Fractional calculus and fractional differential equations have received considerable interest in the recent forty years. Fractional derivative means that the order of differentiation can be an arbitrary real number and even it can be a complex number. Fractional derivative modelling has been applied to many scientific and engineering fields, such as quantum mechanics [1], viscoelasticity and rheology [2], electrical engineering [3], electrochemistry [4], biology [5], biophysics and bioengineering [6], signal and image processing [7], mechatronics [8], and control theory [9][11]. Although few mathematical issues of fractional derivative remain unsolved, most of the difficulties have been overcome, and the applications of fractional calculus in above fields indicate that the fractional models can depict the property and behavior of a realworld problem more accurately. For a comprehensive review of fractional calculus, we refer readers to some monographs [12][14] and references therein. In contrast to integer order derivative, the way of identifying fractional derivative is not unique. There are several types of definitions, such as RiemannLiouville derivative, Caputo derivative, GrünwaldLetnikov derivative, and so on. More details can be found in [13, Chapter 2]. In the recent years, the study of dynamical system with fractional order derivative becomes more and more popular [15][19]. Moreover, the dynamics in fractional dynamical system is more interesting.
Returning back to the fractional derivative, since it has several different definitions, how to develop a generalized form which can unify all the existing fractional derivatives becomes one important topic in fractional calculus [20][22]. Recently, a class of new generalized fractional integral and generalized fractional derivative is introduced in [22]. The new generalized fractional integral and generalized fractional derivative depend on a scale function and a weight function, which makes them more general. When the scale function and the weight function reduce to some specific cases, the generalized fractional operators will reduce to RiemannLiouville fractional integral, RiemannLiouville fractional derivative and Caputo fractional derivative and so on. However, the study of this new generalized fractional integral and generalized fractional derivative are in the very beginning stage now [23][26]. In [24], we show that in generalized fractional diffusion equation, the scale function allows the response domain to be scaled differently. It is required that the scale function should be strictly monotonically increasing or decreasing. A convex increasing scale function will compress the response domain towards to the initial time. A concave increasing scale function will stretch the response domain away from the initial time. The weight function allows the response to be assessed differently at different time, since in many applications, we may require an event to be weighed differently at different time point. For example, modeling of memory of a child may require a heavy weight at current time point, whereas the same for an older person may require more weight on the past. To be an initial attempt of application to chaotic dynamical systems, in this paper, we define a class of new generalized fractional chaotic systems by replacing the original derivatives with the new generalized fractional derivative, then apply a finite difference scheme to study the numerical solutions of two different generalized fractional chaotic systems, namely generalized fractional LotkaVolterra system (GFLVS) and generalized fractional Lorenz system (GFLS). Their complex dynamics will be discussed, and the dynamic behavior depending on the weight and scale function will be shown graphically.
The rest of this paper is organized as follows: In Section 2, the preliminaries of fractional calculus are given. The new generalized fractional integral and generalized fractional derivative are shown. A finite difference approach for solving equations with generalized fractional derivative is carried out. In Section 3, we define the chaotic systems using the generalized fractional derivative of Caputo type, i.e., the GFLVS and GFLS. Some interesting dynamics of those two systems are shown graphically. Finally, the conclusions are drawn in Section 4.
2 Mathematical PreliminariesIn this section, we introduce the preliminaries of generalized fractional derivatives, and show a proper numerical method for differential equations with such derivatives.
2.1 Generalized Fractional CalculusLet us begin with the common fractional operators. In calculus, the
$ I^{n}u(t)=\overbrace{\int^t_0\cdots\int^t_0}^{n\ {\rm times}}u(s)ds\cdots{ds}= \int^t_0\frac{(ts)^{n1}}{(n1)!}u(s)ds $ 
where
Definition 1[13]: The left RiemannLiouville fractional integral of order
$ \begin{align} \left(I^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(\alpha)}\int^t_0(ts)^{\alpha1}u(s)ds \end{align} $  (1) 
provided the integral is finite, where
The RiemannLiouville fractional integral plays an important role in defining fractional derivatives. There are two basic approaches to define the fractional derivative, i.e., "first integration then differentiation" and "first differentiation then integration". The corresponding fractional derivatives are called RiemannLiouville fractional derivative and Caputo fractional derivative, and the definitions are given as follows.
