, Sugihara [1] made tentative distinction between dynamical chaos and measurement error noise due to the basic idea, i.e., if deterministic laws govern the system, then, even if the dynamical behavior is chaotic, the future may to some extent be predicted from the behavior of past values that are similar to those of the present. Recently, the method of surrogate data is used by Small et al. [2–4] to test pseudo-periodic time series data and detect determinism in time series with dynamical noise contamination. As pointed out by Small et al., the method of surrogate data provides a rigorous way to apply statistical hypothesis testing to experimental time series. By the pseudoperiodic surrogate (PPS) algorithm, chaotic and noisy periodic time series are distinguished from the experimental data for the Rössler system and human EGG [2].
E-mail addresses: [email protected], [email protected] (C. Gan). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.040
In [5], Gan has studied the noise-induced dynamics in the softening Duffing oscillator under the Gaussian white noise, where the fractal basin boundary [6,7] and positive leading Lyapunov exponent [8,9] are used to predict the rising of the noise-induced chaos in the softening Duffing oscillator. In this Letter, the PPS algorithm is further employed to test the noisy periodic and the noise-induced chaotic time series in such stochastic oscillatory system. Section 2 briefly describes the stochastic softening Duffing oscillator and the PPS algorithm by Small et al. In Section 3, we present some simulation results of the correlation dimensions for the original sample time series of the system and their corresponding artificial data, where the leading Lyapunov exponents of the original time series are also displayed to support our points. Finally, in Section 4 we conclude. 2. The stochastic softening Duffing oscillator and the PPS algorithm The softening Duffing oscillator under the Gaussian white noise excitation can be written as follows:
C. Gan / Physics Letters A 357 (2006) 204–208
x˙ = y, y˙ = −x + x 3 + ε (−μy + f1 cos ωt + f2 ξ(t)),
(1)
where 0 < ε 1, μ and f1 are the damping coefficient and the amplitude of the harmonic excitation respectively, and ξ(t) is taken as the Gaussian white noise, f2 is the strength of this stochastic excitation. The spectrum density for the Gaussian white noise is assumed to be S0 with the strength D (= 2πS0 ). For system (1), it is well known that the basin surrounded by the stable and unstable manifolds of unperturbed system (ε = 0) will be eroded when ε = 0 and the harmonic excitation is added, and the boundary of the basin can be incursively fractal. Gan in his recent work [5] has shown that fractal basin boundaries can also appear when system (1) is excited only by the Gaussian white noise (ε f1 = 0, ε f2 = 0). From the dynamical theory, fractal basin boundary is generally related to chaotic response, that is, chaos must arise in the system. Based on the necessary condition from the stochastic Melnikov integral and the calculation of the leading Lyapunov exponent from Rosenstein’s algorithm [9], three categories of responses, i.e., almost-harmonic (called noisy periodic here), noise-induced chaotic and thoroughly random responses, are discussed. In the present Letter, the PPS algorithm is further employed to test the first two kinds of noise-contaminated dynamics. Standard surrogate techniques were first suggested by Theiler and his colleagues in 1992 [10]. Using the surrogate data methods, one can test an experimental time series for membership of a specific class of dynamical systems. The algorithm by Theiler et al. tests only for independent noise, linear noise or statically filtered linear noise, while the PPS algorithm presented by Small et al. in [2] can generate surrogates that preserve coarse deterministic features (such as periodic trends) but destroy fine structures (such as deterministic chaos). They showed that PPS method could be applied to differentiate between chaos with dynamic noise and a periodic orbit. The basic algorithm for PPS method presented in [2,3] can be performed according to the following steps: (i) construct the w from the scalar time series vector delay embedding {zt }N−d t=1 N {yt }t=1 according to zt = [yt , yt+τ , yt+2τ , . . . , yt+de τ ], where the embedding dimension de and embedding lag τ can be selected according to Ref. [11], and the embedding window dw = de τ − 1; (ii) choose an initial condition s1 ∈ A at random and let i = 1, where A = {zt | t = 1, 2, . . . , N − dw } is called the reconstructed attractor; (iii) choose a near neighbor zj ∈ A of si according to the probability distribution Prob(zj = zt ) ∝ exp{(−zt − si )/ρ}, where the parameter ρ is the noise radius; (iv) let si+1 = zj +1 be the successor to si and make increment i; (v) if i < N go to step (iii); (vi) the surrogate time series is recorded as {(st )1 } ≡ {(s1 )1 , (s2 )1 , (s3 )1 , . . . , (sN )1 }, i.e., the scalar first components of {st }t . In the above algorithm, the noise radius ρ can be selected according to the suggestions by Small et al. [2] such that the expected number of sequences of length two or more that are identical for data and surrogates is maximized. This selection criteria of noise radius can provide a balance between: (a) too
