Shen's -process on Berwald connection

The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections. Hence Tayebi–Najafi introduced two new processes called Shen's and -processes and showed that the Shen connection is obtained from the Chern connection by Shen's -process. In this paper, we study the Shen's - and -process on Berwald connection and introduce two new torsion-free connections in Finsler geometry. Then, we obtain all of Riemannian and non-Riemannian curvatures of these connections. Using it, we find the explicit form of -curvatures of these connections and prove that -curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian or Riemannian structures. As an application, we consider compact Finsler manifolds and obtain ODEs.

Jacobi Stability Analysis of the Chen System

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

ENERGY β-CONFORMAL CHANGE IN FINSLER GEOMETRY

The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ([Formula: see text] with [Formula: see text]; [Formula: see text] being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

PARALLEL π-VECTOR FIELDS AND ENERGY β-CHANGE

The present paper deals with an intrinsic investigation of the notion of a parallel π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a parallel π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text] being a parallel π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: The Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

CONCURRENT π-VECTOR FIELDS AND ENERGY β-CHANGE

The present paper deals with an intrinsic investigation of the notion of a concurrent π-vector field on the pullback bundle of a Finsler manifold (M, L). The effect of the existence of a concurrent π-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular β-change, namely the energy β-change ([Formula: see text]with[Formula: see text]; [Formula: see text] being a concurrent π-vector field), is established. The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection, and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.