- Email: [email protected]
Chaos in the fractional order periodically forced complex Duffing’s oscillators
Chaos, Solitons and Fractals 24 (2005) 1097–1104 www.elsevier.com/locate/chaos
Chaos in the fractional order periodically forced complex Duffings oscillators Xin Gao a
a,b,*
, Juebang Yu
a
School of Electronic Engineering, University of Electronics Science and Technology of China, Chengdu, Sichuan 610054, PR China b Institute of Electrical and Information Engineering, Southwest University for Nationalities of China, Chengdu, Sichuan 610041, PR China Accepted 15 September 2004
Abstract The occurrence of fractional-order chaotic dynamics have been intensively studied over the last ten years in a large number of real dynamical systems of physical nature. However, a similar study has not yet been carried out for fractional-order chaotic dynamical systems in the complex domain. In this paper, we numerically study the chaotic behaviors in the fractional-order symmetric and non-symmetric periodically forced complex Duffings oscillators. We find that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. Our results are validated by the existence of positive maximal Lyapunov exponent. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic systems have been a focal point of renewed interest for many researchers in the past few decades [1–4]. Such nonlinear systems can occur in various natural and man-made systems, and are known to have great sensitivity to initial conditions. Thus, two trajectories starting at arbitrarily nearby initial conditions in such systems could evolve in drastically different fashions, and soon become uncorrelated and unpredictable. In recent years, a new direction of chaos research has emerged to apply fractional order calculus to dynamic systems. Fractional calculus is a 300-year-old topic, but its applications to physics and engineering are just a recent focus of interest. Many systems are known to display fractional-order dynamics [5–7], such as viscoelastic systems, electrodeelectrolyte polarization, and electromagnetic waves. More recently, many investigations are devoted to fractional order dynamical systems [8–13], such as in model-reference adaptive control, PID controller, Chuas system, Brownian motion, Wien-bridge oscillator, and ‘‘jerk’’ model, etc. In Ref. [10], it is shown that the fractional-order Chuas circuit of order as low as 2.7 can produce a chaotic attractor. In Ref. [13], chaotic behaviors of the fractional-order ‘‘jerk’’ model is studied, in which chaotic attractor is generated with the system orders as low as 2.1. And chaos control [14] and
* Corresponding author at: School of Electronic Engineering, University of Electronics Science and Technology of China, Chengdu, Sichuan 610054, PR China. E-mail address: [email protected] (X. Gao).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.090
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synchronization [15] of fractional-order chaotic systems are investigated. However, to our knowledge, chaotic behaviors of the fractional-order autonomous dynamical systems are intensively studied only in the real dynamical systems, but are not studied in relation to the complex domain. In this paper, we numerically investigate chaotic behavior in the nonlinear models of fractional order, it is symmetric or non-symmetric periodically forced complex fractional order Duffings oscillators. By Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0, 1), based on frequency domain arguments, and the resulting equivalent models are studied. Numerical simulations show that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. This paper is organized as follows: the second section below provides a brief review of fractional calculus (FOC). In Section 3, symmetric or non-symmetric periodically forced complex fractional order Duffings oscillators [17] is introduced, and the use of FOC into Duffings equation to obtain periodically forced complex fractional order Duffings equations. In Section 4, some simulation results are presented. We can find that the phase projection of complex Duffings oscillators changes with the increase of orders from 3.6 to integer. In addition, our results are validated by the existence of positive maximal Lyapunov exponent. 