# Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Shimizu–Morioka System

#### 2.1. Integer Order Shimizu–Morioka System

#### 2.2. Fractional Order Shimizu–Morioka System

**Definition**

**1.**

## 3. Optimal Routh–Hurwitz Conditions for Fractional System

**Theorem**

**1.**

- (i)
- $a>0,\phantom{\rule{3.33333pt}{0ex}}b>0,\phantom{\rule{3.33333pt}{0ex}}0<c<{c}^{-}(a,b;\alpha )$.
- (ii)
- $a>0,\phantom{\rule{3.33333pt}{0ex}}\widehat{b}\le b\le 0,\phantom{\rule{3.33333pt}{0ex}}{c}^{+}(a,b;\alpha )<c<{c}^{-}(a,b;\alpha )$.
- (iii)
- $a\le 0,\phantom{\rule{3.33333pt}{0ex}}b>\overline{b}(a;\alpha ),\phantom{\rule{3.33333pt}{0ex}}0<c<{c}^{-}(a,b;\alpha )$.

**Proof.**

## 4. Stability Analysis of the Fractional Shimizu–Morioka System

**Theorem**

**2.**

**Theorem**

**3.**

## 5. Numerical Results

## 6. Conclusions

- For the $\sigma =[0,\surd 2]$, the new optimal Routh–Hurwitz conditions enable us to detect the critical value of $\alpha $ for the stability criterion of fractional Shimizu–Morioka system when we use $\beta $ as control parameter.
- Furthermore, we introduce the way to calculate the range of adjustable control parameter $\beta $ to obtain the stability criterion for the fractional Shimizu–Morioka system.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Shimizu, T.; Morioka, N. On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model. Phys. Lett. A
**1980**, 76, 201–204. [Google Scholar] [CrossRef] - Shil’nikov, A.L. On bifurcations of the Lorenz attractor in the Shimizu–Morioka model. Phys. D Nonlinear Phenom.
**1993**, 62, 338–346. [Google Scholar] [CrossRef] - Tigan, G.; Turaev, D. Analytical search for homoclinic bifurcations in the Shimizu–Morioka model. Phys. D Nonlinear Phenom.
**2011**, 240, 985–989. [Google Scholar] - Llibre, J.; Pessoa, C. The Hopf bifurcation in the Shimizu–Morioka system. Nonlinear Dyn.
**2015**, 79, 2197–2205. [Google Scholar] [CrossRef] - Capiński, M.J.; Turaev, D.; Zgliczyński, P. Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system. Nonlinearity
**2018**, 31, 5410. [Google Scholar] [CrossRef] - Akinlar, M.A.; Secer, A.; Bayram, M. Stability, synchronization control and numerical solution of fractional Shimizu–Morioka dynamical system. Appl. Math. Inf. Sci.
**2014**, 8, 1699. [Google Scholar] [CrossRef] - Danca, M.F.; Garrappa, R. Suppressing chaos in discontinuous systems of fractional order by active control. Appl. Math. Comput.
**2015**, 257, 89–102. [Google Scholar] [CrossRef] - Čermák, J.; Nechvátal, L. The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system. Nonlinear Dyn.
**2017**, 87, 939–954. [Google Scholar] [CrossRef] - Ahmed, E.; El-Sayed, A.M.A.; El-Saka, H.A. On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. Lett. A
**2006**, 358, 1–4. [Google Scholar] [CrossRef] - Čermák, J.; Nechvátal, L. Local bifurcations and chaos in the fractional Rössler system. Int. J. Bifurc. Chaos
**2018**, 28, 1850098. [Google Scholar] [CrossRef] - Li, C.; Ma, Y. Fractional dynamical system and its linearization theorem. Nonlinear Dyn.
**2013**, 71, 621–633. [Google Scholar] [CrossRef] - Wang, Z.; Yang, D.; Ma, T.; Sun, N. Stability analysis for nonlinear fractional-order systems based on comparison principle. Nonlinear Dyn.
**2014**, 75, 387–402. [Google Scholar] [CrossRef] - Li, H.L.; Muhammadhaji, A.; Zhang, L.; Teng, Z. Stability analysis of a fractional-order predator–prey model incorporating a constant prey refuge and feedback control. Adv. Differ. Equ.
**2018**, 2018, 325. [Google Scholar] [CrossRef] - Li, H.; Cheng, J.; Li, H.B.; Zhong, S.M. Stability analysis of a fractional-order linear system described by the Caputo–Fabrizio derivative. Mathematics
**2019**, 7, 200. [Google Scholar] [CrossRef] - Wu, X.; Wang, H.; Lu, H. Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication. Nonlinear Anal. Real World Appl.
**2012**, 13, 1441–1450. [Google Scholar] [CrossRef] - Gao, Z. Robust stability criterion for fractional-order systems with interval uncertain coefficients and a time-delay. ISA Trans.
**2015**, 58, 76–84. [Google Scholar] [CrossRef] - Liang, S.; Wang, S.G.; Wang, Y. Routh-type table test for zero distribution of polynomials with commensurate fractional and integer degrees. J. Frankl. Inst.
**2017**, 354, 83–104. [Google Scholar] [CrossRef] - Wiggers, S.L.; Pedersen, P. Routh–Hurwitz-Liénard–Chipart criteria. In Structural Stability and Vibration; Springer: Cham, Switzerland, 2018; pp. 133–140. [Google Scholar]
- Shen, Y.; Wang, Y.; Yuan, N. A graphical approach for stability and robustness analysis in commensurate and incommensurate fractional-order systems. Asian J. Control
**2019**. [Google Scholar] [CrossRef] - Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn.
**2002**, 29, 3–22. [Google Scholar] [CrossRef] - Abdullah, F.A.; Liu, F.; Burrage, P.; Burrage, K.; Li, T. Novel analytical and numerical techniques for fractional temporal SEIR measles model. Numer. Algorithms
**2018**, 79, 19–40. [Google Scholar] [CrossRef] - Moustafa, M.; Mohd, M.H.; Ismail, A.I.; Abdullah, F.A. Dynamical analysis of a fractional-order Rosenzweig–MacArthur model incorporating a prey refuge. Chaos Solitons Fractals
**2018**, 109, 1–13. [Google Scholar] [CrossRef] - Toh, Y.T.; Phang, C.; Loh, J.R. New predictor-corrector scheme for solving nonlinear differential equations with Caputo-Fabrizio operator. Math. Methods Appl. Sci.
**2019**, 42, 175–185. [Google Scholar] [CrossRef]

