# Active Control of a Chaotic Fractional Order Economic System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary Tools

#### 2.1. Fractional Calculus

**Definition 1.**A real function $f\left(x\right),\phantom{\rule{0.166667em}{0ex}}x>0$, is said to be in the space ${C}_{\mu},\phantom{\rule{0.166667em}{0ex}}\mu \in \mathbb{R}$ if there exits a real number $\lambda >\mu $, such that $f\left(x\right)={x}^{\lambda}g\left(x\right)$, where $g\left(x\right)\in C[0,\infty )$, and it is said to be in the space ${C}_{\mu}^{m}$ if and only if ${f}^{\left(m\right)}\in {C}_{\mu}$ for $m\in I\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}N$.

**Definition 2.**The Riemann–Liouville fractional integral operator of order α of a real function $f\left(x\right)\in {C}_{\mu},\phantom{\rule{4pt}{0ex}}\mu \ge -1$, is defined as:

- ${J}^{\alpha}{J}^{\beta}f\left(x\right)={J}^{\alpha +\beta}f\left(x\right),$
- ${J}^{\alpha}{J}^{\beta}f\left(x\right)={J}^{\beta}{J}^{\alpha}f\left(x\right),$
- ${J}^{\alpha}{x}^{\xi}=\frac{\Gamma (\xi +1)}{\Gamma (\alpha +\xi +1)}{x}^{\alpha +\xi}$.

**Definition 3.**The Caputo fractional derivative ${D}^{\alpha}$ of a function $f\left(x\right)$ of any real number α, such that $m-1<\alpha \le m$, $m\in I\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}N$, for $x>0$ and $f\in {C}_{-1}^{m}$ in terms of ${J}^{\alpha}$, is:

- ${D}^{\alpha}{J}^{\alpha}f\left(x\right)=f\left(x\right),$
- ${J}^{\alpha}{D}^{\alpha}f\left(x\right)=f\left(x\right)-{\sum}_{k=0}^{m-1}{f}^{\left(k\right)}\left({0}^{+}\right)\frac{{x}^{k}}{k!},\phantom{\rule{1.em}{0ex}}for\phantom{\rule{1.em}{0ex}}x>0$.

#### 2.2. Stability Criterion

**Theorem 1**(See [16]) For a given commensurate fractional order system (3), the equilibria can be obtained by calculating $F\left(X\right)=0$. These equilibrium points are locally-asymptotically stable if all of the eigenvalues λ of the Jacobian matrix $J=\frac{\partial F}{\partial X}$ at the equilibrium points satisfy:

**Figure 1.**Stability region of the fractional order system (3).

#### 2.3. The Adams–Bashforth–Moulton Algorithm

## 3. A Fractional Order Economic System

#### 3.1. Dynamical Behavior

#### 3.2. Numerical Simulations

## 4. Active Control of the Fractional Order Chaotic System

**Theorem 2.**Starting from any initial condition, an equilibrium point ${E}_{i}$ of system (11) is asymptotically stable when the controller ${U}_{j}$, $j=1,2,3$, is active, for $\alpha \ge {\alpha}_{min}$.

**Proof.**As a Lyapunov candidate function associated with System (11), we consider the quadratic function defined by:

- For ${E}_{0}$:$$\left\{\begin{array}{c}U1:=-1.4x+0.98y+0.4x{y}^{2},\hfill \\ U2:=-y-49.9z,\hfill \\ U3:=-10x-z.\hfill \end{array}\right.$$
- For ${E}_{1}$:$$\left\{\begin{array}{c}{U}_{1}=0.904\phantom{\rule{0.166667em}{0ex}}x+1.02608\phantom{\rule{0.166667em}{0ex}}y-0.002304+0.4\phantom{\rule{0.166667em}{0ex}}x{y}^{2}+1.92\phantom{\rule{0.166667em}{0ex}}xy+0.0096\phantom{\rule{0.166667em}{0ex}}{y}^{2},\hfill \\ {U}_{2}=-y-49.9z,\hfill \\ {U}_{3}=-10x-z.\hfill \end{array}\right.$$
- For ${E}_{2}$:$$\left\{\begin{array}{c}{U}_{1}=0.904x+1.02608y+0.002304+0.4\phantom{\rule{0.166667em}{0ex}}x{y}^{2}-1.92xy-0.0096\phantom{\rule{0.166667em}{0ex}}{y}^{2},\hfill \\ {U}_{2}=-y-49.9z,\hfill \\ {U}_{3}=-10x-z.\hfill \end{array}\right.$$

**Figure 5.**Time histories of System (11) for x signal at the equilibrium E

_{0}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 6.**Time histories of system (11) for y signal at the equilibrium E

_{0}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 7.**Time histories of System (11) for z signal at the equilibrium E

_{0}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 8.**Time histories of System (11) for x signal at the equilibrium E

_{1}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 9.**Time histories of System (11) for y signal at the equilibrium E

_{1}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 10.**Time histories of System (11) for z signal at the equilibrium E

_{1}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 11.**Time histories of System (11) for x signal at the equilibrium E

_{2}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 12.**Time histories of System (11) for y signal at the equilibrium E

_{2}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

**Figure 13.**Time histories of System (11) for z signal at the equilibrium E

_{2}with α = 0.9: (

**a**) t

_{max}= 100, (

**b**) t

_{max}= 300.

## 5. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Baskonus, H.M.; Mekkaoui, T.; Hammouch, Z.; Bulut, H.
Active Control of a Chaotic Fractional Order Economic System. *Entropy* **2015**, *17*, 5771-5783.
https://doi.org/10.3390/e17085771

**AMA Style**

Baskonus HM, Mekkaoui T, Hammouch Z, Bulut H.
Active Control of a Chaotic Fractional Order Economic System. *Entropy*. 2015; 17(8):5771-5783.
https://doi.org/10.3390/e17085771

**Chicago/Turabian Style**

Baskonus, Haci Mehmet, Toufik Mekkaoui, Zakia Hammouch, and Hasan Bulut.
2015. "Active Control of a Chaotic Fractional Order Economic System" *Entropy* 17, no. 8: 5771-5783.
https://doi.org/10.3390/e17085771