2. Review of fractional order calculus and its approximation The idea of fractional integrals and derivatives has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz in 1695. Two commonly used definitions for the general fractional differintegral are the Grunwald–Letnikov (GL) definition and the Riemann–Liouviller (RL) definition [16]. The RL definition is given here Z t dq f ðtÞ 1 f ðsÞ ¼ ds; q < 0; ð1Þ dtq CðqÞ 0 ðt sÞqþ1 where q can have non-integer values, C(Æ) is the Eulers gamma function. Notice in many physical systems, system components are forced into some configuration, or initialized. Fractional-order components, however, require a more complicated initialization, as they have an inherent time-varying memory effect [19]. We will assume that the fractional-order integral is beginning at time t = 0. Likewise the fractional derivative is defined as dq f ðtÞ dn dqn f ðtÞ ; q > 0 and n an integer > q: ð2Þ ¼ dtq dtn dtqn A particularly important desire is that Laplace transforms can still be easily used, as they are standard tools in linear system theory. In this regard, the Laplace transform of the fractional-order differintegral is written as q q1k n1 X d f ðtÞ f ðtÞ q k d L ¼ s Lff ðtÞg s ; for all q; ð3Þ dtq dtq1k t¼0 k¼0 where n is an integer such that n 1 < q < n. Upon considering the initial conditions to be zero, this formula reduces to the more expected and comforting form q d f ðtÞ L ¼ sq Lff ðtÞg: ð4Þ dtq The standard definitions of fractional differintegral do not allow direct implementation of the operator in timedomain simulations of complicated systems with fractional elements. Thus, in order to effectively analyze such systems, it is necessary to develop approximations to the fractional operators using the standard integer order operators. One way to study fractional-order systems is through linear approximations. By utilizing frequency domain techniques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams. In this paper, we use this approach to study the behaviors of the fractional-order versions of our chaotic. The approximation approach taken here is that of [20]. Basically the idea is to approximate the system behavior in the frequency domain, using a specified error in decibels and a bandwidth to generate a continuous sequence of pole-zero pairs for the system with a single fractional power pole. This approximation is created by choosing an initial breakpoint, the line with slop of 20 qdB/decade is approximated by a number of zigzag lines connected together with alternate slops of 0 dB/decade and 20 dB/decade. Thus an approximation of any desired accuracy over any frequency band can be achieved. In Table 1 of Ref. [20], approximations for 1/sq with q = 0.1–0.9 in steps of 0.1 were given with errors of approximately 2 dB. We will mainly use these approximations in the following simulations.
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3. Model of fractional order complex Duffing’s oscillators Duffings oscillator with negative linear stiffness, damping and periodic excitation is often written in the form €x x þ a_x þ x3 ¼ d cos xt:
ð5Þ
Mahmoud et al. [17] extended Eq. (5) to the complex domain in order to study strange attractors, chaotic behavior and the problem of chaos control. The periodically forced complex Duffings oscillators of the form €z z þ a_z þ ezjzj2 ¼ c0 cos xt; ð6Þ pffiffiffi 0 where c ¼ 2c expðip=4Þ, c, a, x are positive parameters, z = x + iy is a complex function. Eq. (6) can be reduced to the famous Duffings oscillator (5) when z = x (y = 0, c 0 is real) and e = 1. We substitute z = x + iy into Eq. (6), and real and imaginary parts are separated, our oscillator is a system of two coupled nonlinear second-order differential equations €x x þ a_x þ exðx2 þ y 2 Þ ¼ c cos xt;
ð7aÞ
€y y þ ay_ þ eyðx2 þ y 2 Þ ¼ c cos xt:
ð7bÞ
Eqs. (7) are completely symmetric and it is clear that if one starts from initial conditions ðx0 ; x_ 0 Þ ¼ bðy 0 ; y_ 0 Þ, where b is a constant, the x and y variables behave identically. It is easy to check that Eqs. (7) (c = 0, and let x1 = x, x2 ¼ x_ , _ possess three fixed points (equilibria) at (0, 0, 0, 0) and (±(1/2e)1/2, 0, ±(1/2e)1/2, 0), since our system is symx3 = y, x4 ¼ y) metric. The origin is a saddle point. The three fixed points shall become attractors in the perturbed system (c 5 0). Estimate the maximal Lyapunov exponent using the Kantz algorithm (k 0.0214308) to prove that Eq. (6) has chaotic behavior (see Fig. 1). Another form of the periodically forced complex Duffings oscillators €z kz þ ezjzj2 ¼ c expðixtÞ:
ð8Þ
Eq. (8) is based on the oscillatory solutions of the cubic nonlinear Schrodinger equation (NSL). The cubic nonlinear Schrodinger equation occurs in nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics [18], where c, x, k are positive parameters, z = x + iy is a complex function. We substitute z = x + iy into Eq. (8), and real and imaginary parts are separated, our oscillator is a system of two coupled nonlinear second-order differential equations €x kx þ exðx2 þ y 2 Þ ¼ c cos xt;
Fig. 1. Time history and phase space of (x1, x2), (x3, x4) for Eqs. (7) (e = 1, x = 1, a = 0.15, c = 0.18).