Definition 2[13]:The left RiemannLiouville fractional derivative of order
$ \begin{align} \left(D^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n\alpha)}\left(\frac{d^n}{dt^n}\right) \int^t_0(ts)^{n\alpha1}u(s)ds \end{align} $  (2) 
provided the right side of the identity is finite.
Definition 3[13]: The left Caputo fractional derivative of order
$ \begin{align} \left({^cD}^{\alpha}_{0+}u\right)(t) = \frac{1}{\Gamma(n\alpha)}\int^t_0(ts)^{n\alpha1}u^{(n)}(s)ds \end{align} $  (3) 
provided the right side of the identity is finite.
Besides above, there also exist right RiemannLiouville integral and derivative, and right Caputo fractional derivative [13]. Mathematically, the RiemannLiouville and Caputo fractional operators are used in applications frequently. In most realworld models, we always employ the left Caputo fractional derivative. One reason is that we will study generalized fractional dynamical system later, and the derivative is taken with respect to time variable. In physical models, time is always running forward. The other reason is that in the differential equations with Caputo fractional derivative, the initial conditions are taken in the same form as for integerorder differential equations which have clear physical meanings in the practical application and can be easily measured [14]. In what follows, we will introduce the generalized fractional integral and derivative proposed in [22]. They extend nearly all the existing fractional operators. Now we list the generalized fractional integral and derivative defined on positive half axis. They will be used to define the generalized fractional chaotic systems in next section.
Definition 4[22]: The left generalized fractional integral of order
$ \begin{align} \left(I^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{1}}{\Gamma(\alpha)}\int^{t}_{0} \frac{w(s)\sigma'(s)u(s)}{[\sigma(t)\sigma(s)]^{1\alpha}}ds \end{align} $  (4) 
provided the integral exists, where
Definition 5[22]: The left generalized derivative of order
$ \begin{align} \left(D^m_{[\sigma, w;L]}u\right)(t) = [w(t)]^{1}\left[\left(\frac{1}{\sigma'(t)}D_t\right)^m(w(t)u(t))\right] \end{align} $  (5) 
provided the rightside of equation is finite, where
Definition 6[22]: The Caputo type left generalized fractional derivative of order
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \left(I^{m\alpha}_{0+;[\sigma, w]}D^m_{[\sigma, w;L]}u\right)(t) \end{align} $  (6) 
provided the rightside of equation is finite, where
$ \begin{align} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t) = \frac{[w(t)]^{1}}{\Gamma(1\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)\sigma(s)]^{\alpha}}ds. \end{align} $  (7) 
Now we introduce a finite difference method for solving differential equations with generalized fractional derivative. Consider the following generalized fractional differential equation:
$ \begin{align} \begin{cases} \left(D^{\alpha}_{0+;[\sigma, w]}u\right)(t)=f(t, u(t)), \quad 0 < t\leq T\\ u(0)=u_0 \end{cases} \end{align} $  (8) 
where
$ \begin{align} (D^{\alpha}_{0+;[\sigma, w]}& u)(t_{j+1}) \nonumber\\ &= \frac{[w(t_{j+1})]^{1}}{\Gamma(1\alpha)}\int^{t_{j+1}}_{0}\frac{[w(s)u(s)]'} {[\sigma(t_{{j+1}})\sigma(s)]^{\alpha}}ds \nonumber\\ &= \frac{w_{j+1}^{1}}{\Gamma(1\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{[w(s)u(s)]'}{\left[\sigma(t_{{j+1}})\sigma(s) \right]^{\alpha}}ds \nonumber\\ & \approx \frac{w_{j+1}^{1}}{\Gamma(1\alpha)}\sum^{j}_{k=0}\int^{t_{k+1}}_{t_k} \frac{\frac{w_{k+1}u_{k+1}w_ku_k}{t_{k+1}t_k}} {\left[\sigma_{j+1}\sigma(s)\right]^{\alpha}}ds \nonumber\\ & \approx\sum^{j}_{k=0}\left(A^j_ku_{k+1}B^j_ku_k\right) \end{align} $  (9) 
