时不变分数阶系统反周期解的存在性

杨绪君; 宋乾坤

doi:10.3879/j.issn.1000-0887.2014.06.010

时不变分数阶系统反周期解的存在性

doi: 10.3879/j.issn.1000-0887.2014.06.010

杨绪君¹,
宋乾坤²

¹重庆交通大学信息科学与工程学院, 重庆 400074;²重庆交通大学理学院,重庆 400074

基金项目: 国家自然科学基金(61273021);重庆市自然科学基金（重点）(CQcstc2013jjB40008)

详细信息

作者简介:
杨绪君(1989—)，男，江苏徐州人，硕士生(E-mail: xujunyangcquc@163.com)

中图分类号: O175.13
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- 被引次数: 0
出版历程
- 收稿日期: 2014-03-24
- 修回日期: 2014-05-06
- 刊出日期: 2014-06-11

On the Existence of Anti-Periodic Solutions in Time-Invariant Fractional Order Systems

YANG Xu-jun¹,
SONG Qian-kun²

¹School of Information Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, P.R.China;²School of Science, Chongqing Jiaotong University, Chongqing 400074, P.R.China

Funds: The National Natural Science Foundation of China(61273021)

摘要

摘要: 反周期解问题是非线性微分系统动力学的重要特征.近年来，非线性整数阶微分系统的反周期解问题得到了广泛的研究，非线性分数阶微分系统的反周期解问题也得到了初步的讨论.不同于已有的工作，该文研究时不变分数阶系统反周期解的存在性问题.证明了时不变分数阶系统在有限时间区间内不存在反周期解，而当分数阶导数的下限趋近于无穷大时，时不变分数阶系统却存在反周期解.
- 分数阶微积分 /
- 分数阶系统 /
- 反周期解
Abstract: The anti-periodic solution problem makes an important characteristic of dynamics for nonlinear differential systems. In recent years, the anti-periodic solution problem in integer order nonlinear differential systems had been widely studied, while the anti-periodic solution problem in fractional order nonlinear differential systems had been preliminarily discussed. Other than the previous work, the existence of anti-periodic solutions in time-invariant fractional order systems was investigated. It is shown that although within a finite time interval the solutions do not show any anti-periodic behavior, when the lower limit of the fractional order derivative tends to infinity the anti-periodic orbits will be obtained.
- fractional order calculus /
- fractional order system /
- anti-periodic solution

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参考文献(22)

[1]	Butzer P L, Engels W,Wipperfürth U. An extension of the dyadic calculus with fractional order derivatives. further theory and applications[J].Computers & Mathematics With Applications,1986,12A(8): 921-943.
[2]	Diethelm K, Ford N J. Analysis of fractional differential equations[J].Journal of Mathematical Analysis and Applications,2002,265(2): 229-248.
[3]	Tavazoei M S, Haeri M. Describing function based methods for predicting chaos in a class of fractional order differential equations[J].Nonlinear Dynamics,2009,57(3): 363-373.
[4]	Tavazoei M S, Haeri M. A proof for non existence of periodic solutions in time invariant fractional order systems[J].Automatica,2009,45(8): 1886-1890.
[5]	Tavazoei M S. A note on fractional-order derivatives of periodic functions[J].Automatica,2010,46(5): 945-948.
[6]	Yazdani M, Salarieh H. On the existence of periodic solutions in time-invariant fractional order systems[J].Automatica,2011,47(8): 1834-1837.
[7]	Kaslik E, Sivasundaram S. Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions[J].Nonlinear Analysis: Real World Applications,2012,13(3): 1489-1497.
[8]	Rahimi M A, Salarieh H, Alasty A. Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory[J].Applied Mathematical Modelling,2012,36(3): 863-877.
[9]	Wang J R, Feckan M, Zhou Y. Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations[J].Communications in Nonlinear Science and Numerical Simulation,2013,18(2): 246-256.
[10]	Okochi H. On the existence of periodic solutions to nonlinear abstract parabolic equations[J].Journal of the Mathematical Society of Japan,1988,40(3): 541-553.
[11]	Haraux A. Anti-periodic solutions of some nonlinear evolution equations[J].Manuscripta Mathematica,1989,63(4): 479-505.
[12]	Okochi H. On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains[J].Nonlinear Analysis: Theory, Methods & Applications,1990,14(9): 771-783.
[13]	Chen Y Q. Anti-periodic solutions for semilinear evolution equations[J].Journal of Mathematical Analysis and Applications,2006,315(1): 337-348.
[14]	Liu B W. Anti-periodic solutions for forced Rayleigh-type equations[J].Nonlinear Analysis: Real World Applications,2009,10(8): 2850-2856.
[15]	Liu A M, Feng C H. Anti-periodic solutions for a kind of high order differential equations with multi-delay[J].Communications in Mathematical Analysis,2011,11(1): 137-150.
[16]	Liu Y J. Anti-periodic solutions of nonlinear first order impulsive functional differential equations[J].Mathematica Slovaca,2012,62(4): 695-720.
[17]	Tian Y, Henderson J. Anti-periodic solutions of higher order nonlinear difference equations: a variational approach[J].Journal of Difference Equations and Applications,2013,19(8): 1380-1392.
[18]	Ahmad B, Nieto J J. Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory[J].Topological Methods in Nonlinear Analysis,2010,35(2): 295-304.
[19]	Ahmad B, Nieto J J. Existence of solutions for impulsive anti-periodic boundary value problems of fractional order[J].Taiwanese Journal of Mathematics,2011,15(3): 981-993.
[20]	Chen A P, Chen Y. Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations [J].Differential Equations and Dynamical Systems,2011,19(3): 237-252.
[21]	Cernea A. On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions[J].Journal of Applied Mathematics and Computing,2012,38(1/2): 133-143.
[22]	Kilbas A, Srivastava H M,Trujillo J J.Theory and Applications of Fractional Differential Equations(North-Holland Mathematics Studies 204) [M]. Amsterdam: Elsevier, 2006.