$\mathit{\alpha}$ | Range of $\mathit{\beta}$ | Stability Condition |
---|---|---|

$\alpha >{\alpha}_{cr}$ | Depend on Equation (30) | Stable |

$\alpha <{\alpha}_{cr}$ | $0<\beta <\infty $ | stable |

**Table 2.**The stability condition for the system under certain range of $\beta $ when ${\alpha}_{cr}=0.9001093006$.

$\mathit{\alpha}$ | Range of $\mathit{\beta}$ | Stability Condition |
---|---|---|

$0.95>{\alpha}_{cr}$ | $[0,0.01114995609],[1.025145474,\infty ]$ | stable |

$0.901>{\alpha}_{cr}$ | $[0,0.1393510901],[0.2364214161,\infty ]$ | stable |

$0.89<{\alpha}_{cr}$ | $[0,\infty ]$ | stable |

$0.85<{\alpha}_{cr}$ | $[0,\infty ]$ | stable |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ng, Y.X.; Phang, C.
Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions. *Computation* **2019**, *7*, 23.
https://doi.org/10.3390/computation7020023

**AMA Style**

Ng YX, Phang C.
Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions. *Computation*. 2019; 7(2):23.
https://doi.org/10.3390/computation7020023

**Chicago/Turabian Style**

Ng, Yong Xian, and Chang Phang.
2019. "Computation of Stability Criterion for Fractional Shimizu–Morioka System Using Optimal Routh–Hurwitz Conditions" *Computation* 7, no. 2: 23.
https://doi.org/10.3390/computation7020023