Stability for manifolds of the equilibrium state of fractional Birkhoffian systems

He, Jin–Man; Shao-Kai, Luo

doi:10.1007/s00707-015-1307-5

Cited by 19 publications

(5 citation statements)

References 50 publications

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“…When α → 1, β → 1, the fractional Euler-Poinsot model of a rigid body which rotates with respect to a fixed point (57) or (58) is reduced to the traditional Euler-Poinsot model (54). Based on Eqs.…”

Section: Application A: Fractional Euler-poinsot Model Of Rigid Body mentioning

confidence: 99%

“…Since then, the study of the basic theories and methods for fractional dynamics has become a hot topic, and various types of dynamical systems with fractional derivatives have been widely studied in the literature, such as fractional Lagrangian mechanics, fractional Hamiltonian mechanics and fractional nonholonomic mechanics [35][36][37][38][39][40][41][42][43][44][45][46][47]. Recently, we established fractional generalized Hamiltonian mechanics and fractional Birkhoffian mechanics and constructed their basic theoretical framework, respectively [48][49][50][51][52][53][54]. The basic theories and methods of fractional dynamics have been extensively applied in many fields of science and engineering.…”

Section: Introductionmentioning

confidence: 99%

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Fractional Nambu dynamics

Shao-Kai

2015

Acta Mech

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In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Nambu system with fractional derivatives (the fractional Nambu dynamics), and study its applications to mechanical systems. By using the definition of combined fractional derivative, we construct unified fractional Nambu equations and, respectively, give four new kinds of fractional Nambu equations based on the different definitions of fractional derivative. Further, we study the relations between the fractional Nambu system, the fractional generalized Hamiltonian system, the fractional Birkhoffian system, the fractional Hamiltonian system, the fractional Lagrangian system and a series of integer-order dynamical systems and present the transformation conditions. And furthermore, as applications of the fractional Nambu method, we construct two kinds of fractional dynamical models, which include the fractional Euler-Poinsot model of rigid body which rotates with respect to a fixed point and the fractional Yamaleev oscillator model. This work provides a general method for studying a fractional dynamical problem which is related to science and engineering.

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Section: Application A: Fractional Euler-poinsot Model Of Rigid Body mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional Nambu dynamics

Shao-Kai

2015

Acta Mech

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“…This important discovery caused the shock of science, and scientists began to study many problems about the dynamical system with fractional derivatives. Since then, the study of the basic theories and methods for fractional dynamics has become a hot topic, and won wide development in theories and applications [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Recently, we established the fractional generalized Hamiltonian mechanics, which include its gradient representation, Lie algebraic structure, generalized Poisson conservation law, variation equations, construction method of integral invariants, and so on [40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications

Shao-Kai

2015

Applied Mathematics and Computation

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“…Fractional order phenomenon is ubiquitous in the real world and has strong memory and hereditary characteristic, so fractional model can better describe the dynamical properties and internal structure of many classical problems than integer ones. In recent years, many valuable results of fractional order dynamical systems have been obtained and widely applied in many areas, such as mathematical physics [1][2][3][4][5][6][7][8][9][10][11], optimum theory [12], financial problems [13], anomalous diffusion [14], secure communication [15,16], biological systems [17,18], and heat transfer process [19]. These research works illustrate the practicality and importance of fractional calculus and promote its development.…”

Section: Introductionmentioning

confidence: 99%

Quasi‐Matrix and Quasi‐Inverse‐Matrix Projective Synchronization for Delayed and Disturbed Fractional Order Neural Network

Chen

2019

Complexity

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This paper is concerned with the quasi-matrix and quasi-inverse-matrix projective synchronization between two nonidentical delayed fractional order neural networks subjected to external disturbances. First, the definitions of quasi-matrix and quasi-inverse-matrix projective synchronization are given, respectively. Then, in order to realize two types of synchronization for delayed and disturbed fractional order neural networks, two sufficient conditions are established and proved by constructing appropriate Lyapunov function in combination with some fractional order differential inequalities. And their estimated synchronization error bound is obtained, which can be reduced to the required standard as small as what we need by selecting appropriate control parameters. Because of the generality of the proposed synchronization, choosing different projective matrix and controllers, the two synchronization types can be reduced to some common synchronization types for delayed fractional order neural networks, like quasi-complete synchronization, quasi-antisynchronization, quasi-projective synchronization, quasi-inverse projective synchronization, quasi-modified projective synchronization, quasi-inverse-modified projective synchronization, and so on. Finally, as applications, two numerical examples with simulations are employed to illustrate the efficiency and feasibility of the new synchronization analysis.

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Stability for manifolds of the equilibrium state of fractional Birkhoffian systems

Cited by 19 publications

References 50 publications

Fractional Nambu dynamics

Fractional Nambu dynamics

A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications

Quasi‐Matrix and Quasi‐Inverse‐Matrix Projective Synchronization for Delayed and Disturbed Fractional Order Neural Network

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