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Integral conditions for nonuniform $μ$-dichotomy on the half-line
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 |
$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$ |
$α∈ (0,1)$ |
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. |
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. |
[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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