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

An instability theorem for nonlinear fractional differential systems

Nguyen Dinh Cong 1, , Doan Thai Son 1,2, , Stefan Siegmund 3, and  Hoang The Tuan 1,

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.
Keywords: Fractional differential equations, qualitative theory, stability theory, instability condition.
Mathematics Subject Classification: 34A08, 34K20.
Citation: Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. A. Coddington and N. Levinson, Theory of Differential Equations, McCrow–Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Lancaster and M. Tismenetsky, The Theory of Matrices. Second Edition, Academic Press, San Diego, 1985.  Google Scholar

[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.  Google Scholar

[17]

D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., 2 (1996), 963-968.   Google Scholar

[18]

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. A. Coddington and N. Levinson, Theory of Differential Equations, McCrow–Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Lancaster and M. Tismenetsky, The Theory of Matrices. Second Edition, Academic Press, San Diego, 1985.  Google Scholar

[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.  Google Scholar

[17]

D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., 2 (1996), 963-968.   Google Scholar

[18]

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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