205
much randomization (few identical sequences of length); and (b) too little (data and surrogate near identical). 3. Simulation results on the noise-contaminated time series In this section, the PPS algorithm is employed to test the noise-contaminated dynamics in system (1). As studied by Small et al., the PPS algorithm is a robust method to test noisy periodic time series data against the null hypothesis of a periodic orbit with uncorrelated noise. In Figs. 1 and 2, we all set ε = 0.1, ω = 1.0, D = 1.0 for simulating the sample time series from the system, unless otherwise indicated. The x-axis in Figs. 1(a)–(d) and 2(a)–(d) means dynamical evolving time t in seconds, while the y-axis in Figs. 1(a)–(b) and 2(a)–(b) and Figs. 1(c)–(d) and 2(c)–(d) represents the displacement x in meter in system (1) and mean divergence of the system’s sample response respectively. Here, the leading Lyapunov exponent of the system’s response can be evaluated by the least-square fit from the curve in Figs. 1(c)–(d) and 2(c)–(d) (see [9]). An obvious linear slope in Fig. 2(c) or (d) means the response is a chaotic one. It deserves to notify that, the values of the leading Lyapunov exponents (the slopes of the obvious linear regions in the mean divergence curves) are not accurately calculated, the main goal in current work is just to show nonchaotic and chaotic responses in stochastic system (1). Due to the particularity of the system discussed in this Letter, the initial state is hard to be chosen for performing the simulation of the system’s response since the basin is readily eroded by the external excitation. For all the original sample time series presented in Figs. 1 and 2, the initial states are picked up from the results of safe basins given in [5]. For each sample time series of system (1) and 30 PPS data sets, we estimated the correlation dimension according to the algorithm by Judd in [12]. The algorithm described in [12] estimates correlation dimension dc as a function of viewing scale ε0 , hence dc (ε0 ). Therefore, the correlation dimension for each time series is not a single number, but a curve. For the periodic signals contaminated with the Gaussian white noise (see Fig. 1(a), (b)), the data and surrogates are indistinguishable and in this case the null hypothesis of a periodic orbit with uncorrelated noise cannot be rejected, see Fig. 1(e), (f). This means that periodic nature still dominates the dynamical behavior though it is contaminated by the dynamical noise and seems irregular. These results are consistent with what the leading Lyapunov exponents predicted (see Fig. 1(c), (d)), from which one learns that the slope of the curve is almost flat and the leading Lyapunov exponent is nearly zero. In Fig. 2, we also tested this algorithm on noise-induced chaotic time series. For these chaotic ones, the original signals and their corresponding surrogates are clearly distinct when ε0 tends to zero, and in this case the null hypothesis of a periodic orbit with uncorrelated noise should be rejected. From Fig. 2(c), (d), apparent incline appears in the curve which tells one that the leading Lyapunov exponent is positive and finite and the time series is a chaotic one. From Figs. 1 and 2, the noisy periodic time series can be easily picked out from
206
C. Gan / Physics Letters A 357 (2006) 204–208
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 1. PPS test for the noisy periodic time series. The top panels show the original time series from system (1) for (a) non-zero damping force case, μ = 5.0, f1 = 3.8, f2 = 0.02 and (b) zero damping force case, μ = 0.0, f1 = 0.0, f2 = 0.02. Also shown in (c) and (d) are the leading Lyapunov exponents by Rosenstein’s algorithm for each (see [4]). Comparisons of the correlation dimensions for pseudoperiodic time series and surrogate data are presented in (e) ρ = 0.002 and (f) ρ = 0.001, where the thick real line and the thin scatter lines represent the results from the original and the surrogate data respectively. In these noisy periodic cases, there is no distinction between the data and surrogates.
