Bifurcation for the generalized Birkhoffian system

Abstract

The system described by the generalized Birkhoff equations is called a generalized Birkhoffian system. In this paper, the condition under which the generalized Birkhoffian system can be a gradient system is given. The stability of equilibrium of the generalized Birkhoffian system is discussed by using the properties of the gradient system. When there is a parameter in the equations, its influences on the stability and the bifurcation problem of the system are considered.

Skew-gradient representations of constrained mechanical systems

“…(14) has been nondimensionalized, and we try to transform it into a skew-gradient system. The differential equations of motion arë…”

Perturbation to Noether Symmetries and Adiabatic Invariants for Birkhoffian Systems

“…Notice that there is the relationship [8] between the nonisochronous variation Δ and the isochronous variation as follows:…”

“…Theorem 5. For the nonconservative Lagrangian system (8) under the El-Nabulsi dynamics model, if there is a gauge function ( , ,̇) that makes the generators 0 ( , ,̇) and ( , ,̇) of infinitesimal transformations (9) satisfy the following conditions, 0 + +̇(̇−̇̇0 )…”

“…Mei pointed out that the Birkhoffian mechanics is a new development after quantum mechanics appears [6]. The study on the dynamics of Birkhoffian system has made a series of important results [7,8].…”

“…For the nonconservative Lagrange system(8), which is disturbed by the small perturbation , if there is a gauge function ( , ,̇) that makes the generators 0 ( , ,̇) and ( , ,̇) of infinitesimal transformations satisfy (cosh − cosh ) 1− , ( = 0, 1, 2, . .…”