 Research
 Open Access
 Published:
Discretization of forced Duffing system with fractionalorder damping
Advances in Difference Equations volume 2014, Article number: 66 (2014)
Abstract
In this paper we are interested in studying the effect of the fractionalorder damping in the forced Duffing oscillator before and after applying a discretization process to it. Fixed points and their stability are discussed for the discrete system obtained. Finally, numerical simulations using Matlab are carried out to investigate the dynamic behavior such as bifurcation, chaos, and chaotic attractors. We note that on increasing the value of the fractionalorder parameter, the resulting discrete system is stabilized.
1 Introduction
In recent years differential equations with fractionalorder have attracted the attention of many researchers because of their applications in many areas of science and engineering. Analytical and numerical techniques have been implemented to study such equations. The fractional calculus has allowed the operations of integration and differentiation to be applied upon any fractionalorder. For the existence of solutions for fractional differential equations, see [1, 2].
As regards the development of existence theorems for fractional functionaldifferential equations, many contributions exist and can be referred to [3–5]. Many applications of fractional calculus amount to replacing the time derivative in a given evolution equation by a derivative of fractionalorder.
We recall the basic definitions (Caputo) and properties of fractionalorder differentiation and integration.
Definition 1 The fractional integral of order $\beta \in {\mathbb{R}}^{+}$ of the function $f(t)$, $t>0$, is defined by
and the fractional derivative of order $\alpha \in (0,1)$ of $f(t)$, $t>0$, is defined by
To solve fractionalorder differential equations there are two famous methods: frequency domain methods [6] and time domain methods [7]. In recent years it has been shown that the second method is more effective because the first method is not always reliable in detecting chaos [8] and [9].
Often it is not desirable to solve a differential equation analytically, and one turns to numerical or computational methods. In [10], a numerical method for nonlinear fractionalorder differential equations with constant or timevarying delay was devised. It should be noticed that the fractional differential equations tend to lower the dimensionality of the differential equations in question, however, introducing delay in differential equations makes it infinite dimensional. So, even a single ordinary differential equation with delay could display chaos.
A lot of differential equations with Caputo fractional derivative were simulated by the predictorcorrector scheme, such as the fractional Chua system, the fractional Chen system, the Lorenz system, and so on. We should note that the predictorcorrector method is an approximation for the fractionalorder integration, however, our approach is an approximation for the righthand side.
Indeed, fractionalorder systems are useful in studying the anomalous behavior of dynamical systems in electrochemistry, biology, viscoelasticity, and chaotic systems; see for example [11]. Dealing with fractionalorder differential equations as dynamical systems is somewhat new and has motivated the leading research literature recently; see for example [12–26]. The nonlocal property of fractional differential equations means that the next state of a system not only depends on its current state but also on its history states. This property is very closely resembling to the real world and thus fractional differential equations have become popular and have been applied to dynamical systems.
On the other hand, some examples of dynamical systems generated by piecewise constant arguments have been studied in [27–30]. Here we propose a discretization process to obtain the discrete version of the system under study. Meanwhile, we apply the discretization process to discretize the fractionalorder logistic differential equation.
2 Forced Duffing oscillator with fractionalorder damping
The Duffing oscillator is an example of a periodically forced oscillator with nonlinear elasticity. It is one of the prototype systems of nonlinear dynamics. It first became popular for studying harmonic oscillations and, later, chaotic nonlinear dynamics in the wake of early studies by the engineer Georg Duffing [31]. The system has been successfully used to model a variety of physical processes, such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionization waves in plasmas. Despite the simplicity of the Duffing oscillator, the dynamical behavior is extremely rich and research is still going on today [32].
Forced Duffing oscillators are much harder to analyze analytically, because of the periodic force involved. It is far better to use computer approximations of the system to analyze how the forced Duffing oscillator is behaving under certain conditions. The Duffing equation, a wellknown nonlinear differential equation, is used for describing many physical, engineering, and even biological problems [33].
