Abstract
In this paper we are interested in the fractionalorder form of Chua’s system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, bifurcation and chaos for different values of the fractionalorder parameter are discussed. We show that the proposed discretization method is different from other discretization methods, such as predictorcorrector and Euler methods, in the sense that our method is an approximation for the righthand side of the system under study.
Keywords:
Chua’s system; fractionalorder differential equations; fixed points; asymptotic stability; chaotic attractor; bifurcation; chaosIntroduction
In recent years differential equations with fractional order have attracted many researchers’ attention because of their applications in many areas of science and engineering; see, for example, [1,2], and [3]. The need for fractionalorder differential equations stems in part from the fact that many phenomena cannot be modeled by differential equations with integer derivatives. The fractional calculus has allowed the operations of integration and differentiation to be applied upon any fractional order. Recently, the theory of fractional differential equations attracted many scientists and mathematicians to work on [48]. For stability conditions and synchronization of a system of fractionalorder differential equations, one can see [911].
We recall the basic definitions (Caputo) and properties of fractional order differentiation and integration.
Definition 1 The fractional integral of order of the function , , is defined by
and the fractional derivative of order of , , is defined by
In addition, the following results are the main ones in fractional calculus. Let , ,
• If is absolutely continuous on , then .
To solve fractionalorder differential equations, there are two famous methods: frequency domain methods [12] and time domain methods [13]. 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 [14] and [15].
Often it is not desirable to solve a differential equation analytically, and one turns to numerical or computational methods.
In [16], 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 them infinite dimensional. So, even a single ordinary differential equation with delay could display chaos.
Dealing with fractionalorder differential equations as dynamical systems is somehow new and has motivated the leading research literature recently; see, for example, [17,18] and [19]. 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 historical states. This property is very close 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 were studied in [2024].
Discretization process
In [25], a discretization process is introduced to discretize the fractionalorder differential equations, and we take Riccati’s fractionalorder differential equations as an example. We noticed that when the fractionalorder parameter , Euler’s discretization method is obtained. In [26], the same discretization method is applied to the logistic fractionalorder differential equation. We concluded that Euler’s method is able to discretize firstorder difference equations; however, we succeeded in discretizing a secondorder difference equation.
Here, we are interested in applying the discretization method to a system of differential equations like Chua’s system which is one of the autonomous differential equations capable of generating chaotic behavior. This system is well known and has been studied widely.
Let and consider the differential equation of fractional order
The corresponding equation with a piecewise constant argument
Thus
Thus
Thus
Repeating the process, we get when , then . So, we get
Thus
Consider Chua’s dynamical system with cubic nonlinearity (see [27,28])
In [28], the author studied the effect of the fractional dynamics in Chua’s system. It has been demonstrated that the usual idea of system order must be modified when fractional derivatives are present.
Here, we are concerned with fractionalorder Chua’s system given by
Actually, we are interested in discretizing fractionalorder Chua’s system with piecewise constant arguments given in the form
with initial conditions , , and .
The proposed discretization method has the following steps.
and the solution of (7) is given by
and the solution of (7) is given by
Repeating the process, we can easily deduce that the solution of (7) is given by