ð9aÞ
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Fig. 2. Time history and phase space of (x1, x2) for Eqs. (9), (x3, x4) (e = 1.3, x = 1, k = 1, c = 0.001).
€y ky þ eyðx2 þ y 2 Þ ¼ c sin xt:
ð9bÞ
Eqs. (9) are not symmetric. It is easy to check that Eqs. (9) (c = 0, and let x1 = x, x2 ¼ x_ , x3 = y, x4 ¼ y_ ) possess three fixed points (equilibria). The three fixed points shall become attractors in the perturbed system (c 5 0). Estimate the maximal Lyapunov exponent (k 0.0012) to prove that Eq. (8) has chaotic behavior (see Fig. 2). To get the fractional order Duffings equation, (7) can be rewritten as a system of first-order autonomous differential equations, as follows: x_ 1 ¼ x2 ; x_ 2 ¼ x1 ax2 ex1 ðx21 þ x23 Þ þ c cos xt; x_ 3 ¼ x4 ;
ð10Þ
x_ 4 ¼ x3 ax4 ex3 ðx21 þ x23 Þ þ c cos xt: According to the fractional order calculus definitions, the conventional derivative is replaced by a fractional derivative. Eq. (10) can be written as a symmetric periodically forced complex fractional-order Duffings equation, as follows: 8 d q x1 ¼ x2 ; > dtq > > > d p x2 < ¼ x1 ax2 ex1 ðx21 þ x23 Þ þ c cos xt; dtp ð11Þ m d x3 > ¼ x4 ; > dtm > > : d n x4 ¼ x3 ax4 ex3 ðx21 þ x23 Þ þ c cos xt; dtn where oscillator parameter a, e, c, and x are allowed to be varied, and q, p, m, n are the fractional-order. Similarly, (9) can be written as a non-symmetric complex fractional-order Duffings equations, as follows: 8 l d x1 > ¼ x2 ; > dtl > > > < dm xm2 ¼ kx1 ex1 ðx2 þ x2 Þ þ c cos xt; 1 3 dt ð12Þ d n x3 > ¼ x ; > 4 n > dt > > : d 1 x4 ¼ kx3 ex3 ðx21 þ x23 Þ þ c sin xt; dt1 where oscillator parameter k, e, c, and x are allowed to be varied, and l, m, n, 1 are the fractional order. Simulations were performed as follows.
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4. Simulating analyzed We use Matlab to simulate to (11), and let q = p = m = n = 0.9, and the total mathematical system order becomes 3.6 with a = 0.22, x = 1, e = 2.3, c = 0.18. For the phase space of (x1, x2) and (x3, x4), see Fig. 3. The maximal Lyapunov exponent is calculated to prove that complex Duffings equation (11) has chaotic behavior. It is clear that k is positive (k 0.01788295), which shows that the attractor is chaotic. Against, let q = 1, p = m = n = 0.9, we simulate to (11) and the total mathematical system order becomes 3.7 with a = 0.18, x = 1.2, e = 1.8, c = 0.43. The phase space of (x1, x2) and (x3, x4), see Fig. 4. The maximal Lyapunov exponent k is positive (k 0.0241057), which shows that the attractor is chaotic.
Fig. 3. Time history and phase space of (x1, x2) and (x3, x4) (a = 0.22, e = 2.3, x = 1, c = 0.18).
Fig. 4. Time history and phase space for q = 1, p = m = n = 0.9, total mathematical system order 3.7.
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Next, we consider the non-symmetric complex fractional-order Duffings equation (12), and given l = m = n = 1 = 0.9, total mathematical system order is 3.6. Based upon the approximation approach and use Matlab to simulate Eq. (12). For the phase space of (x1, x2) and (x3, x4), see Fig. 5. The maximal Lyapunov exponent is calculated to prove that fractional order complex Duffings equation (12) has chaotic behavior. It is clear that k is positive (k 0.04956385), which shows that the attractor is chaotic. We further consider finding that the phase projection of system (12) changes with the increase of orders from 3.7, 3.8 to integer so as to demonstrate that the continuance of chaotic behaviors exists in the fractional-order periodically forced complex Duffings oscillators. The 3D phase space of (x1, x2, x3) from 3.7 to integer order, see Figs. 6–8.
