On Some Properties and Symmetries of the 5-Dimensional Lorenz System

Abstract

The 5-dimensional Lorenz system for the gravity-wave activity is considered. Some stability problems and the existence of periodic orbits are studied. Also, a symplectic realization and some symmetries are given.

Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method

“…. The initial approximation is u 0 (t) = u(0) • cos(ω 0 t) + u(0) ω 0 • sin(ω 0 t), solution of the equation L u(t) = 0, with initial conditions given by Equation (21).…”

“…k will be optimally identified. In this way the approximate analytic solutions of the nonlinear problems Equations ( 15), ( 18), (21), can be constructed, via the OAFM method.…”

Symmetries and Properties of the Energy-Casimir Mapping in the Ball-Plate Problem

“…More precisely, taking into account the chain rule for the computation of,,̇1,̈1,̇2,̈2 and replacing̈1,̈2 by using (21), the relations obtained by applying the second prolongation of k on (21) become two equations in , 1 ,̇1, 2 ,̇2, which are all independent. These equations must be satisfied identically in , 1 ,̇1, 2 ,̇2, which leads to the finding of k. Detailed utilization of the aforementioned method can be found, for example, in [17,18]. Proposition 7.…”

Approximate Closed-Form Solutions for the Maxwell-Bloch Equations via the Optimal Homotopy Asymptotic Method

“…The Rikitake two-disk dynamo system was studied by [2] and applied in [3,4]. Other geometrical properties of the dynamical systems, such as the Hamiltonian realization, the equilibria points, and integrable deformations, were analyzed in [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”