This paper presents a novel 3D fractional-ordered chaotic system. The dynamical behavior of this system is investigated. An analog circuit diagram is designed for generating strange attractors. Results have been observed using Electronic Workbench Multisim software, they demonstrate that the fractional-ordered nonlinear chaotic attractors exist in this new system. Moreover, they agree very well with those obtained by numerical simulations. Keywords Circuit design · Chaotic system · Fractional derivative · Stability Introduction be applied to secure communication and control processing, e.g., the transmitted signals can be masked by chaotic signals Recently, the study of fractional calculus have become a in secure communications and the image messages can be focus of interest [1–12]. Because the applications of frac- covered by chaotic signals in image encryption. In addition, tional calculus were found in many scientiﬁc ﬁelds, such the circuit implementation can verify the chaotic character- as rheology, diffusive transport, electrical networks, elec- istics of the chaotic systems physically, provide support for tromagnetic theory, quantum evolution of complex systems, the application of chaos, and promote their technological colored noise, etc. Compared with the classical well-known application in the future. Therefore, the circuit implemen- models, it was found that fractional derivatives provide a bet- tation of the chaotic systems has also attracted more and ter tool for modeling memory and heredity properties of var- more attention for engineering applications. Especially, for ious phenomena. Various types of fractional derivatives and those fractional-order attractors, the circuit implementations their applications can be found in the literature, for instance, for them are more important [24–30]. the Caputo derivative , the recently introduced fractional In this work, we construct a new 3D fractional-order derivative without singular kernel (Caputo–Fabrizio deriva- chaotic system. Through studying its dynamical behavior tive)  and the Atangana–Baleanu derivative which is by numerical simulation based on the improved Adams– based upon the well-known generalized Mittag–Lefﬂer func- Bashforth–Moulton method  and designs chain ship tion [15,16]. fractional-order chaotic circuit based on frequency-domain Besides, many scientists and engineers have been attracted approximation method . Besides, we realize the fractional- to the theory of chaos since the discovery of the Lorenz attrac- order chaotic system through Multisim software 13.0 circuit tor . It was found that fractional-order chaos has useful simulation platform. application in many ﬁeld of science like engineering, physics, mathematical biology, psychological, and life sciences [18– 23]. On the other hand, chaotic signal is a key issue for future Preliminaries applications of chaos-based information systems, and can In what follows, Caputo derivatives are considered, taking the advantage that this allows for traditional initial and boundary B Z. Hammouch Hammouch.email@example.com conditions to be included in the formulation of the considered problem. T. Mekkaoui touﬁk_mekkaoui@yahoo.fr Deﬁnition 1 A real function f (x ), x > 0, is said to be in the E3MI, FSTE Moulay Ismail University, Errachidia, Morocco space C ,μ ∈ R if there exits a real number λ>μ, such 123 252 Complex & Intelligent Systems (2018) 4:251–260 that f (x ) = x g(x ), where g(x ) ∈ C [0, ∞) and it is said to m (m) be in the space C if and only if f ∈ C for m ∈ IN. Deﬁnition 2 The Riemann–Liouville fractional integral oper- ator of order α of a real function f (x ) ∈ C ,μ ≥−1, is deﬁned as α α−1 J f (x ) = (x − t ) f (t )dt , Γ(α) α> 0, x > 0 and J f (x ) = f (x ). (1) The operators J has some properties, for α, β ≥ 0 and ξ ≥−1: α β α+β – J J f (x ) = J f (x ), α β β α – J J f (x ) = J J f (x ), Fig. 1 Stability region of the fractional-order system (3) Γ(ξ +1) α ξ α+ξ – J x = x . Γ(α+ξ +1) Circuit implementation and numerical Deﬁnition 3 The Caputo fractional derivative D of a func- simulations tion f (x ) of any real number α such that m − 1 <α ≤ m, m α m ∈ IN,for x > 0 and f ∈ C in the terms of J is −1 Adams–Bashforth (PECE) algorithm α m−α m D f (x ) = J D f (x ) x We recall here the improved