Bifurcation analysis and chaos control of periodically driven discrete fractional order memristive Duffing Oscillator
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Bifurcation analysis and chaos control ofperiodically driven discrete fractional ordermemristive Duffing Oscillator
Samuel Ogunjo , a and Ibiyinka Fuwape , Department of Physics, Federal University of Technology, Akure, Ondo State, Nigeria. Michael and Cecilia Ibru University, Ughelli, Delta State, Nigeria.
Abstract.
Discrete fractional order chaotic systems extends the mem-ory capability to capture the discrete nature of physical systems. Inthis research, the memristive discrete fractional order chaotic system isintroduced. The dynamics of the system was studied using bifurcationdiagrams and phase space construction. The system was found chaoticwith fractional order 0 . < n < . The study of chaos arose from the study of integer order systems [13]. This wassoon extended to discrete systems to explain the behaviour in natural systems andmodels especially population models [7]. Many methods used in integer order anddiscrete systems have been developed over the years to study observational datasuch as climate [9], economic and financial data [11] and communication systems [8].Although it has the capability to capture memory effect in models, fractional ordersystems have not gained attention over the years due to inadequate solution technique[7]. New developments in recent times such as computation methods and computingpowers have brought attention to fractional order systems [20].There exist situations where it is desired that systems be chaotic. This can beachieved through chaotification or anti-control [18]. At other times when the exis-tence of chaos in a system is undesirable, the methods of chaos control can be usedto remove the effect of chaos in such system. Chaos control involves obtaining achaotic, periodic or stationary behaviour in a chaotic system by the application oftiny perturbations or controllers [7,4]. Synchronization is a form of control wherebythe trajectory of a chaotic system is made to track the trajectory of another sys-tem. Different forms of synchronization such as projective synchronization, function a Corresponding author: e-mail: [email protected] b e-mail: [email protected]; [email protected] Will be inserted by the editor projective synchronization, phase synchronization, lag synchronization, complete syn-chronization, anticipated synchronization have been proposed [16]. Synchronizationof chaotic systems has been found useful in secure communication [17,14].The practical application of chaos to secure communication involves building elec-tronic circuits for its implementation. Chua [5] proposed a nonlinear device, the mem-ristor, as the fourth circuit element in addition to the resistor, capacitor and inductor.The memristor is considered as a nonlinear resistor with memory [6]. Novel chaoticsystems which have been developed based on the memristor include the two compo-nent circuit [22], memristor biomembranes [23], memristive oscillatory systems [21],and Muthuswamy and Chua system [15]. Memristors have practical potential applica-tions in high performance computing, dynamic memory elements and neural synapses[19].Discrete fractional order implementation of systems have been investigated includ-ing: Extension of integer order chaotic systems have been carried for systems suchas sine and standard maps [25], Logistic map [24], Chua system [2] and Henon map[10,12]. In this paper, we extend the study of the Duffing oscillators to the discretefractional order model. By introducing a memristor into the system, the differentresponses of the system to varying parameters were investigated using bifurcationdiagrams. Control of the memristive discrete fractional order model was also carriedout using the method of active control.
Different forms of flux dependent rate of change of charge have been proposed. In thiswork, we study a flux controlled memristor of the form φ ( q ) = ω q + βq (1)The memductance is obtained as M ( q ) = dφ ( q ) dq = ω + 3 βq (2)where ω and β are constants. The fractional sum of order ν < ∆ − νa x ( t ) := 1 Γ ( ν ) t − ν X s = a ( t − σ ( s )) ( ν − x ( s ) , t ∈ { a, a + 1 , a + 2 , . . . } (3) σ ( s ) = s + 1, and t ( ν ) is the falling function defined as t ( ν ) = Γ ( t + 1) Γ ( t + 1 − ν ) (4)This definition was extended by [1] for the ν -order ( ν > , ν N ) Caputo-likedelta difference as ∆ νC x ( t ) := 1 Γ ( n − ν ) t − ( n − ν ) X s = a ( t − σ ( s )) ( n − ν − ∆ ns x ( s ) (5) ill be inserted by the editor 3 where n = [ ν ] + 1.Using this transformation, a discrete fractional system can be written as [7] ∆ νa x ( t ) = f ( t + v − , x ( t + ν −
1) (6)This has a solution given by x ( t ) = x + 1 Γ ( a ) t − ν X s =1 − ν ( t − s − ( ν − f ( s + ν − , x ( s + v −
1) (7)
Duffing oscillator is defined as m ¨ x + c ˙ x + kx + ax = f sin( ̟t ) (8)where m , c , k , and a are system constants. Substituting Equation 2 and rewriting asa two dimensional system, the integer order Duffing oscillator can be written as˙ x = y ˙ y = f sin( ̟t ) − M ( q ) x − f y (9)where f = fm , 3 β = am , ω = km , M ( q ) = ω + 3 βx and f = cm . System 9 can bewritten in fractional order form as d n xdt n = yd n ydt n = f sin( ̟t ) − M ( q ) x − f y (10)where n is the order of the system. Using the transformation described in section 2.2,the discrete fractional form of the memristive Duffing oscillator is written as ∆ νa x = y ( t + ν − ∆ νa y = f sin( ̟t ) − M ( q ) x ( t + ν − − f y ( t + ν −
1) (11)Sabarathinam et al [19] transformed equation 9 into a four dimensional integer ordersystem for analysis.