other noise-contaminated dynamics of the system, while for the noise-induced chaotic ones, other measures, such as the time history and the leading Lyapunov exponents, etc., should be complemented for a proper description on the data’s dynamical nature. All the results shown in Figs. 1 and 2 coincide with what Small et al. predicted about noisy data in [2].
Based on the results presented in [5] and the above correlation dimension comparisons, one knows that some quantitative methods, such as the fractal basin boundary, the leading Lyapunov exponent and the surrogate test, etc., can be improved to identify the noise-contaminated dynamical responses.
C. Gan / Physics Letters A 357 (2006) 204–208
207
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 2. PPS test for the noise-induced chaotic time series. The top panels show the original time series from system (1) for (a) non-zero damping force case, μ = 5.0, f1 = 3.88, f2 = 0.01 and (b) zero damping force case, μ = 0.0, f1 = 0.0, f2 = 0.08. Also shown in (c) and (d) are the leading Lyapunov exponents by Rosenstein’s algorithm for each (see [4]). Comparisons of the correlation dimensions for noise-induced chaotic time series and surrogate data are presented in (e) ρ = 0.002 and (f) ρ = 0.0008, where the thick real line and the thin scatter lines represent the results from the original and the surrogate data respectively. In these cases, the data and surrogates are clearly distinguishable.
4. Summary and discussion To date, surrogate techniques have largely been applied within the nonlinear time series community to screen data prior to analysis with nonlinear methods. The present Letter further demonstrates applications of PPS algorithm to the noise-
induced dynamics in the softening Duffing oscillator under the Gaussian white noise excitation. From the numerical results presented in Section 3, it is interesting to find that the noisy periodic time series (zero leading Lyapunov exponent) and the noise-induced chaotic ones (finite positive leading Lyapunov exponent) can be easily identified
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C. Gan / Physics Letters A 357 (2006) 204–208
by the PPS algorithm. The surrogate test is still in its infancy, it needs to be further improved to test the random-predominant (infinite leading Lyapunov exponent, not given in the present Letter) time series arising from a nonlinear stochastic system. Acknowledgement The author acknowledges the financial support from the National Natural Science Foundation of China under Grant No. 10302025. References [1] G. Sugihara, R.M. May, Nature 344 (19) (1990) 734. [2] M. Small, D. Xu, R.G. Harrison, Phys. Rev. Lett. 87 (18) (2001) 188101.
[3] M. Small, C.K. Tse, IEEE Trans. Circuit Systems I: Fund. Theory Appl. 50 (5) (2003) 663. [4] M. Small, C.K. Tse, Physica D 164 (2002) 187. [5] C. Gan, Chaos Solitons Fractals 25 (2005) 1069. [6] J. Xu, Q.S. Lu, K.L. Huang, Acta Mech. Sinica 12 (1996) 281. [7] C.B. Gan, Q.S. Lu, K.L. Huang, Acta Mech. Solida Sinica 11 (3) (1998) 253. [8] H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, Cambridge Univ. Press, Cambridge, 1997. [9] M.T. Rosenstein, J.J. Collins, C.J. Luca, Physica D 65 (1993) 117. [10] J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J.D. Farmer, Physica D 58 (1992) 77. [11] H.D.I. Abarbanel, Analysis of Observed Chaotic Data, Springer-Verlag, New York, 1996. [12] K. Judd, Physica D 56 (1992) 216.