version of Adams–Bashforth– m−α−1 (m) = (x − t ) f (t )dt (2) Moulton algorithm [31,34] for the fractional-order systems. Γ(m − α) Consider the fractional-order initial value problem: and has the following properties for m −1 <α ≤ m, m ∈ IN, m D x = f (x (t )) 0 ≤ t ≤ T , μ ≥−1 and f ∈ C : t (5) (k) (k) α α x (0) = x , k = 0, 1,... , m − 1. – D J f (x ) = f (x ), m−1 α α (k) + It is equivalent to the Volterra integral equation: – J D f (x ) = f (x ) − f (0 ) , for x > 0, k! k=0 [α]−1 k t t 1 (k) α−1 x (t ) = x + (t − s) f (s, x (s))ds. (6) Stability criterion k! Γ(α) k=0 To investigate the dynamics and to control the chaotic behav- Diethelm et al. have given a predictor–corrector scheme (see ior of a fractional-order dynamic system: ), based on the Adams–Bashforth–Moulton algorithm to integrate Eq. (6). By applying this scheme to the fractional- D x (t ) = f (x (t )), (3) order system (5), and setting we need the following indispensable stability theorem (Fig. h = , t = nh, n = 0, 1,..., N . 1). Theorem 1 (See [32,33]) For a given commensurate fracti- Equation (6) can be discretized as follows: onal-order system (3), the equilibria can be obtained by [α]−1 calculating f (x ) = 0. These equilibrium points are locally k α t h (k) p asymptotically stable if all the eigenvalues λ of the Jacobian x (t ) = x + f (t , x (t )) h n+1 n+1 n+1 0 h k! Γ(α + 2) ∂ f k=0 matrix J = at the equilibrium points satisfy ∂ x α + a f (t , x (t )), (7) j ,n+1 j h j π Γ(α + 2) j =0 |arg(λ)| > α. (4) where 123 Complex & Intelligent Systems (2018) 4:251–260 253 1 1 Fig. 2 Chain ship unit of: a and b 0.98 0.9 s s Table 1 Equilibrium points and corresponding eigenvalues Equilibrium Eigenvalues points E (0, 0, 0)λ = 3,λ =−7,λ = 0 1 2 3 −2 E (−0.923250, 1.35886, −0.889584)λ =−5.478102, 1 1 λ = 0.418480 ± 5.245549I 2,3 α+1 α n − (n − α)(n + 1) , j = 0, α+1 α+1 α+1 a = (n − j +2) +(n − j ) −2(n − j + 1) , 1 ≤ j ≤ n j ,n+1 1, j = n + 1, (8) and the predictor is given by [α]−1 n t 1 (k) x (t ) = x + b f (t , x (t )), n+1 j ,n+1 j h j h 0 k! Γ(α) Fig. 3 Chaotic attractors of the fractional-order system (16) obtained k=0 j =0 by numerical simulations: a x − y, b y − z, c x − z,for α = 0.98 (9) α α time-domain simulations. To study such systems, it is nec- where b = ((n + 1) − j ) − (n − j ) . j ,n+1 essary to develop approximations to the fractional operators The error estimate of the above scheme is using the standard integer order operators. According to cir- cuit theory, the approximation formulation of α,from0.1to max |x (t ) − x (t )| = O(h ), j =0,1,...,N j h j 0.9, in reference , bode plot approximation chart, can be realized by the complex-frequency domain of the chain ship in which p = min(2, 1 + α). equivalent circuit. When α = 0.98, it can be worked out that the approximation formula of is The fractional frequency-domain approximation 0.98 The standard deﬁnition of fractional differintegral does not 1 1.2974(s + 1125) = . (10) allow the direct implementation of the fractional operators in 0.98 s (s + 1423)(s + 0.01125) 123 254 Complex & Intelligent Systems (2018) 4:251–260 Fig. 4 Asymptotically stable orbits of the fractional-order system (16) by numerical simulations: a x − y, b x − z, c y − z,for α = 0.9 Fig. 5 Time series of the fractional-order system (16) by numerical simulations: a x, b y, c z for α = 0.9 In formula (10), s = j ω, its complex frequency and the chain ship circuit unit is described in Fig. 2a. The transfer R = 91.1873 MΩ, R = 190.933 ω, 1 2 function between A and B can be obtained as follows: C = 975.32 nF, and C = 3.6806 µF. (12) 1 2 R R 1 2 H (s) = + 0.98 Similarly, for α = 0.9, we can reach that the approxima- sR C + 1 sR C + 1 1 1 2 2 1 1 1 R R C C tion formula of is 0 0 1 2 + s + 0.9 C C C +C 1 2 1 2 = . (11) 1 1 s + s + R C R C 1 2.2675(s + 1.292)(s + 215.4) 1 1 2 2 = . (13) 0.9 s (s + 0.01292)(s + 2.154)(s + 359.4) Taking C = 1νF. Since H (s)C = , we can reach 0 0 0.98 123 Complex & Intelligent Systems (2018) 4:251–260 255 Fig. 6 Circuit diagram for the realization of the fractional-order chaotic system (16)for α = 0.98 123 256 Complex & Intelligent Systems (2018) 4:251–260 Fig. 7 Circuit diagram for the realization of the fractional-order chaotic system (16)for α = 0.9 123 Complex & Intelligent Systems (2018) 4:251–260 