Originally the Duffing equation was introduced by German electrical engineer Duffing in 1918. The equation is given by
where the constants μ, λ, b, and γ are the damping coefficient, linear stiffness, nonlinear stiffness, excitation amplitude, and excitation frequency, respectively. All previous constants, assumed to be positive except λ, can also be negative.
Here we are concerned with the forced Duffing oscillator with fractionalorder damping given by
where $\alpha \in (0,1)$ is the fractionalorder parameter. First of all, we split (2.2) into a system of three equations, as follows:
The solution of system (2.3) with $\alpha =0.85$ is simulated using the GrünwaldLetnikov method described in [34] and is shown in Figures 1 and 2.
2.1 Bifurcation and chaos
The nonlinear dynamic of the duffing oscillator with fractionalorder damping is simulated using Matlab. We firstly fix the parameters $\mu =0.9$, $\lambda =1$, $b=1$, $\omega =1$, $\gamma =0.6$, and the initial state is $x(0)=y(0)=z(0)=0$. When $\alpha =1$, the system is described by the classical Duffing equation. Now vary the fractionalorder parameter α from 0 to 1 with step size $\mathrm{\Delta}\alpha =0.005$, and the bifurcation can easily be detected by examining the relationship between α and x. Figure 3 shows the effect of the fractionalorder parameter on the dynamics of the system.
Bifurcation diagrams with other control parameters, ω with $\alpha =0.80$; step size $\mathrm{\Delta}\omega =0.1$ and γ with $\alpha =0.85$; step size $\mathrm{\Delta}\gamma =0.005$ are shown in Figures 4 and 5. Figure 3 shows the strong effect of the fractionalorder parameter on the dynamic of system (2.3); it stabilizes the system as $\alpha \to 1$. Figure 4 shows an oscillating behavior of the bifurcation diagram as expected because ω is the excitation frequency inside the sine function.
3 Discretization process
In [35], a discretization process is introduced to discretize the fractionalorder differential equations and we take Riccati’s fractionalorder differential equations as an example. We notice that when the fractionalorder parameter $\alpha \to 1$, Euler’s discretization method is obtained. In [36], the same discretization method is applied to the logistic fractionalorder differential equation. We conclude that Euler’s method is able to discretize first order difference equations, however, we succeeded in discretizing a second order difference equation. Finally, in [37], we applied the same procedure to the fractionalorder Chua system to get the same results as in the two previous papers, and we showed that when $\alpha \to 1$ the system will be stabilized.
Here we are very interested in applying the dicretization method to a system of differential equations like the Chua system.
Now let $\alpha \in (0,1)$ and consider the differential equation of fractional order
The corresponding equation with piecewise constant argument is
Let $t\in [0,r)$, then $\frac{t}{r}\in [0,1)$. So, we get
Thus
Let $t\in [r,2r)$, then $\frac{t}{r}\in [1,2)$. Thus, we get
Thus
Let $t\in [2r,3r)$, then $\frac{t}{r}\in [2,3)$. So, we get
Thus
Repeating the process we get this result: when $t\in [nr,(n+1)r)$, then $\frac{t}{r}\in [n,n+1)$. So, we get
Thus
We are interested in discretizing (3.1) with piecewise constant arguments given in the form
with initial conditions $x(0)=0$, $y(0)=0$, and $z(0)=0$.
Applying the above mentioned discretization process and letting $t\to (n+1)r$ we obtain the discrete system
Indeed, there are other discretization methods for discretizing fractionalorder differential equations, for example:

The GrünwaldLetnikov definition (GL) which is a generalization of the derivative. The idea behind is that h, the step size, should approach 0 as n approaches infinity.

The predictorcorrector method is an approximation for the fractionalorder integration.
As a matter of fact, our approach is an approximation for the righthand side.