where
$ \begin{align*} A^j_k =&\ \frac{w^{1}_{j+1}w_{k+1}}{\Gamma(2\alpha)(\sigma_{k+1}\sigma_k)} \\ & \times \left[(\sigma_{j+1}\sigma_k)^{1\alpha} (\sigma_{j+1}\sigma_{k+1})^{1\alpha}\right]\\ B^j_k =&\ \frac{w^{1}_{j+1}w_{k}}{\Gamma(2\alpha)(\sigma_{k+1}\sigma_k)} \\ & \times \left[(\sigma_{j+1}\sigma_k)^{1\alpha} (\sigma_{j+1}\sigma_{k+1})^{1\alpha}\right] \end{align*} $ 
Therefore, we obtain the finite difference scheme:
$ \begin{align} \sum^{j}_{k=0}\left(A^j_ku_{k+1}B^j_ku_k\right)=f(t_{j+1}, u_{j+1}) \end{align} $  (10) 
and the corresponding iteration scheme as
$ \begin{align} u_{j+1}=\begin{cases} \frac{1}{A^j_j}\left[f_j\sum\limits^{j1}_{k=0} \left(A^j_ku_{k+1}B^j_ku_k\right)+B^j_ju_j \right], \\ \qquad \qquad \qquad \qquad \qquad\qquad j=1, 2, \ldots, N1\\ \frac{1}{A^0_0}\left(f_0+B^0_0u_0\right), \qquad\qquad \ \, j=0 \end{cases} \end{align} $  (11) 
where
In what follows, we will apply this method to solve the generalized fractional chaotic systems. The numerical analysis of the above scheme can be found in [26].
3 Dynamic Behavior of Generalized Fractional Chaotic SystemsIn this section, we introduce two nonlinear dynamical systems but redefine them with Caputo type generalized fractional derivative. The classical and fractional senses are special cases of the new generalized fractional system below.
3.1 Generalized Fractional LotkaVolterra and Generalized Fractional Lorenz SystemReplacing the derivative with the generalized fractional derivative defined by (7), we define the generalized fractional LotkaVolterra system (GFLVS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = ax  bxy + mx^2  sx^2z\\ D^{\alpha_2}_{0+;[\sigma, w]}y = cy +dxy\\ D^{\alpha_3}_{0+;[\sigma, w]}z = pz + sx^2z \end{cases} \end{align} $  (12) 
where
By selecting the parameters
Similarly, we define the generalized fractional Lorenz system (GFLS) as
$ \begin{align} \begin{cases} D^{\alpha_1}_{0+;[\sigma, w]}x = r(yx)\\ D^{\alpha_2}_{0+;[\sigma, w]}y = x(\rhoz)y\\ D^{\alpha_3}_{0+;[\sigma, w]}z = xy\beta{z} \end{cases} \end{align} $  (13) 
where
By taking the parameters
Now we analyze the influence of the scale and weight functions on the responses of generalized fractional differential equation. For simplicity, we consider
$ \begin{align} D^{\alpha}_{0+;[\sigma, w]}u(t) = Au(t) + f(t) \end{align} $  (14) 
where
Equation (14) is equivalent to
$ \begin{align} \frac{[w(t)]^{1}}{\Gamma(1\alpha)}\int^{t}_{0} \frac{[w(s)u(s)]'}{[\sigma(t)\sigma(s)]^{\alpha}}ds = Au(t) + f(t). \end{align} $  (15) 
Let
$ \begin{align} \frac{1}{\Gamma(1\alpha)}\int^{t}_{0} \frac{v(s)'}{[\sigma(t)\sigma(s)]^{\alpha}}ds = Av(t) + w(t)f(t). \end{align} $  (16) 
According to [13], we deduce the solution of (16) as:
$ \begin{align} v(t) =&\ E_{\alpha}\left(A[\sigma(t)\sigma(0)]^{\alpha}\right)v_0 \nonumber\\ & +\int^t_0(\sigma(t)\sigma(s))^{\alpha1} \nonumber\\ &\times E_{\alpha, \alpha}[A(\sigma(t)\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $  (17) 
which implies that
$ \begin{align} u(t)=&\ \frac{w(0)}{w(t)}E_{\alpha}\left(A[\sigma(t)\sigma(0)]^{\alpha}\right)u_0 \nonumber\\ & +\frac{1}{w(t)}\int^t_0(\sigma(t)\sigma(s))^{\alpha1} \nonumber\\ & \times E_{\alpha, \alpha}[A(\sigma(t)\sigma(s))^{\alpha}]w(s)f(s)ds \end{align} $  (18) 
where
