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For a disturbed dynamical system that can be transformed into fractional Birkhoffian representation with all dynamical information, under a more general fractional infinitesimal transformation of Lie group with high time extension and fractional extension, a new kind of fractional Mei symmetrical perturbation method which is most universal significance is presented and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant; as the special cases of new method, we, respectively, reveal an autonomous disturbed fractional Birkhoffian system possesses more adiabatic invariants, a new kind of non-Noether exact invariant directly led by fractional Mei symmetry and a new kind of non-Noether exact and adiabatic invariant of integer Birkhoffian systems. Also, as the new method’s applications to nonlinear dynamical problems, we, respectively, explore the symmetrical perturbation and adiabatic invariant of a disturbed fractional general relativistic Buchdahl model and a disturbed fractional Emden model. It is worth pointing out that, for a disturbed dynamical system, this work reveals intrinsic relation between the fractional symmetrical perturbation and the adiabatic invariant, and provides a general method for finding adiabatic invariants of an actual disturbed fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

For a dynamical system that can be transformed into fractional Birkhoffian representation, under a more general kind of fractional infinitesimal transformation of Lie group, we present the fractional conformal invariance method and it is found that, using the new method, we can find a new kind of non-Noether conserved quantity; and we find that, as a special case, an autonomous fractional Birkhoffian system possesses more conserved quantities. Also, as the fractional conformal invariance method’s applications, we, respectively, explore the conformal invariance and conserved quantities of a fractional Lotka biochemical oscillator and a fractional Hojman–Urrutia model. This work constructs a basic theoretical framework of fractional conformal invariance method, and provides a general method for finding conserved quantities of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

In this paper we present a new method for the equilibrium stability of a dynamical system, i.e., the fractional generalized Hamilton method. We reveal the uncertainty and its mathematical representation for the fractional and nonlinear problem and find a general method of constructing a family of fractional dynamical models. And, six criterions of fractional generalized Hamilton method of equilibrium stability are presented. As applications, we explore the equilibrium stability of a fractional Duffing oscillator model and a fractional Whittaker model. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the paper, we present fractional generalized Hamiltonian method of dynamical stability, in terms of Riesz–Riemann–Liouville derivative, and study its applications. For an actual dynamical system, the fractional generalized Hamiltonian method of constructing a fractional dynamical model is given, and then the six criterions for fractional generalized Hamiltonian method of dynamical stability are presented. As applications, by using the fractional generalized Hamiltonian method, we construct five kinds of actual fractional dynamical models, which include a fractional Euler–Poinsot model of rigid body that rotates with respect to a fixed-point, a fractional Hojman–Urrutia model, a fractional Lorentz–Dirac model, a fractional Whittaker model and a fractional Robbins–Lorenz model, and we explore the dynamical stability of these models, respectively. This work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]