4 Stability of fixed points
Now we study the asymptotic stability of the fixed points of the system (3.4) in the unforced case, that is, $sin(\mathrm{\Omega}t)=0$, which has three fixed point namely, ${\mathit{fix}}_{1}=(0,0,0)$, ${\mathit{fix}}_{2}=(\frac{\lambda}{b}i,0,0)$, and ${\mathit{fix}}_{3}=(\frac{\lambda}{b}i,0,0)$.
By considering the Jacobian matrix for these fixed points and calculating its eigenvalues, we can investigate the stability of it based on the roots of the system’s characteristic equation [38]. The Jacobian matrix is given by
Linearizing the unforced system about ${\mathit{fix}}_{1}$ yields the following characteristic equation:
where $s=\frac{{r}^{\alpha}}{\mathrm{\Gamma}(1+\alpha )}$ and $l=\frac{{r}^{1\alpha}}{\mathrm{\Gamma}(2\alpha )}$. Let
From the Jury test, if $P(1)>0$, $P(1)<0$, and ${a}_{3}<1$, ${b}_{3}>{b}_{1}$, ${c}_{3}>{c}_{2}$, where ${b}_{3}=1{a}_{3}^{2}$, ${b}_{2}={a}_{1}{a}_{3}{a}_{2}$, ${b}_{1}={a}_{2}{a}_{3}{a}_{1}$, ${c}_{3}={b}_{3}^{2}{b}_{1}^{2}$, and ${c}_{2}={b}_{3}{b}_{2}{b}_{1}{b}_{2}$, then the roots of $P(\mathrm{\Lambda})$ satisfy $\mathrm{\Lambda}<1$ and thus ${\mathit{fix}}_{1}$ is asymptotically stable. This is not satisfied here since ${a}_{3}>1$. That is, ${\mathit{fix}}_{1}$ is unstable.
While linearizing the same unforced system about ${\mathit{fix}}_{2}$ or ${\mathit{fix}}_{3}$ yields the following characteristic equation:
We let ${a}_{11}=3$, ${a}_{22}=3+r\mu l$, and ${a}_{33}=(1+r\mu l+2sr\lambda l)$. From the Jury test, if $F(1)>0$, $F(1)<0$, and ${a}_{33}<1$, ${b}_{33}>{b}_{11}$, ${c}_{33}>{c}_{22}$, where ${b}_{33}=1{a}_{33}^{2}$, ${b}_{22}={a}_{11}{a}_{33}{a}_{22}$, ${b}_{11}={a}_{22}{a}_{33}{a}_{11}$, ${c}_{33}={b}_{33}^{2}{b}_{11}^{2}$, and ${c}_{22}={b}_{33}{b}_{22}{b}_{11}{b}_{22}$, then the roots of $F(\mathrm{\Lambda})$ satisfy $\mathrm{\Lambda}<1$ and thus ${\mathit{fix}}_{2}$ or ${\mathit{fix}}_{3}$ is asymptotically stable. We can check easily that $F(1)<0$, that is, both ${\mathit{fix}}_{2}$ and ${\mathit{fix}}_{3}$ are unstable.
5 Attractors, bifurcation, and chaos
In this section we show some dynamic behavior of the system (3.4) such as attractors, bifurcation and chaos for different α and r (see Figures 614). The effect of taking a fractional order on the system dynamics is investigated using phase diagrams and bifurcation diagrams. The bifurcation diagram is also used to examine the effect of the excitation amplitude and the frequency on the Duffing system with fractionalorder damping. We show bifurcation and chaos of the dynamical system (3.4) by means of bifurcation diagrams first with respect to the parameter μ, then with respect to the fractionalorder parameter α, and finally with respect to the parameter γ.
Let $r=0.25$ be fixed and vary α from 0.70 to 0.95 and μ from 2 to 8. The initial state of the system (3.4) is $({x}_{0},{y}_{0},{z}_{0})=(0,0,0)$. The step size for ρ is 0.01, the resulting bifurcation diagrams are shown in Figure 12(a)(f). It is observed from the figures that increasing the fractionalorder parameter α and fixing the parameter r stabilize the chaotic system.