In (18), we observe that how the weight and scale functions influence the behavior of (14). First of all, the weight function cannot be zero in the domain, otherwise solution
The fractional chaotic systems are sufficiently generalized by using the generalized fractional derivative, since many existing fractional derivatives, as well as integer order derivatives, are special cases of the generalized fractional derivative. In our numerical experiments, we find many interesting dynamical behaviors of generalized fractional chaotic systems which are never found in common fractional or integer order chaotic systems. Here we present some particular simulation results. However, our discussion depends on Figs. 2 and 3, and others figures are not shown here.
First, we simulate the influence of scale function on dynamics of chaotic systems. In GFLVS, we take fractional order
Second, we simulate the influence of weight function on dynamics of chaotic systems. In GFLVS, we take fractional order
Finally, to end this section, we make some remarks based on the numerical experiments above. Some other figures are not listed here for shortening the length of paper.
1) The GFLVS is chaotic with scale function
2) The GFLS is chaotic with scale function
3) Our previous work [23][26] verified that in generalized fractional integral and generalized fractional derivative, the basic property of scale function
4) A similar observation to weight function can be found in [23][26], which shows that in generalized fractional integral and generalized fractional derivative, the basic property of weight function
5) In Figs. 2 and 3, one can observe that both changing the scale and weight functions make the systems change between different dynamical behavior (e.g., limit cycle and stable equilibrium point). These phenomena can be regarded as general cases for generalized fractional chaotic systems. We shall guess that either scale function or weight function would influence the dynamics of generalized fractional chaotic systems. In Fig. 2, the weight function is fixed so that the influence of scale function on GFLVS and GFLS is presented. Similarly, in Fig. 3, the scale function is fixed so that the influence of weight function on GFLVS and GFLS is shown. From (18), we clearly see that the scale function plays an important role in scaling the long time behavior of dynamics since it is located in the generalized exponential function, and the weight function provides a different average since it lies inside the integral, and it is a variable coefficient simultaneously. Apparently, the behavior of function
In this paper, we presented a class of new generalized fractional chaotic system, using the new generalized fractional derivative proposed recently. Many dynamical systems with integer or fractional order derivatives can be extended by replacing the derivative with the generalized fractional derivative. Therefore, the new generalized fractional dynamical systems considered in this paper can exhibit more complex dynamic behaviors. In simulations, we show that the dynamical behaviors of such systems not only depend on fractional order, but also depend on the scale and weight functions.
Acknowledgement:The author is grateful to Professor O. P. Agrawal (SIUC, USA) for introducing him theory of generalized fractional calculus, suggesting the basic idea of this paper, as well as his kind help and continuous encouragement in the recent years.
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