# American Institute of Mathematical Sciences

October  2017, 22(8): 3079-3090. doi: 10.3934/dcdsb.2017164

## An instability theorem for nonlinear fractional differential systems

 1 Institute of Mathematics, Vietnam Academy of Science and Technology, Viet Nam 2 Department of Mathematics, Hokkaido University, Japan 3 Department of Mathematics, Technische Universität Dresden, Dresden, Germany

Received  August 2016 Revised  October 2016 Published  June 2017

In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector
 $\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$
where
 $α∈ (0,1)$
is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.
Citation: Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3079-3090. doi: 10.3934/dcdsb.2017164
##### References:
 [1] R. Abu-Saris and Q. Al-Mdallal, On the asymptotic stability of linear system of fractionalorder difference equations, Fract. Calc. Appl. Anal., 16 (2013), 613-629.  doi: 10.2478/s13540-013-0039-2. [2] R. Agarwal, D. O'Regan and S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60 (2015), 653-676.  doi: 10.1007/s10492-015-0116-4. [3] E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542-553.  doi: 10.1016/j.jmaa.2006.01.087. [4] J. Audounet, D. Matignon and G. Montseny, Semi-linear diffusive representations for nonlinear fractional differential systems, Nonlinear control in the year 2000, Vol. 1 (Paris), Lecture Notes in Control and Inform. Sci. , 258, Springer, London, (2001), 73–82. doi: 10.1007/BFb0110208. [5] B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coefficients, Applied Mathematics and Computation, 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104. [6] J. Čermák, T. Kisela and L. Nechvátal, Stability regions for linear fractional differential systems and their discretizations, Applied Mathematics and Computation, 219 (2013), 7012-7022.  doi: 10.1016/j.amc.2012.12.019. [7] J. Čermák, I. Győri and L. Nechvátal, On explicit stability conditions for a linear fractional difference system, Fract. Calc. Appl. Anal., 18 (2015), 651-672.  doi: 10.1515/fca-2015-0040. [8] E. A. Coddington and N. Levinson, Theory of Differential Equations, McCrow–Hill, New York, 1955. [9] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for planar fractional differential equations, Applied Mathematics and Computation, 226 (2014), 157-168.  doi: 10.1016/j.amc.2013.10.010. [10] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, Linearized asymptotic stability for fractional differential equations, Electronic Journal of Qualitative Theory of Differential Equations, (2016), 1-13. [11] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for fractional differential equations in high dimensional spaces, Nonlinear Dynamics, 86 (2016), 1885-1894.  doi: 10.1007/s11071-016-3002-z. [12] W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1768-1777.  doi: 10.1016/j.na.2009.09.018. [13] K. Diethelm, The Analysis of Fractional Differential Equations. An Application–Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2. [14] J. Y. Kaminski, R. Shorten and E. Zeheb, Exact stability test and stabilization for fractional systems, Systems Control Lett., 85 (2015), 95-99.  doi: 10.1016/j.sysconle.2015.08.005. [15] P. Lancaster and M. Tismenetsky, The Theory of Matrices. Second Edition, Academic Press, San Diego, 1985. [16] C. Li and Y. Ma, Fractional dynamical system and its linearization theorem, Nonlinear Dynamics, 71 (2013), 621-633.  doi: 10.1007/s11071-012-0601-1. [17] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., 2 (1996), 963-968. [18] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [19] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc. , San Diego, CA, 1999. [20] S. G. Samko, A. A. Kilbas and O. I. Maritchev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Swizerland, 1993. [21] J. Shen and J. Lam, Non-existence of finite stable eqilibria in fractional-order nonlinear systems, Automatica, 50 (2014), 547-551.  doi: 10.1016/j.automatica.2013.11.018.

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##### References:
 [1] R. Abu-Saris and Q. Al-Mdallal, On the asymptotic stability of linear system of fractionalorder difference equations, Fract. Calc. Appl. Anal., 16 (2013), 613-629.  doi: 10.2478/s13540-013-0039-2. [2] R. Agarwal, D. O'Regan and S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math., 60 (2015), 653-676.  doi: 10.1007/s10492-015-0116-4. [3] E. Ahmed, A. M. A. El-Sayed and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542-553.  doi: 10.1016/j.jmaa.2006.01.087. [4] J. Audounet, D. Matignon and G. Montseny, Semi-linear diffusive representations for nonlinear fractional differential systems, Nonlinear control in the year 2000, Vol. 1 (Paris), Lecture Notes in Control and Inform. Sci. , 258, Springer, London, (2001), 73–82. doi: 10.1007/BFb0110208. [5] B. Bonilla, M. Rivero and J. J. Trujillo, On systems of linear fractional differential equations with constant coefficients, Applied Mathematics and Computation, 187 (2007), 68-78.  doi: 10.1016/j.amc.2006.08.104. [6] J. Čermák, T. Kisela and L. Nechvátal, Stability regions for linear fractional differential systems and their discretizations, Applied Mathematics and Computation, 219 (2013), 7012-7022.  doi: 10.1016/j.amc.2012.12.019. [7] J. Čermák, I. Győri and L. Nechvátal, On explicit stability conditions for a linear fractional difference system, Fract. Calc. Appl. Anal., 18 (2015), 651-672.  doi: 10.1515/fca-2015-0040. [8] E. A. Coddington and N. Levinson, Theory of Differential Equations, McCrow–Hill, New York, 1955. [9] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for planar fractional differential equations, Applied Mathematics and Computation, 226 (2014), 157-168.  doi: 10.1016/j.amc.2013.10.010. [10] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, Linearized asymptotic stability for fractional differential equations, Electronic Journal of Qualitative Theory of Differential Equations, (2016), 1-13. [11] N. D. Cong, T. S. Doan, S. Siegmund and H. T. Tuan, On stable manifolds for fractional differential equations in high dimensional spaces, Nonlinear Dynamics, 86 (2016), 1885-1894.  doi: 10.1007/s11071-016-3002-z. [12] W. Deng, Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 72 (2010), 1768-1777.  doi: 10.1016/j.na.2009.09.018. [13] K. Diethelm, The Analysis of Fractional Differential Equations. An Application–Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2. [14] J. Y. Kaminski, R. Shorten and E. Zeheb, Exact stability test and stabilization for fractional systems, Systems Control Lett., 85 (2015), 95-99.  doi: 10.1016/j.sysconle.2015.08.005. [15] P. Lancaster and M. Tismenetsky, The Theory of Matrices. Second Edition, Academic Press, San Diego, 1985. [16] C. Li and Y. Ma, Fractional dynamical system and its linearization theorem, Nonlinear Dynamics, 71 (2013), 621-633.  doi: 10.1007/s11071-012-0601-1. [17] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., 2 (1996), 963-968. [18] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [19] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc. , San Diego, CA, 1999. [20] S. G. Samko, A. A. Kilbas and O. I. Maritchev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Swizerland, 1993. [21] J. Shen and J. Lam, Non-existence of finite stable eqilibria in fractional-order nonlinear systems, Automatica, 50 (2014), 547-551.  doi: 10.1016/j.automatica.2013.11.018.
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