The bifurcation analysis of the discrete fractional order Duffing oscillator with respectto different system parameters was investigated and the results presented in Figure 1.The system behaviour with respect to the fractional order, n , shows a reverse perioddoubling route to chaos (Figure 1a). Chaotic regions were found in the region 0 . 562 with transition to period-2 and period-3 bifurcation at n = 0 . 86 and n = 0 . 57 respectively (Figure 1b). A period doubling route to chaos was also observedwhen the parameter ω was varied. The system was found to be chaotic in only smallpositive region 0 . < ω < . f , n, β, f , ̟ ) =(0 . , . , , . , . Will be inserted by the editor chaotic when − . < ω < − . f , the system wasfound to be chaotic when 0 . < f < . 45 (Figure 1c). The parameter ̟ presentsan interesting bifurcation structure in the region 0 < ̟ < 1. In the small region0 < ̟ < . 01, the system was found to be chaotic with a reverse period doublingbifurcation. In Figure 2, the system showed periodic behaviour in the region 0 . <̟ < . < ̟ < . 01 but aquasiperiodic route when ̟ > . According to Fuwape and Ogunjo [7], tracking control is obtained when the individualcomponents of a system followed different predefined rules. The aim is to control the x and y component of System (11) to follow different trajectories. Rewriting thememristive discrete fractional order system (System 11) and adding controllers, wehave ∆ νa x = y ( t + ν − 1) + u ∆ νa y = f sin( ̟t ) − M ( x ) x ( t + ν − − f y ( t + ν − 1) + u (12)The error term is written as ˙ e = ˙ x − ˙ c ˙ e = ˙ y − ˙ c (13)where c ( c , c ) are user defined functions. If c = c , mixed tracking is obtained.Substituting Equation 13 into Equation 11, we obtain˙ e = e + c − c ˙ e = f sin( ̟t ) − ω e − ω c − βx − f e − f c − c (14)Eliminating nonlinear terms in e , e gives the subcontroller functions u = − c + c + v u = − f sin( ̟t ) + ω c + 3 βx + f c + c + v (15)where the parameters v i will be obtained later. Substituting Equation 15 into Equa-tion 14, we obtain (cid:18) ˙ e ˙ e (cid:19) = (cid:18) − ω f (cid:19) (cid:18) e e (cid:19) + (cid:18) v v (cid:19) (16)The method of active control requires that a constant matrix K is chosen which willcontrol the error dynamics such that the feedback matrix becomes (cid:18) v v (cid:19) = K (cid:18) e e (cid:19) Thus, a matrix of the form K = (cid:18) λ − ω λ + f (cid:19) (17) ill be inserted by the editor 5 Fig. 1. Bifurcation diagram for y-component with (a) fractional order, n with pa-rameters ( f , ω , β, f , ̟ ) = (0 . , − . , . , . , . ω with system parame-ters ( f , n, β, f , ̟ ) = (0 . , . , , . , . f using ( f , ω , β, n, ̟ ) =(0 . , − . , , . , . ̟ with system parameters ( f , ω , β, f , n ) =(0 . , − . , , . , . To verify the effectiveness of the tracking controllers designed above, numericalsimulations were carried out. The x component was made to track and exponentialfunction while the y component tracked a sine wave. The result obtained is shownin Figure 5. The trajectories of the x and y component were found to track thedesired set functions, hence, we conclude that tracking control of the system has beenachieved. Recently, there has been renewed interest in fractional order systems. In this research,we extend the chaotic fractional order Duffing system to a discrete-fractional ordersystem with a memristor. Bifurcation diagrams were produced to show the differentbehaviour of the system under different parameter changes. The possibility of controlthe two state space of the discrete fractional order memristive Duffing oscillator totwo different trajectory was investigated. The effectiveness of the controllers weredemonstrated by numerical simulations. Will be inserted by the editor Fig. 2. Bifurcation diagram for y-component with ̟ in the range 0 < ̟ < Fig. 3. Bifurcation diagram for y-component with the memristor M ( q )ill be inserted by the editor 7 Fig. 4. x − y phase space realization of the driven Duffing oscillator obtained with parameter( f , ω , β, f , ̟ ) = (0 . , − . , . , . , . < n < Time -0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25 x , y Fig. 5. Tracking control of the driven Duffing oscillator obtained with parameter( f , ω , β, f , ̟, n ) = (0 . , − . , . , . , . , . x component (red line) is madeto track the function A sin( wt ) and the y component (black line) tracks the function Ae kt ,where A, w, k are constants. Will be inserted by the editor References 1. T Abdeljawad Comput. Math. with Appl. Advances in Difference Equations 320 (2013).3. F M Atici, P W Eloe Proc. Am. Math. Soc. 981 (2009)4. S Boccaletti Physics Reports 103 (2000).5. L O Chua IEEE Transactions on Circuit Theory 507 (1971)6. M F Danca, W K Tang, G Chen Chaos, Solitons & Fractals 31 (2016).7. I A Fuwape, S Ogunjo International Journal of Dynamics and Control In press Journal of Atmosphericand Solar-Terrestrial Physics 46 (2016).9. I A Fuwape, S T Ogunjo, S S Oluyamo, A B Rabiu Theoretical and Applied Climatology 119 (2017)10. T Hu Applied Mathematics Statistical Mechanics and its Applications 389 (2017).12. Y Liu Indian Journal of Physics 313 (2016)13. 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