Fig. 5. Time history and phase space of (x1, x2) and (x3, x4) (k = 2.2, x = 2.4, e = 2.83, c = 0.32, t0 = 1.22).
Fig. 6. 3D phase projection for Eq. (12) (k = 1.98, x = 2.4, e = 2.79, c = 0.24, t0 = 1.22) and total mathematical system order 3.7 (l = 1, m = f = 1 = 0.9).
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Fig. 7. 3D phase projection for Eq. (12) (k = 0.98, x = 2.4, e = 2.86, c = 0.24, t0 = 1.22) and total mathematical system order 3.8 (l = m = 1, f = 1 = 0.9).
Fig. 8. 3D phase projection for Eqs. (9) (e = 1.3, x = 1, k = 1, c = 0.001, t0 = 1.2566) and total mathematical system order 4 (l = m = f = 1 = 1).
5. Conclusions Our main aim in this paper is to study chaotic behavior in the symmetric and non-symmetric periodically forced complex fractional order Duffings oscillators (11) and (12). Numerical simulations show that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. To our knowledge, this paper is the first report of complex domain fractional-order systems. Control and synchronization of chaos in fractional-order complex domain dynamical systems are also interesting topics for future studies.
References [1] Hartley TT. The Duffing double scroll. In: Proc Amer Cont Conf. Pittsburgh, PA; June 1989. p. 419–23. [2] Chiang H-D, Liu C-W et al. Chaos in a simple power system. IEEE Trans Power Syst 1993;8(4):1407–17. [3] Stupnicka WS, Rudowski J. The coexistence of periodic, almost-periodic and chaotic attractors in the Van Der Pol–Duffing oscillator. J Sound Vib 1997;199(2):165–75.
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[4] Ajjarapu V, Lee B. Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system. IEEE Trans Power Syst 1992;7(1):424–31. [5] Koeller RC. Application of fractional calculus to the theory of viscoelasticity. J Appl Mech 1984;51(June):299. [6] Mandelbrot B. Some noises with 1/f spectrum, a bridge between direct current and white noise. IEEE Trans Inform Theory 1967;IT-13(2):289–98. [7] Sun HH et al. Linear approximation of transfer function with a pole of fractional order. IEEE Trans Automat Contr 1984;AC29(5):441–4. [8] Podlubny I. Fractional-order systems and PIkDl-controllers. IEEE Trans Automat Contr 1999;44(1):208–14. [9] Vinagre BM, Petra´sˇ I, Podlubny I, Chen YQ. Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn 2002;29:269–79. [10] Hartley TT, Lorenzo CF, Killory Qammer H. Chaos in a fractional order Chuas system. IEEE Trans Circ Syst I: Fund Theory Appl 1995;42(8):485–90. [11] Perrin E, Harba R, Berzin-Joseph C, Iribarren I, Bonami A. nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans Signal Process 2001;49(5):1049–59. [12] Ahmad W, Ei-Khazali R, Elwakil AS. Fractional order Wien-bridge oscillator. Electron Lett 2001;37(18/30):1110–2. [13] Ahmad WM, Sprott JC. Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 2003;16:339–51. [14] Ahmad WM, Harb AM. On nonlinear control design for autonomous chaotic systems of integer and fractional orders. Chaos, Solitons & Fractals 2003;18:693–701. [15] Li C, Liao X, Yu J. The synchronization of fractional-order systems. Phys Rev E 2003;68:067203. [16] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974. [17] Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffings oscillators. Physica A 2001;292:193–206. [18] If F, Berg P. Split-step spectral method for nonlinear Schrodinger equation with absorbing boundaries. J Comput Phys 1987;72:501–3. [19] Hartley TT, Lorenzo CF. Dynamics and control of initialized fractional-order systems. Nonlinear Dyn 2002;29:201–33. [20] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Autom Control 1992;37(Sept.):1465–70.
