ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 29 Number 7 July 2012 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/cpl http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIETY CHIN. PHYS. LETT. Vol. 29, No. 7 (2012) 070501 Wavelet Phase Synchronization of Fractional-Order Chaotic Systems * CHEN Feng(陈¶)1, XIA Lei(gX)2, LI Chun-Guang(oS1)3** 1School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054 2Xi’an Institute of Electromechanical Information Technology, Xi’an 710065 3 Department of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027 (Received 21 March 2012) Nowadays, fractional-order systems are attracting more and more attention. There are several ways available for analyzing fractional-order systems, among which wavelet transform is an efficient method for analyzing system dynamics in both time and frequency domains. We investigate the wavelet phase synchronization employing wavelet transform to explore the phase synchronization behaviors of fractional-order chaotic oscillators. We analyze in detail the synchronization behaviors with changes to the coupling strength, the central frequency Ω 0, and the time scale of the wavelet. PACS: 05.45.-a, 05.45.Xt DOI: 10.1088/0256-307X/29/7/070501 The chaotic dynamics of fractional order sys- edge, wavelet phase synchronization in fractional or- tems began to attract more and more attention re- der chaotic systems has not been discussed yet. In cently. Fractional calculus has a history of more this Letter, we address it in detail. than 300 years,[1] the applications of which to physics In Refs. [23-26] the researchers proposed to use a and engineering are just a recent focus of interest. wavelet approach to investigate the synchronization Many systems are known to display fractional-order of chaotic systems. It was approved that synchroniza- dynamics, such as viscoelastic systems,[2] dielectric tion can be detected by introducing a complex wavelet polarization,[3] electrode-electrolyte polarization,[4] transform, as demonstrated in Ref. [23], W (a; t) = and electromagnetic waves.[5] Many investigations jW (a; t)j exp(i'(a; t)), where '(a; t) is the phase de- were devoted to dynamics[6−15] of fractional-order sys- fined. The corresponding condition for the synchro- tems. Specifically, in Ref. [6] chaos and hyperchaos in nization between chaotic systems with phases '1(a; t) the fractional order Rössler equations were studied, in and '2(a; t) on a time scale a is j'1(a; t) − '2(a; t)j < which we showed that chaos can exist in the fractional const. In the present study, this wavelet approach order Rössler equation with order as low as 2.4, and is applied to the coupled fractional-order chaotic sys- hyperchaos can exist in the fractional order Rössler tems. As in many studies of phase synchronization, hyperchaos equation with order as low as 3.8. the chaotic system is selected as the Rössler oscilla- In the study of chaotic systems, synchronization is tor. a focus of interest.[16−18] Besides the identical (com- First, let us build the system. Our system con- plete) synchronization of chaotic systems, some other sists of two coupled fractional-order Rössler oscilla- types of synchronization also have very interesting tors. There are several definitions of fractional deriva- cooperative behaviors of chaotic systems, including tives in the literature.[1] The best known may be the phase synchronization,[19;20] lag synchronization,[21] Riemann–Liouville definition, which is given by projective synchronization,[22] and wavelet phase d f(t) 1 dn Z t f(휏) synchronization/time-scale synchronization of chaotic 훼 훼 = n 훼−n+1 푑휏; (1) oscillators.[23−26] dt Γ (n − 훼) dt 0 (t − 휏) In Ref. [27] we have studied the synchronization of where Γ is the gamma function and n − 1 ≤ 훼 < n. fractional order chaotic systems, and since then, the Upon considering all the initial conditions to be zero, synchronization of fractional order chaotic systems has the Laplace transform of the Riemann–Liouville frac- 훼 d f(t) 훼 begun to attract the attention of some researchers, see tional derivative is Lf dt훼 g = s Lff(t)g: Thus, the e.g. Ref. [28]. However, in the literature, the authors fractional integral operator of order 훼 can be repre- are mostly concerned with the identical synchroniza- sented by the transfer function F (s) = 1=s훼 in the tion of fractional order chaotic systems. In Ref. [29] frequency domain. The standard definition of frac- we studied the phase and