Now vary the fractionalorder parameter α from 0.70 to 0.95, but with a fixed system parameter μ and change the parameter r from 0.15 to 0.30; the resulting bifurcation diagrams are shown in Figure 12(a)(d).
Then vary the fractionalorder parameter α from 0.85 to 0.95, but with a fixed system parameter $r=0.25$ and vary the parameter γ from 0 to 2; the resulting bifurcation diagrams are shown in Figure 12(a)(d).
It is clear from Figure 13 that increasing the value of the fractionalorder parameter α transforms the system into its stable behavior, exactly as in the case of the fractionalorder system (3.1).
6 Conclusion
A discretization method is applied in this paper to the forced Duffing oscillator with fractionalorder damping. The dynamics of the discretized fractionally damped Duffing equation has been examined numerically. Also, the conclusion of bifurcation of the parameterdependent system has been drawn numerically. Increasing the value of the fractionalorder damping term stabilizes the system under study in both cases: fractionalorder system and discretized system.
References
 1.
ElSayed A, ElMesiry A, ElSaka H: On the fractionalorder logistic equation. Appl. Math. Lett. 2007, 20: 817–823. 10.1016/j.aml.2006.08.013
 2.
ElSayed AMA: Nonlinear functionaldifferential equations of arbitrary orders. Nonlinear Anal. 1998, 33(2):181–186. 10.1016/S0362546X(97)005257
 3.
Das S: Functional Fractional Calculus for System Identification and Controls. Springer, Berlin; 2007.
 4.
Podlubny I Mathematics in Science and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.
 5.
Podlubny I: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 2002, 5(4):367–386. Dedicated to the 60th anniversary of Prof. Francesco Mainardi
 6.
Sun H, Abdelwahed A, Onaral B: Linear approximation for transfer function with a pole of fractionalorder. IEEE Trans. Autom. Control 1984, 29: 441–444. 10.1109/TAC.1984.1103551
 7.
Diethelm K, Ford NJ, Freed AD: A predictorcorrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002, 29: 3–22. 10.1023/A:1016592219341
 8.
Tavazoei MS, Haeri M: Unreliability of frequency domain approximation in recognizing chaos in fractionalorder systems. IET Signal Process. 2007, 1: 171–181. 10.1049/ietspr:20070053
 9.
Tavazoei MS, Haeri M: Limitation of frequency domain approximation for detecting chaos in fractionalorder systems. Nonlinear Anal., Theory Methods Appl. 2008, 69: 1299–1320. 10.1016/j.na.2007.06.030
 10.
Wang Z: A numerical method for delayed fractionalorder differential equations. J. Appl. Math. 2013., 2013: Article ID 256071
 11.
Monje CA, Chen Y, Vinagre B, Xue D, Feliu V Advanced Industrial Control Series. In Fractional Order Systems and Controls: Fundamentals and Applications. Springer, Berlin; 2010.
 12.
Erjaee GH: On analytical justification of phase synchronization in different chaotic systems. Chaos Solitons Fractals 2009, 39(3):1195–1202. 10.1016/j.chaos.2007.06.006
 13.
Wu GC, Dumitru B: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 2014, 75(1–2):283–287. 10.1007/s1107101310657
 14.
Wu GC, Dumitru B, Zeng SD: Discrete chaos in fractional sine and standard maps. Phys. Lett. A 2014. 10.1016/j.physleta.2013.12.010
 15.
Faieghi M, Kuntanapreeda S, Delavari H, Baleanu D: LMIbased stabilization of a class of fractionalorder chaotic systems. Nonlinear Dyn. 2013, 72(1–2):301–309. 10.1007/s1107101207146
 16.
Yan JP, Li CP: On chaos synchronization of fractional differential equations. Chaos Solitons Fractals 2007, 32(2):725–735. 10.1016/j.chaos.2005.11.062
 17.
