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In this paper, we present a new method of fractional dynamics, i.e., the fractional Mei symmetrical perturbation method of a disturbed system, and explore the adiabatic invariant directly led by the perturbation. For a dynamical system which is disturbed by small forces of perturbation, the disturbed fractional generalized Hamiltonian equation is investigated and, under a more general kind of fractional infinitesimal transformation of a Lie group, the fractional Mei symmetrical definition and determining equation of a disturbed dynamical system are given; then, the determining equation of fractional Mei symmetrical perturbation is obtained. In particular, we present the fractional Mei symmetrical perturbation method of a disturbed dynamical system and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant directly led by the perturbation. As special cases, we obtain a new fractional Mei symmetrical conservation law of the undisturbed dynamical system and a Mei symmetrical perturbation theorem of the disturbed integer dynamical system. Also, as the fractional Mei symmetrical perturbation method’s applications, we find the adiabatic invariants of a disturbed fractional Duffing oscillator and a disturbed fractional Lotka biochemical oscillator. This work constructs a basic theoretical framework of fractional Mei symmetrical perturbation method and provides a general method of fractional dynamics.

In this paper, we reveal the uncertainty and its fractional generalized Hamiltonian representation for the fractional and nonlinear problem and find general methods of constructing a family of fractional dynamical models. By using the definition of combined fractional derivative, we construct a unified fractional generalized Hamiltonian equation, a fractional generalized Hamiltonian equation with combined Riemann–Liouville derivative, and a fractional generalized Hamiltonian equation with combined Caputo derivative; also, as special cases, under the different definitions of fractional derivatives, we, respectively, obtain a series of different kinds of fractional generalized Hamiltonian equations. In particular, we present a new concept of the family of fractional dynamical models, and it is found that, using the fractional generalized Hamiltonian method, we can construct a series of families of fractional dynamical models. And then, as the new method’s application, we construct three new kinds of families of fractional dynamical models, which include a family of fractional Lorentz–Dirac models, a family of fractional Lotka–Volterra models and a family of fractional Henon–Heiles models.

In this paper, we present a general method of finding conserved quantities, i.e., fractional Jacobi last multiplier method, and explore its applications to fractional dynamical system. Based on the definition of Riesz–Riemann–Liouville fractional derivative, we study general fractional dynamical equations, construct its fractional Jacobi last multiplier, and, respectively, give the determining equation and three important properties of the multiplier. And then, we present the fractional Jacobi last multiplier method, which includes three theorems of finding conserved quantities of fractional dynamical systems. Further, in fractional framework, we explore the relation between Lie symmetry and Jacobi last multiplier. Furthermore, the fractional Jacobi last multiplier method is applied to the fractional generalized Hamiltonian system, the complete fractional Hamiltonian system, the standard fractional Hamiltonian system, the fractional Nambu system and the fractional Birkhoffian system, and five corresponding propositions are given. Also by using the fractional Jacobi last multiplier method, we respectively find the conserved quantities of a fractional relativistic Buchdahl model, a fractional Euler–Poinsot model and a fractional Duffing oscillator model. This work constructs a basic theoretical framework of fractional Jacobi last multiplier method, and provides a general method of fractional dynamics.

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.

In the paper, we present a new stability theory of fractional dynamics, i.e., the stability for manifolds of equilibrium state of a fractional Birkhoffian system, in terms of Riesz derivatives, and explore its applications. For an autonomous fractional Birkhoffian system, the gradient representation and second-order gradient representation are studied, and the conditions under which the system can be considered as a gradient system and a second-order gradient system are given, respectively. Then, equilibrium equations, perturbation equations and first approximate equations are obtained, and the stability theorem for manifolds of equilibrium state of the general autonomous system is used to a fractional Birkhoffian system, and three criterions on the stability for manifolds of the equilibrium state of the system are investigated. As special cases of the fractional Birkhoffian method, the three criterions can also be applied to a traditional Birkhoffian system, a fractional Hamiltonian system and a classical Hamiltonian system, respectively. As applications, by using the fractional Birkhoffian method of this paper, we construct three kinds of fractional dynamical models, which include a four-dimensional fractional dynamical model, a fractional Henon–Heiles model and a fractional Lorentz–Dirac model, and we explore the stability for manifolds of the equilibrium state of these models, respectively. This work provides a general method for studying the dynamic stability of an actual fractional dynamical model that is related to science and engineering.

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Henon–Heiles model and the fractional Emden model.

Based on definition of Riemann–Liouville fractional derivatives, we find that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the generalized Poisson conservation law of the fractional generalized Hamiltonian system is investigated. As special cases of this paper, the conditions under which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system and a classical Hamiltonian system are given, and the Lie algebraic structure and Poisson conservation law of these special dynamical systems are obtained, respectively. Then, by using the method and results of this paper, we construct three kinds of new fractional dynamical model, i.e. a fractional Henon–Heiles model, a fractional Euler–Poinsot model of a rigid body and a fractional Volterra model of the three species groups, and their Poisson conserved quantities are obtained.

In this paper, we present a new fractional theory of dynamics, i.e., the dynamics of generalized Hamiltonian system with fractional derivatives (fractional generalized Hamiltonian mechanics). Based on the definition of Riemann–Liouville fractional derivatives, the fractional generalized Hamiltonian equations are obtained, the gradient representation and second-order gradient representation of the fractional generalized Hamiltonian system are studied, and then the conditions on which the system can be considered as a gradient system and a second-order gradient system are given, respectively. By using the method and results of this paper, the conditions under which a fractional generalized Hamiltonian equation can be reduced to a generalized Hamiltonian equation, a fractional Hamiltonian equation and a Hamiltonian equation are given, respectively, and then the existing conditions and their form of gradient equation and second-order gradient equation are investigated. Finally, an example of a fractional dynamical system is given to illustrate the method and results of the application.

For a generalized Hamiltonian system, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. And a number of important relationships of the Lie symmetrical method for a generalized Hamiltonian system are investigated, which reveal the interior properties of the system. By using the relationships, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for generalized Hamiltonian systems is presented. The basic integral variable relation not only can be used for a linear dynamic system but can also be used for a nonlinear dynamic system. The new conserved quantity is constructed in terms of infinitesimal generators of Lie symmetry and the interior properties of the system itself without solving the structural equation that may be very difficult to solve. Then, the method is applied in the generalized Hamiltonian system of even dimensions and the Hamiltonian system, respectively. Furthermore, the relationship between the generalized Hamiltonian system and a Birkhoffian system is studied, and a Lie symmetrical conservation law for a semiautonomous (or autonomous) Birkhoffian system is obtained. Finally, one example is given to illustrate the method and results of the application.

For a dynamical system in phase space, a new Lie symmetrical method to find a conserved quantity is presented in a general infinitesimal transformation of Lie groups. Based on the invariance of the differential equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, an important relationship that reveals the interior properties of a dynamical system in phase space is given. By using the relationship, a Lie symmetrical basic integral variable relation and a new Lie symmetrical conservation law for the dynamical system in phase space are given. The new conserved quantity is constructed in terms of the infinitesimal generators of Lie symmetry and the system itself without solving the structural equation. Furthermore, the method is applied in the Hamiltonian system, the nonconservative Hamiltonian system and the nonholonomic Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.