lag synchronization of cou- tional differintegral does not allow direct implementa- pled fractional order chaotic oscillators. In Ref. [30] tion of the fractional operators in time-domain simula- we investigated the projective synchronization of frac- tions. An efficient method to circumvent this problem tional order chaotic systems. However, to our knowl- is to approximate fractional operators by using stan- *Supported by the National Natural Science Foundation of China under Grant Nos 61171153 and 61101045, the Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No 2007B42, the Zhejiang Provincial Natural Science Foundation under Grant No LR12F01001, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars. **Corresponding author. Email: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd 070501-1 CHIN. PHYS. LETT. Vol. 29, No. 7 (2012) 070501 dard integer order operators. In the following simu- not grow with time, i.e. j'1(a; t) − '2(a; t)j < const, lations, we use the approximation method proposed thus the phase synchronization condition is satisfied. in Ref. [31], which was also adopted in many studies. In Table1 of Ref. [7], the authors presented the ap- 40 Ω π proximations for 1=sq with q = 0:1 − 0:9 in steps 0.1 0=0.5 30 | 2 Ω0=1π with errors of approximately 2 dB. We will mainly use φ 20 - these approximations in the following simulations. In 1 φ our model, the conventional derivative is replaced by | 10 Ω =1.5π a fractional derivative as follows: 0 0 Ω0=2π 훼 0 200 400 600 800 1000 1200 1400 1600 1800 d x1;2 = −! y − z + "(x − x ); Time dta 1;2 1;2 1;2 2;1 1;2 훼 Fig. 2. Phase differences calculated for different central- d y1;2 = ! x + by ; frequency values. dta 1;2 1;2 1;2 훼 d z1;2 30 a = 0:2 + z1;2(x1;2 − 10); (2) (a) dt 25 | 2 20 where 훼 is the fractional-order, b is a constant, and φ - 15 ε=0.05 1 !1;2 are system parameters for the oscillators 1 and 2, φ | which control the frequency. The parameter " is the 10 ε=0.1 coupling strength between the two oscillators. From 5 0 Ref. [6], we set 훼 = 0:9, b = 0:35, and ! ≈ 1, then ε=0.5 1;2 −5 the chaotic behavior is found. The chaotic attractor 0 200 400 600 800 1000 1200 1400 1600 1800 Time is shown in Fig. 1. 3 (b) 2.5 25 20 2 15 2 1.5 Α z 10 5 1 0 0.5 20 10 15 20 0 0 5 10 y −10 −5 0 0 0.5 1 1.5 2 2.5 3 −20 −15 −10 x Α1 Fig. 1. Chaotic attractor of the fractional-order Rössler Fig. 3. (a) Phase differences of the coupled fractional- oscillator. order Rössler equations at time scale a = 5. (b) The Next, let us consider to use the continuous Morlet- correlation between the amplitude A1, A2. Here a = 5, " = 0:5. wavelet to obtain the wavelet transform of chaotic time series,[23;24] so that we can analyze the wavelet In our simulation, the time series used for wavelet phase synchronization of this coupled system. transform are generated by the two coupled Rössler equations. In Ref. [24] the authors proved that only Z +1 at a certain time scale are the two integer-order os- W (a; t ) = x(t) * (t)dt; (3) 0 a;t0 −∞ cillators synchronized with each other. We set the time scale a = 5 for the wavelet function, and set where a;t0 (t) is the wavelet function related to the !1;2 = 1:03 and 0.97 for the two oscillators. For p1 t−t0 mother-wavelet 0(t) as a;t0 (t) = s 0( s ). The an integer order chaotic system, it was mentioned time scale a corresponds to the width of the wavelet in Ref. [25] that there will be no phase synchroniza- function a;t0 (t), and t0 is the time shift of the tion if the central frequency Ω0 is lower than a cer- wavelet. The Morlet-wavelet[32] tain critical value. In Ref. [26], the author adopted an interesting method to investigate the influence of 1 −휂2 (휂) = p exp(jΩ 휂) exp( ) (4) the central frequency Ω0 on the phase synchroniza- 0 4 0 휋 2 tion. Now, we study the impact of the central fre- is regarded as a mother-wavelet function. The wavelet quency Ω0 on phase synchronization in the fractional- surface function is order chaotic system. We set the coupling parameter " = 0:5. The simulation result is shown in Fig. 2. We W (a; t) = jW (a; t)j exp(i'(a; t)): (5) can see that the phase difference increases with time when Ω0 = 0:5휋; 휋 and remains bounded if Ω0 > 1:5휋. The phase '(a; t) = arg W (a; t). Thus, in the fractional-order chaotic system, the cen- We define the mean frequency difference as ∆Ω = tral frequency Ω0 is also important for determining _ _ [19] h휑1(a; t) − 휑2(a; t)i, where ⟨·⟩ denotes the average the phase synchronization.