Chaos in the fractional order periodically forced complex Duffings oscillators Xin Gao a
a,b,*
, Juebang Yu
a
School of Electronic Engineering, University of Electronics Science and Technology of China, Chengdu, Sichuan 610054, PR China b Institute of Electrical and Information Engineering, Southwest University for Nationalities of China, Chengdu, Sichuan 610041, PR China Accepted 15 September 2004
Abstract The occurrence of fractional-order chaotic dynamics have been intensively studied over the last ten years in a large number of real dynamical systems of physical nature. However, a similar study has not yet been carried out for fractional-order chaotic dynamical systems in the complex domain. In this paper, we numerically study the chaotic behaviors in the fractional-order symmetric and non-symmetric periodically forced complex Duffings oscillators. We find that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. Our results are validated by the existence of positive maximal Lyapunov exponent. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic systems have been a focal point of renewed interest for many researchers in the past few decades [1–4]. Such nonlinear systems can occur in various natural and man-made systems, and are known to have great sensitivity to initial conditions. Thus, two trajectories starting at arbitrarily nearby initial conditions in such systems could evolve in drastically different fashions, and soon become uncorrelated and unpredictable. In recent years, a new direction of chaos research has emerged to apply fractional order calculus to dynamic systems. Fractional calculus is a 300-year-old topic, but its applications to physics and engineering are just a recent focus of interest. Many systems are known to display fractional-order dynamics [5–7], such as viscoelastic systems, electrodeelectrolyte polarization, and electromagnetic waves. More recently, many investigations are devoted to fractional order dynamical systems [8–13], such as in model-reference adaptive control, PID controller, Chuas system, Brownian motion, Wien-bridge oscillator, and ‘‘jerk’’ model, etc. In Ref. [10], it is shown that the fractional-order Chuas circuit of order as low as 2.7 can produce a chaotic attractor. In Ref. [13], chaotic behaviors of the fractional-order ‘‘jerk’’ model is studied, in which chaotic attractor is generated with the system orders as low as 2.1. And chaos control [14] and
* Corresponding author at: School of Electronic Engineering, University of Electronics Science and Technology of China, Chengdu, Sichuan 610054, PR China. E-mail address: [email protected] (X. Gao).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.090
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synchronization [15] of fractional-order chaotic systems are investigated. However, to our knowledge, chaotic behaviors of the fractional-order autonomous dynamical systems are intensively studied only in the real dynamical systems, but are not studied in relation to the complex domain. In this paper, we numerically investigate chaotic behavior in the nonlinear models of fractional order, it is symmetric or non-symmetric periodically forced complex fractional order Duffings oscillators. By Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0, 1), based on frequency domain arguments, and the resulting equivalent models are studied. Numerical simulations show that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. This paper is organized as follows: the second section below provides a brief review of fractional calculus (FOC). In Section 3, symmetric or non-symmetric periodically forced complex fractional order Duffings oscillators [17] is introduced, and the use of FOC into Duffings equation to obtain periodically forced complex fractional order Duffings equations. In Section 4, some simulation results are presented. We can find that the phase projection of complex Duffings oscillators changes with the increase of orders from 3.6 to integer. In addition, our results are validated by the existence of positive maximal Lyapunov exponent. 