257 Fig. 9 Circuit simulation asymptotically stable orbits of the fractional- order system (16) observed by the oscilloscope 1V/Div: a x − y, b x − z, Fig. 8 Chaotic attractors of the fractional-order system (16) observed c y − z,for α = 0.9 by the oscilloscope 1V/Div: a x − y, b y − z, c x − z with α = 0.98 A new 3D fractional-order chaotic system The chain ship circuit unit for this case is shown in Fig. 2b. The transfer function between A and B is We introduce the following system: 1 1 1 C C C 1 2 3 H (s) = + + , (14) 0.9 ⎧ 1 1 1 α 2 s + s + s + D x =−2x − y , R C R C R C 1 1 2 2 3 3 α 2 (16) D y =−4xz + 3y − z , we can reach D z = 4xy − 7z + yz, R = 62.84 MΩ, R = 250 kΩ, R = 2.5kΩ, 1 2 3 C = 1.23 µF, C = 1.83 µF, and C = 1.1 µF. (15) 1 2 3 where the fractional-order α ∈ (0, 1]. 123 258 Complex & Intelligent Systems (2018) 4:251–260 Fig. 10 Time series of the fractional-order system (16) observed by the oscilloscope 1V/Div: a x, b y, c z,for α = 0.9 123 Complex & Intelligent Systems (2018) 4:251–260 259 Dynamical analysis The derived results between numerical simulation and circuit experimental simulation are in agreement with each other. To reveal dynamical properties of the nonlinear system (16), the equilibria should be considered at ﬁrst Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, −2x − y = 0, ⎪ and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative −4xz + 3y − z = 0, (17) Commons license, and indicate if changes were made. 4xy − 7z + yz = 0. The obtained equilibrium points from (17) and the corre- References sponding eigenvalues are given in Table 1. Hence, E is unstable, and E is a saddle point of index 0 1 1. Caponetto R, Dongola G, Fortuna L (2010) Fractional order 2. With the aid of Theorem 1, a necessary condition for the systems: modeling and control application. World Scientiﬁc, Sin- fractional-order systems (16) to remain chaotic is keeping at gapore least one eigenvalue λ in the unstable region, i.e., |arg(λ )| > 2. Miller KS, Rosso B (1993) An introduction to the fractional cal- i i απ culus and fractional differential equations. Wiley, New York , It means that when α> 0.949318 system (16) exhibits 3. Oldham KB, Spanier J (1974) The fractional calculus. Academic a chaotic behavior. Press, New York 4. Petras I (2011) Fractional-order nonlinear systems: modeling, anal- ysis and simulation. Springer, Berlin Circuit designs and numerical simulations 5. Podlubny I (1999) Fractional differential equations: mathematics in science and engineering. Academic Press, New York 6. Belgacem FBM et al (2017) New and extended applications of Applying the improved version of Adams–Bashforth–Moulton the natural and Sumudu transforms: fractional diffusion and stokes numerical algorithm described above with a step size h = ﬂuid ﬂow realms. Chapter no. 6 in book: Advances in real and 0.01, system (16) can be discretized. It is found that chaos complex analysis with applications, pp 107–120. Springer Link https://link.springer.com/chapter/10.1007/978-981-10-4337-6-6 exists in the fractional-order system (16) when α> 0.94 7. Hammouch Z, Mekkaoui T, Belgacem FB (2014) Numerical sim- with the initial condition (x , y , z ) = (0.7, 0.1, 0).Fig- 0 0 0 ulations for a variable order fractional Schnakenberg model. AIP ure 3a–c demonstrate that the systems has chaotic behavior Conf Proc 1637(1):1450–1455 (AIP) for α = 0.98. On the other hand, when we take some val- 8. Singh J et al (2018) A fractional epidemiological model for com- puter viruses pertaining to a new fractional derivative. Appl Math ues of α ≤ 0.94, the fractional system (16) can display Comput 316:504–515 the periodic attractors, and asymptotically stable orbits (see 9. Hammouch Z, Mekkaoui T (2012) Travelling-wave solutions for Figs. 4, 5). Moreover, using Multisim software 13 to con- some fractional partial differential equation by means of general- duct simulations on the 3D fractional-order system (16), ized trigonometry functions. Int J Appl Math Res 1(2):206–212 10. Hammouch Z, Mekkaoui T (2014) Traveling-wave solutions of the analog circuits are designed to realize the behavior of (16). generalized Zakharov equation with time-space fractional deriva- Three state variables x, y and z are implemented by three tives. Math Eng Sci Aerosp MESA 5(4):1–11 