Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Li, Yan; Chen, YangQuan; Podlubný, Igor

doi:10.1016/j.camwa.2009.08.019

Cited by 1,353 publications

(667 citation statements)

References 6 publications

Supporting

Mentioning

656

Contrasting

Unclassified

Order By: Relevance

“…The following definitions and Lemmas for the next consideration are easily based on the results of Ref. [12,13,14,20]. Definition 2.1.…”

Section: Problem Formulationmentioning

confidence: 99%

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Zhang¹,

Liu²

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

The full state hybrid projective synchronization (FSHPS) definition for variable-order fractional chaotic/hyperchaotic systems with nonlinear external disturbances and unknown parameters is firstly presented. Then by introducing a compensator and a nonlinear controller, the FSHPS scheme is generated to eliminate the influence of nonlinear external disturbances effectively. Moreover, the parameters are estimated validly. Based on these control methods, appropriate parameters and controller to achieve FSHPS for the variable-order fractional chaotic/hyperchaotic systems are chosen impactfully. Simulations of variable-order fractional Chen and Lü system and fractional order hyperchaotic Lorenz system in the sense of FSHPS are performed and results show the effectiveness of our method.

show abstract

“…The following definitions and Lemmas for the next consideration are easily based on the results of Ref. [12,13,14,20]. Definition 2.1.…”

Section: Problem Formulationmentioning

confidence: 99%

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Zhang¹,

Liu²

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

show abstract

“…Recently, the problems of fractional calculus have been paid more and more attention since fractional operators have been broadly applied in more and more scientific *Corresponding Author: Hongyong Yang: School of Information and Electrical Engineering, Ludong University, Yantai 264025, China; Email: hyyang@yeah.net Fujun Han, Mei Zhao, Shuning Zhang, Jun Yue: School of Information and Electrical Engineering, Ludong University, Yantai 264025, China fields, for examples: in mechanics, physics, material science, informatics and engineering [1][2][3][4][5][6][7]. Not only can fractional order controllers be used to add robustness of control systems, but controlled more easily than that of integer-order systems [8,9].…”

Section: Introductionmentioning

confidence: 99%

“…In the actual situation, many dynamical systems can not be depicted with the integer-order differential equation and only be characterized with the fractional-order differentials [1,2,8]. For example, flocking movement by means of the individual secretions, food searching of microbial in the complex environments, formation of submarine robots in the seabed with a mass number of microorganisms [3,9].…”

Section: Introductionmentioning

confidence: 99%

Distributed containment control of heterogeneous fractional-order multi-agent systems with communication delays

Yang¹,

Han²,

Zhao³

et al. 2017

Open Physics

View full text Add to dashboard Cite

Because many networked systems can only be characterized with fractional-order dynamics in complex environments, fractional-order calculus has been studied deeply recently. When diverse individual features are shown in different agents of networked systems, heterogeneous fractional-order dynamics will be used to describe the complex systems. Based on the distinguishing properties of agents, heterogeneous fractional-order multi-agent systems (FOMAS) are presented. With the supposition of multiple leader agents in FOMAS, distributed containment control of FOMAS is studied in directed weighted topologies. By applying Laplace transformation and frequency domain theory of the fractional-order operator, an upper bound of delays is obtained to ensure containment consensus of delayed heterogenous FOMAS. Consensus results of delayed FOMAS in this paper can be extended to systems with integer-order models. Finally, numerical examples are used to verify our results.

show abstract

“…In consequence, many meaningful results in these fields have been obtained. See [1,2,3,4,5,12,13,15] for a good overview.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions for a class of fractional differential coupled system with integral boundary value conditions

Deng-feng¹,

Liu²

2016

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

This paper investigates the existence of positive solutions for the following high-order nonlinear fractional differential boundary value problem (BVP, for short)where n − 1 < α ≤ n, n ≥ 3, 0 ≤ λ < 2, D α 0 + is the Caputo fractional derivative. By using the monotone method, the theory of fixed point index on cone for differentiable operators and the properties of Green's function, some new uniqueness and existence criteria for the considered fractional BVP are established. As applications, some examples are worked out to demonstrate the main results. c 2016 All rights reserved.Keywords: Fractional differential equations, differentiable operators, fixed point index theorems on cone, integral boundary value conditions. 2010 MSC: 34A08, 34B15, 34B18.

show abstract

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability

Cited by 1,353 publications

References 6 publications

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Full state hybrid projective synchronization of variable-order fractional chaotichyperchaotic systems with nonlinear external disturbances and unknown parameters

Distributed containment control of heterogeneous fractional-order multi-agent systems with communication delays

Positive solutions for a class of fractional differential coupled system with integral boundary value conditions

Contact Info

Product

Resources

About