2. Review of fractional order calculus and its approximation The idea of fractional integrals and derivatives has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz in 1695. Two commonly used definitions for the general fractional differintegral are the Grunwald–Letnikov (GL) definition and the Riemann–Liouviller (RL) definition [16]. The RL definition is given here Z t dq f ðtÞ 1 f ðsÞ ¼ ds; q < 0; ð1Þ dtq CðqÞ 0 ðt sÞqþ1 where q can have non-integer values, C(Æ) is the Eulers gamma function. Notice in many physical systems, system components are forced into some configuration, or initialized. Fractional-order components, however, require a more complicated initialization, as they have an inherent time-varying memory effect [19]. We will assume that the fractional-order integral is beginning at time t = 0. Likewise the fractional derivative is defined as dq f ðtÞ dn dqn f ðtÞ ; q > 0 and n an integer > q: ð2Þ ¼ dtq dtn dtqn A particularly important desire is that Laplace transforms can still be easily used, as they are standard tools in linear system theory. In this regard, the Laplace transform of the fractional-order differintegral is written as q q1k n1 X d f ðtÞ f ðtÞ q k d L ¼ s Lff ðtÞg s ; for all q; ð3Þ dtq dtq1k t¼0 k¼0 where n is an integer such that n 1 < q < n. Upon considering the initial conditions to be zero, this formula reduces to the more expected and comforting form q d f ðtÞ L ¼ sq Lff ðtÞg: ð4Þ dtq The standard definitions of fractional differintegral do not allow direct implementation of the operator in timedomain simulations of complicated systems with fractional elements. Thus, in order to effectively analyze such systems, it is necessary to develop approximations to the fractional operators using the standard integer order operators. One way to study fractional-order systems is through linear approximations. By utilizing frequency domain techniques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams. In this paper, we use this approach to study the behaviors of the fractional-order versions of our chaotic. The approximation approach taken here is that of [20]. Basically the idea is to approximate the system behavior in the frequency domain, using a specified error in decibels and a bandwidth to generate a continuous sequence of pole-zero pairs for the system with a single fractional power pole. This approximation is created by choosing an initial breakpoint, the line with slop of 20 qdB/decade is approximated by a number of zigzag lines connected together with alternate slops of 0 dB/decade and 20 dB/decade. Thus an approximation of any desired accuracy over any frequency band can be achieved. In Table 1 of Ref. [20], approximations for 1/sq with q = 0.1–0.9 in steps of 0.1 were given with errors of approximately 2 dB. We will mainly use these approximations in the following simulations.
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3. Model of fractional order complex Duffing’s oscillators Duffings oscillator with negative linear stiffness, damping and periodic excitation is often written in the form €x x þ a_x þ x3 ¼ d cos xt:
ð5Þ
Mahmoud et al. [17] extended Eq. (5) to the complex domain in order to study strange attractors, chaotic behavior and the problem of chaos control. The periodically forced complex Duffings oscillators of the form €z z þ a_z þ ezjzj2 ¼ c0 cos xt; ð6Þ pffiffiffi 0 where c ¼ 2c expðip=4Þ, c, a, x are positive parameters, z = x + iy is a complex function. Eq. (6) can be reduced to the famous Duffings oscillator (5) when z = x (y = 0, c 0 is real) and e = 1. We substitute z = x + iy into Eq. (6), and real and imaginary parts are separated, our oscillator is a system of two coupled nonlinear second-order differential equations €x x þ a_x þ exðx2 þ y 2 Þ ¼ c cos xt;
ð7aÞ
€y y þ ay_ þ eyðx2 þ y 2 Þ ¼ c cos xt:
ð7bÞ
Eqs. (7) are completely symmetric and it is clear that if one starts from initial conditions ðx0 ; x_ 0 Þ ¼ bðy 0 ; y_ 0 Þ, where b is a constant, the x and y variables behave identically. It is easy to check that Eqs. (7) (c = 0, and let x1 = x, x2 ¼ x_ , _ possess three fixed points (equilibria) at (0, 0, 0, 0) and (±(1/2e)1/2, 0, ±(1/2e)1/2, 0), since our system is symx3 = y, x4 ¼ y) metric. The origin is a saddle point. The three fixed points shall become attractors in the perturbed system (c 5 0). Estimate the maximal Lyapunov exponent using the Kantz algorithm (k 0.0214308) to prove that Eq. (6) has chaotic behavior (see Fig. 1). Another form of the periodically forced complex Duffings oscillators €z kz þ ezjzj2 ¼ c expðixtÞ:
ð8Þ
Eq. (8) is based on the oscillatory solutions of the cubic nonlinear Schrodinger equation (NSL). The cubic nonlinear Schrodinger equation occurs in nonlinear optics, deep-water wave theory, plasma physics and bimolecular dynamics [18], where c, x, k are positive parameters, z = x + iy is a complex function. We substitute z = x + iy into Eq. (8), and real and imaginary parts are separated, our oscillator is a system of two coupled nonlinear second-order differential equations €x kx þ exðx2 þ y 2 Þ ¼ c cos xt;
Fig. 1. Time history and phase space of (x1, x2), (x3, x4) for Eqs. (7) (e = 1, x = 1, a = 0.15, c = 0.18).