channels, respectively. The implementations use resistors, 11. Touﬁk M, Atangana A (2017) New numerical approximation of capacitors, analog multipliers, and analog operational ampli- fractional derivative with non-local and non-singular kernel: appli- cation to chaotic models. Eur Phys J Plus 132(10):444 ﬁers, as shown in Figs. 6 and 7. A comparison of Figs. 3, 4, 5, 12. Hammouch Z, Mekkaoui T (2015) Control of a new chaotic 6, 7, and 8 (resp. 4–9 and 5–10) proves that analog circuit for fractional-order system using Mittag-Lefﬂer stability. Nonlinear system (16) is well coincident with numerical simulations. Stud 22(4):565–577 A conclusion can be made that the chaotic and non-chaotic 13. Caputo M (1967) Linear models of dissipation whose Q is almost frequency independent. J R Astral Soc 13:529–539 behaviors exist in the fractional-order system (16), which 14. Caputo M, Fabrizio M (2015) A new deﬁnition of fractional deriva- veriﬁes its existence and validity (Figs. 9, 10). tive without singular kernel. Prog Fract Differ Appl 1(2):1–13 15. Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel, theory and application to heat transfer model. Therm Sci 20(2):763–769 Conclusion 16. Atangana A, Koca I (2016) Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos Solitons Fractals 89:447–454 In this paper, we introduce a new three-dimensional fractional- 17. Lorenz EN (1963) Deterministic nonperiodic ﬂow. J Atmos Sci order chaotic system and its existence and stability. By 20:130–141 adopting a chain ship circuit form , the circuit experimen- 18. Grigorenko I, Grigorenko E (2003) Chaotic dynamics of the frac- tal simulation of this fractional-order system is presented. tional Lorenz system. Phys Rev Lett 91(3):034–101 123 260 Complex & Intelligent Systems (2018) 4:251–260 19. Mekkaoui T et al (2015) Fractional-order nonlinear systems: 29. Ma T, Guo D Circuit simulation and implementation for synchro- chaotic dynamics, numerical simulation and circuit design. Fract nization. http://www.paper.edu.cn Dyn 343 30. Liu CX (2011) Fractional-order chaotic circuit theory and applica- 20. Hammouch Z, Mekkaoui T (2014) Chaos synchronization of a tions. Xian Jiaotong University Press, Xian fractional nonautonomous system. Nonauton Dyn Syst 1:6171 31. Diethelm K, Ford N (2002) Analysis of fractional differential equa- 21. Jun-Guo L (2005) Chaotic dynamics and synchronization of tions. J Math Anal Appl 265:229–248 fractional-order GenesioTesi systems. Chin Phys 14(8):1517 32. Matignon D (1996) Stability results for fractional differential equa- 22. Li C, Chen G (2004) Chaos and hyperchaos in the fractional-order tions with applications to control processing. In: Proceedings of Rossler equations. Physica A 341:55–61 computational engineering in systems applications, pp 963–968 23. Baskonus HM et al (2015) Active control of a chaotic fractional 33. Tavazoei M, Haeri M (2007) A necessary condition for double order economic system. Entropy 17(8):5771–5783 scroll attractor existence in fractional-order systems. Phys Lett A 24. Baskonus HM et al (2016) Chaos in the fractional order logistic 367(1):102–113 delay system: circuit realization and synchronization. AIP Conf 34. Diethelm K, Ford N, Freed A, Luchko Y (2005) Algorithms for Proc 1738(1):290005 (AIP Publishing) the fractional calculus: a selection of numerical method. Comput 25. Zhou P, Huang K (2014) A new 4-D non-equilibrium fractional- Methods Appl Mech Eng 94:743–773 order chaotic system and its circuit implementation. Commun Nonlinear Sci Numer Simul 19:2005–2011 26. Kammara AC, Palanichamy L, Knig A (2016) Multi-objective opti- Publisher’s Note Springer Nature remains neutral with regard to juris- mization and visualization for analog design automation. Complex dictional claims in published maps and institutional afﬁliations. Intell Syst 2.4:251–267 27. Hartley TT, Lorenzo CF, Killory HQ (1995) Chaos in a fractional order Chua’s system. IEEE Trans Circuits Syst I Fundam Theory Appl 42(8):485–490 28. Zhang X, Sun Q, Cheng P (2014) Design of a four-wing hetero- geneous fractional-order chaotic system and its circuit simulation. Int J Smart Design Intell Syst 7
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Published: May 29, 2018
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