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Fractionalorder Chua’s system: discretization, bifurcation and chaos
Advances in Difference Equations volume 2013, Article number: 320 (2013)
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.
Introduction
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 [4–8]. For stability conditions and synchronization of a system of fractionalorder differential equations, one can see [9–11].
We recall the basic definitions (Caputo) and properties of fractional order 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 (n1,n) of f(t), t>0, is defined by
In addition, the following results are the main ones in fractional calculus. Let \beta ,\gamma \in {\mathbb{R}}^{+}, \alpha \in (0,1),

{I}_{a}^{\beta}:{L}^{1}\to {L}^{1}, and if f(x)\in {L}^{1}, then {I}_{a}^{\gamma}{I}_{a}^{\beta}f(x)={I}_{a}^{\gamma +\beta}f(x).

{lim}_{\beta \to n}{I}_{a}^{\beta}f(x)={I}_{a}^{n}f(x) uniformly on [a,b], n=1,2,3,\dots , where {I}_{a}^{1}f(x)={\int}_{a}^{x}f(s)\phantom{\rule{0.2em}{0ex}}ds.

{lim}_{\beta \to 0}{I}_{a}^{\beta}f(x)=f(x) weakly.

If f(x) is absolutely continuous on [a,b], then {lim}_{\alpha \to 1}{D}_{a}^{\alpha}f(x)=\frac{df(x)}{dx}.
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 [20–24].
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 \alpha \to 1, 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 \alpha \in (0,1) and consider the differential equation of fractional order
The corresponding equation with a piecewise constant argument
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). So, we get
Thus
Let t\in [2r,3r), then \frac{t}{r}\in [2,3). So, we get
Thus
Repeating the process, we get when t\in [nr,(n+1)r), then \frac{t}{r}\in [n,n+1). 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 x(0)={x}_{o}, y(0)={y}_{o}, and z(0)={z}_{o}.
The proposed discretization method has the following steps.
(1) Let t\in [0,r), then \frac{t}{r}\in [0,1). So, we get
and the solution of (7) is given by
(2) Let t\in [r,2r), then \frac{t}{r}\in [1,2). So, we get
and the solution of (7) is given by
Repeating the process, we can easily deduce that the solution of (7) is given by
Let t\to (n+1)r, we obtain the discretization
which can be rewritten as
Remark 1 It should be noticed that if \alpha \to 1 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:

fi{x}_{1}=(0,0,0),

fi{x}_{2}=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}),

fi{x}_{3}=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}).
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 fi{x}_{1} yields the following characteristic equation:
where s=\frac{{r}^{\alpha}}{\mathrm{\Gamma}(1+\alpha )}. Let
Now, let {a}_{1}=A, {a}_{2}=B, and {a}_{3}=C. 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(\lambda ) satisfy \lambda <1 and thus fi{x}_{1} is asymptotically stable. This is not satisfied here since γ and β are positive and so {b}_{3}<{b}_{1}. That is, fi{x}_{1} is unstable.
While linearizing system (8) about fi{x}_{2} or fi{x}_{3} yields the following characteristic equation:
We let {a}_{11}=s(1+\frac{2}{7}\gamma )3, {a}_{22}=3s(2+\frac{4}{7}\gamma ){s}^{2}(\beta +\gamma \frac{2}{7}\gamma ), and {a}_{33}=1+s(1+\frac{2}{7}\gamma )+{s}^{2}(\beta +\gamma \frac{2}{7}\gamma )\frac{2}{7}\gamma \beta {s}^{3}. 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(\lambda ) satisfy \lambda <1 and thus fi{x}_{2} or fi{x}_{3} is asymptotically stable. We can check easily that F(1)<0, that is, both fi{x}_{2} and fi{x}_{3} 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 r=0.0105, \beta =\frac{100}{7}, and \alpha =0.75 with initial conditions ({x}_{0},{y}_{0},{z}_{0})=(0.2,0.4,0.2).
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 \alpha =0.75 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.
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 \alpha \to 1, the discretization will be Euler’s discretization [29]. In addition, we carried out the numerical simulation when \alpha \to 1, 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 \alpha =0.75 will not produce any bifurcation diagrams.
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Agarwal, R.P., ElSayed, A.M.A. & Salman, S.M. Fractionalorder Chua’s system: discretization, bifurcation and chaos. Adv Differ Equ 2013, 320 (2013). https://doi.org/10.1186/168718472013320
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DOI: https://doi.org/10.1186/168718472013320
Keywords
 Chua’s system
 fractionalorder differential equations
 fixed points
 asymptotic stability
 chaotic attractor
 bifurcation
 chaos