ð9aÞ
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Fig. 2. Time history and phase space of (x1, x2) for Eqs. (9), (x3, x4) (e = 1.3, x = 1, k = 1, c = 0.001).
€y ky þ eyðx2 þ y 2 Þ ¼ c sin xt:
ð9bÞ
Eqs. (9) are not symmetric. It is easy to check that Eqs. (9) (c = 0, and let x1 = x, x2 ¼ x_ , x3 = y, x4 ¼ y_ ) possess three fixed points (equilibria). The three fixed points shall become attractors in the perturbed system (c 5 0). Estimate the maximal Lyapunov exponent (k 0.0012) to prove that Eq. (8) has chaotic behavior (see Fig. 2). To get the fractional order Duffings equation, (7) can be rewritten as a system of first-order autonomous differential equations, as follows: x_ 1 ¼ x2 ; x_ 2 ¼ x1 ax2 ex1 ðx21 þ x23 Þ þ c cos xt; x_ 3 ¼ x4 ;
ð10Þ
x_ 4 ¼ x3 ax4 ex3 ðx21 þ x23 Þ þ c cos xt: According to the fractional order calculus definitions, the conventional derivative is replaced by a fractional derivative. Eq. (10) can be written as a symmetric periodically forced complex fractional-order Duffings equation, as follows: 8 d q x1 ¼ x2 ; > dtq > > > d p x2 < ¼ x1 ax2 ex1 ðx21 þ x23 Þ þ c cos xt; dtp ð11Þ m d x3 > ¼ x4 ; > dtm > > : d n x4 ¼ x3 ax4 ex3 ðx21 þ x23 Þ þ c cos xt; dtn where oscillator parameter a, e, c, and x are allowed to be varied, and q, p, m, n are the fractional-order. Similarly, (9) can be written as a non-symmetric complex fractional-order Duffings equations, as follows: 8 l d x1 > ¼ x2 ; > dtl > > > < dm xm2 ¼ kx1 ex1 ðx2 þ x2 Þ þ c cos xt; 1 3 dt ð12Þ d n x3 > ¼ x ; > 4 n > dt > > : d 1 x4 ¼ kx3 ex3 ðx21 þ x23 Þ þ c sin xt; dt1 where oscillator parameter k, e, c, and x are allowed to be varied, and l, m, n, 1 are the fractional order. Simulations were performed as follows.
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4. Simulating analyzed We use Matlab to simulate to (11), and let q = p = m = n = 0.9, and the total mathematical system order becomes 3.6 with a = 0.22, x = 1, e = 2.3, c = 0.18. For the phase space of (x1, x2) and (x3, x4), see Fig. 3. The maximal Lyapunov exponent is calculated to prove that complex Duffings equation (11) has chaotic behavior. It is clear that k is positive (k 0.01788295), which shows that the attractor is chaotic. Against, let q = 1, p = m = n = 0.9, we simulate to (11) and the total mathematical system order becomes 3.7 with a = 0.18, x = 1.2, e = 1.8, c = 0.43. The phase space of (x1, x2) and (x3, x4), see Fig. 4. The maximal Lyapunov exponent k is positive (k 0.0241057), which shows that the attractor is chaotic.
Fig. 3. Time history and phase space of (x1, x2) and (x3, x4) (a = 0.22, e = 2.3, x = 1, c = 0.18).
Fig. 4. Time history and phase space for q = 1, p = m = n = 0.9, total mathematical system order 3.7.
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Next, we consider the non-symmetric complex fractional-order Duffings equation (12), and given l = m = n = 1 = 0.9, total mathematical system order is 3.6. Based upon the approximation approach and use Matlab to simulate Eq. (12). For the phase space of (x1, x2) and (x3, x4), see Fig. 5. The maximal Lyapunov exponent is calculated to prove that fractional order complex Duffings equation (12) has chaotic behavior. It is clear that k is positive (k 0.04956385), which shows that the attractor is chaotic. We further consider finding that the phase projection of system (12) changes with the increase of orders from 3.7, 3.8 to integer so as to demonstrate that the continuance of chaotic behaviors exists in the fractional-order periodically forced complex Duffings oscillators. The 3D phase space of (x1, x2, x3) from 3.7 to integer order, see Figs. 6–8.