Fahd J, Thabet A, Dumitru B: Stability of q fractional nonautonomous systems. Nonlinear Anal., Real World Appl. 2013, 14(1):780–784. 10.1016/j.nonrwa.2012.08.001
 18.
Wu GC, Dumitru B: New applications of the variational iteration method  from differential equations to q fractional difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 21 10.1186/16871847201321
 19.
Khalili GA, Moslemi YA, Dumitru B: On the fractional Hamilton and Lagrange mechanics. Int. J. Theor. Phys. 2012, 51(9):2909–2916. 10.1007/s1077301211698
 20.
Podlubny I: Numerical solution of ordinary fractional differential equations by the fractional difference method. In Advances in Difference Equations (Veszprém, 1995). Gordon & Breach, Amsterdam; 1997:507–515.
 21.
Podlubny I: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 2000, 3(4):359–386.
 22.
Podlubny, I: The Laplace transform method for linear differential equations of fractionalorder.arXiv:functan/9710005 (1997)
 23.
Sun K, Wang X, Sprott JC: Bifurcations and chaos in fractionalorder simplified Lorenz system. Int. J. Bifurc. Chaos 2010, 20(4):1209–1219. 10.1142/S0218127410026411
 24.
Bhalekar S, DaftardarGejji V, Baleanu D, Magin R: Generalized fractionalorder Bloch equation with extended delay. Int. J. Bifurc. Chaos 2012., 22(4): Article ID 1250071
 25.
Gejji VD, Bhalekar S, Gade P: Dynamics of fractionalorder Chen systems with delays. Pramana J. Phys. 2012, 79(1):61–69. 10.1007/s1204301202918
 26.
Chen YQ, Vinagre BM, Podlubny I: A new discretization method for fractionalorder differentiators via continued fraction expansion. 3. Proceedings of DETC 2003, 761–769.
 27.
Akhmet MU: Stability of differential equations with piecewise constant arguments of generalized type. Nonlinear Anal. 2008, 68(4):794–803. 10.1016/j.na.2006.11.037
 28.
Akhmet, MU, Altntana, D, Ergenc, T: Chaos of the logistic equation with piecewise constant arguments.arXiv:1006.4753 (2010)
 29.
ElSayed AMA, Salman SM: Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments. Malaya J. Mat. 2012, 1(1):14–18.
 30.
ElSayed AMA, Salman SM: Chaos and bifurcation of the logistic discontinuous dynamical systems with piecewise constant arguments. Malaya J. Mat. 2013, 3(1):14–20.
 31.
Duffing G: Erzwungene Schwingung bei veränderlicher Eigenfrequenz und ihre technische Bedeutung. Vieweg, Braunschweig; 1918.
 32.
Korsch HJ, Jodl HJ: Chaos: A Program Collection for the PC. Springer, Berlin; 1999.
 33.
Moon FC: Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Wiley, New York; 1987.
 34.
Petras I Nonlinear Physical Science. In FractionalOrder Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin; 2011.
 35.
ElSayed AMA, Salman SM: On a discretization process of fractionalorder Riccati differential equation. J. Fract. Calc. Appl. 2013, 4(2):251–259.
 36.
ElSayed AMA, Salman SM, ElRaheem ZF: On a discretization process of fractionalorder logistic differential equations. J. Egypt. Math. Soc. 2013. 10.1016/j.joems.2013.09.001
 37.
ElSayed AMA, Agarwal RP, Salman SM: Fractionalorder Chua’s system: discretization, bifurcation and chaos. Adv. Differ. Equ. 2013., 2013: Article ID 320
 38.
Elaidy SN Undergradute Texts in Mathematics. In An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.
Acknowledgements
The authors would like to thank the referees of this manuscript for their valuable comments and suggestions.
Author information
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Received
Accepted
Published
DOI
Keywords
 Duffing oscillator
 fractionalorder damping
 fixed points
 stability
 chaotic attractor
 bifurcation
 chaos