Let , we obtain the discretization
which can be rewritten as
Remark 1 It should be noticed that if in (8), we deduce the Euler discretization method of Chua’s system [29].
It is worth to mention here that many discretization methods, such as Euler’s method and predictorcorrector method, have been applied to Chua’s system (4). Euler’s method discretization is an approximation for the derivative while the predictorcorrector method is an approximation for the integral. However, our proposed discretization method here is an approximation for the righthand side as it is pretty clear from formula (8).
Fixed points and their asymptotic stability
Now we study the asymptotic stability of the fixed points of system (8) which has three fixed points:
By considering a Jacobian matrix for one of these fixed points and calculating their eigenvalues, we can investigate the stability of each fixed point based on the roots of the system characteristic equation [30].
Linearizing system (8) about yields the following characteristic equation:
Now, let , , and . From the Jury test, if , , and , , , where , , , , and , then the roots of satisfy and thus is asymptotically stable. This is not satisfied here since γ and β are positive and so . That is, is unstable.
While linearizing system (8) about or yields the following characteristic equation:
We let , , and . From the Jury test, if , , and , , , where , , , , and , then the roots of satisfy and thus or is asymptotically stable. We can check easily that , that is, both and are unstable.
Attractors, bifurcation and chaos
Since the Lyapunov exponent is a good indicator for existence of chaos, we compute the Lyapunov characteristic exponents (LCEs) via the householder QR based methods described in [31]. LCEs play a key role in the study of nonlinear dynamical systems and they are the measure of sensitivity of solutions of a given dynamical system to small changes in the initial conditions. One feature of chaos is the sensitive dependence on initial conditions; for a chaotic dynamical system, at least one LCE must be positive. Since for nonchaotic systems all LCEs are nonpositive, the presence of a positive LCE has often been used to help determine if a system is chaotic or not. We find that LCE1 = 0.0263, LCE2 = −0.0077, and LCE3 = −0.4160. Figure 1 shows the LCEs for system (8) for parameter values , , and with initial conditions .
Figure 1. Lyapunov characteristic exponents (LCEs) for system (8).
On the other hand, we show some attractors of system (8) for different α. The numerical experiments show that playing with the parameter α away from will not produce any bifurcation diagrams. Figures 2 and 3 show attractors of system (8), while Figures 413 show bifurcation diagrams for the same system.
Figure 2. Strange attractor of (8) with,,,.
Figure 3. Attractor of (8) with,,,.
Figure 4. Bifurcation diagram of (8) with,,.
Figure 5. Bifurcation diagram of (8) with,,.
Figure 6. Bifurcation diagram of (8) with,,.
Figure 7. Bifurcation diagram of (8) with,,.
Figure 8. Bifurcation diagram of (8) with,,.
Figure 9. Bifurcation diagram of (8) with,,.
Figure 10. Bifurcation diagram of (8) with,,.
Figure 11. Bifurcation diagram of (8) with,,.
Figure 12. Bifurcation diagram of (8) with,,.
Figure 13. Bifurcation diagram of (8) with,,.
Conclusion
A discretization method is introduced to discretize fractionalorder differential equations and we take Chua’s system with cubic nonlinearity for our purpose. We have noticed that when , the discretization will be Euler’s discretization [29]. In addition, we carried out the numerical simulation when , we did not get any bifurcation at all. Actually, this is not surprising since we did the same in Rössler’s system in its discrete version. When we contacted Prof. Dr. Rössler himself about why we were not getting any bifurcation diagrams, he assured our results. Finally, it is not clear in this situation why the parameter α takes one value only to produce bifurcation and chaos diagrams.
On the other hand, we show some attractors of system (8) for different α. The numerical experiments show that playing with the parameter α away from will not produce any bifurcation diagrams.
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.
Acknowledgements
The authors would like to thank the referees of this manuscript for their valuable comments and suggestions.
References