Fig. 5. Time history and phase space of (x1, x2) and (x3, x4) (k = 2.2, x = 2.4, e = 2.83, c = 0.32, t0 = 1.22).
Fig. 6. 3D phase projection for Eq. (12) (k = 1.98, x = 2.4, e = 2.79, c = 0.24, t0 = 1.22) and total mathematical system order 3.7 (l = 1, m = f = 1 = 0.9).
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Fig. 7. 3D phase projection for Eq. (12) (k = 0.98, x = 2.4, e = 2.86, c = 0.24, t0 = 1.22) and total mathematical system order 3.8 (l = m = 1, f = 1 = 0.9).
Fig. 8. 3D phase projection for Eqs. (9) (e = 1.3, x = 1, k = 1, c = 0.001, t0 = 1.2566) and total mathematical system order 4 (l = m = f = 1 = 1).
5. Conclusions Our main aim in this paper is to study chaotic behavior in the symmetric and non-symmetric periodically forced complex fractional order Duffings oscillators (11) and (12). Numerical simulations show that chaotic behaviors exist in the fractional-order periodically forced complex Duffings oscillators with orders less than 4. To our knowledge, this paper is the first report of complex domain fractional-order systems. Control and synchronization of chaos in fractional-order complex domain dynamical systems are also interesting topics for future studies.
References [1] Hartley TT. The Duffing double scroll. In: Proc Amer Cont Conf. Pittsburgh, PA; June 1989. p. 419–23. [2] Chiang H-D, Liu C-W et al. Chaos in a simple power system. IEEE Trans Power Syst 1993;8(4):1407–17. [3] Stupnicka WS, Rudowski J. The coexistence of periodic, almost-periodic and chaotic attractors in the Van Der Pol–Duffing oscillator. J Sound Vib 1997;199(2):165–75.
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[4] Ajjarapu V, Lee B. Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system. IEEE Trans Power Syst 1992;7(1):424–31. [5] Koeller RC. Application of fractional calculus to the theory of viscoelasticity. J Appl Mech 1984;51(June):299. [6] Mandelbrot B. Some noises with 1/f spectrum, a bridge between direct current and white noise. IEEE Trans Inform Theory 1967;IT-13(2):289–98. [7] Sun HH et al. Linear approximation of transfer function with a pole of fractional order. IEEE Trans Automat Contr 1984;AC29(5):441–4. [8] Podlubny I. Fractional-order systems and PIkDl-controllers. IEEE Trans Automat Contr 1999;44(1):208–14. [9] Vinagre BM, Petra´sˇ I, Podlubny I, Chen YQ. Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn 2002;29:269–79. [10] Hartley TT, Lorenzo CF, Killory Qammer H. Chaos in a fractional order Chuas system. IEEE Trans Circ Syst I: Fund Theory Appl 1995;42(8):485–90. [11] Perrin E, Harba R, Berzin-Joseph C, Iribarren I, Bonami A. nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Trans Signal Process 2001;49(5):1049–59. [12] Ahmad W, Ei-Khazali R, Elwakil AS. Fractional order Wien-bridge oscillator. Electron Lett 2001;37(18/30):1110–2. [13] Ahmad WM, Sprott JC. Chaos in fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 2003;16:339–51. [14] Ahmad WM, Harb AM. On nonlinear control design for autonomous chaotic systems of integer and fractional orders. Chaos, Solitons & Fractals 2003;18:693–701. [15] Li C, Liao X, Yu J. The synchronization of fractional-order systems. Phys Rev E 2003;68:067203. [16] Oldham KB, Spanier J. The fractional calculus. New York: Academic Press; 1974. [17] Mahmoud GM, Mohamed AA, Aly SA. Strange attractors and chaos control in periodically forced complex Duffings oscillators. Physica A 2001;292:193–206. [18] If F, Berg P. Split-step spectral method for nonlinear Schrodinger equation with absorbing boundaries. J Comput Phys 1987;72:501–3. [19] Hartley TT, Lorenzo CF. Dynamics and control of initialized fractional-order systems. Nonlinear Dyn 2002;29:201–33. [20] Charef A, Sun HH, Tsao YY, Onaral B. Fractal system as represented by singularity function. IEEE Trans Autom Control 1992;37(Sept.):1465–70.