Bhalekara, S, DaftardarGejjib, V, Baleanuc, D, Magine, R: Transient chaos in fractional Bloch equations. Comput. Math. Appl.. 64, 3367–3376 (2012). Publisher Full Text

Faieghi, M, Kuntanapreeda, S, Delavari, H, Baleanu, D: LMIbased stabilization of a class of fractionalorder chaotic systems. Nonlinear Dyn.. 72, 301–309 (2013). Publisher Full Text

Golmankhaneh, AK, Arefi, R, Baleanu, D: The proposed modified Liu system with fractional order. Adv. Math. Phys.. 2013, (2013) Article ID 186037

Das, S: Functional Fractional Calculus for System Identification and Controls, Springer, Berlin (2007)

ElSayed, AMA, ElMesiry, A, ElSaka, H: On the fractionalorder logistic equation. Appl. Math. Lett.. 20, 817–823 (2007). Publisher Full Text

ElSayed, AMA: On the fractional differential equations. J. Appl. Math. Comput.. 49(23), 205–213 (1992). Publisher Full Text

ElSayed, AMA: Nonlinear functionaldifferential equations of arbitrary orders. Nonlinear Anal.. 33, 181–186 (1998). Publisher Full Text

Podlubny, I: Fractional Differential Equations, Academic Press, London (1999)

Matouk, AE: Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system. Phys. Lett. A. 373, 2166–2173 (2009). Publisher Full Text

Matouk, AE: Chaos synchronization between two different fractional systems of Lorenz family. Math. Probl. Eng.. 2009, (2009) Article ID 572724

Matouk, AE: On some stability conditions and hyperchaos synchronization in the new fractional order hyperchaotic Chen system. LeHavreNormandy, France, June 29July 02. (2009)

Sun, H, Abdelwahed, A, Onaral, B: Linear approximation for transfer function with a pole of fractional order. IEEE Trans. Autom. Control. 29, 441–444 (1984). Publisher Full Text

Diethelm, K, Ford, NJ, Freed, AD: A predictorcorrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn.. 29, 3–22 (2002). Publisher Full Text

Tavazoei, MS, Haeri, M: Unreliability of frequency domain approximation in recognizing chaos in fractional order systems. IET Signal Process.. 1, 171–181 (2007). Publisher Full Text

Tavazoei, MS, Haeri, M: Limitation of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Anal., Theory Methods Appl.. 69, 1299–1320 (2008). Publisher Full Text

Wang, Z: A numerical method for delayed fractionalorder differential equations. J. Appl. Math.. 2013, (2013) Article ID 256071

Erjaee, GH: On analytical justification of phase synchronization in different chaotic systems. Chaos Solitons Fractals. 3, 1195–1202 (2009)

Yan, JP, Li, CP: On chaos synchronization of fractional differential equations. Chaos Solitons Fractals. 32, 725–735 (2007). Publisher Full Text

Wu, GC, Baleanu, D: Discrete fractional logistic map and its chaos. Nonlinear Dyn. (2013). doi:10.1007/s1107101310657 Publisher Full Text

Akhmet, MU: Stability of differential equations with piecewise constant arguments of generalized type. Nonlinear Anal.. 68, 794–803 (2008). Publisher Full Text

Akhmet, MU, Altntana, D, Ergenc, T: Chaos of the logistic equation with piecewise constant arguments. arXiv:1006.4753 (2010)

ElSayed, AMA, Salman, SM: Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments. Malaya J. Mat.. 1, 14–18 (2012)

ElSayed, AMA, Salman, SM: Chaos and bifurcation of the logistic discontinuous dynamical systems with piecewise constant arguments. Malaya J. Mat.. 3, 14–20 (2013)

ElSayed, AMA, Salman, SM: The unified system between Lorenz and Chen systems: a discretization process. Electron. J. Math. Anal. Appl.. 1, 318–325 (2013)

ElSayed, AMA, Salman, SM: On a discretization process of fractional order Riccati’s differential equation. J. Fract. Calc. Appl.. 4, 251–259 (2013)

ElSayed, AMA, Salman, SM: On a discretization process of fractionalorder Logistic differential equation. J. Egypt. Math. Soc. (accepted)

Stegemann, C, Albuquerque, HA, Rech, PC: Some two dimensional parameter space of a Chua system with cubic nonlinearity. Chaos, Interdiscip. J. Nonlinear Sci.. 20, 817–823 (2007)

Hartely, TT, Lorenzo, CF, Qammer, HK: Chaos in a fractional order Chau’s system. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.. 42, 485–490 (1995). Publisher Full Text

Elaidy, SN: An Introduction to Difference Equations, Springer, New York (2005)

Holmgren, R: A First Course in Discrete Dynamical Systems, Springer, New York (1994)

Udwadia, FE, von Bremen, H: A note on the computation of the largest pLyapunov characteristic exponents for nonlinear dynamical systems. J. Appl. Math. Comput.. 114, 205–214 (2000). Publisher Full Text