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Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

Ndogmo J.-C.

Journal of Applied Analysis

2018

Vol. 24, nr 1

17--26

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Ndogmo J.-C.

Journal of Applied Analysis

2018

Vol. 24, nr 2

175--183

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Adiscussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

Lyapunov function for cosmological dynamical system

Szydłowski M., Krawiec A.

Demonstratio Mathematica

2017

Vol. 50, nr 1

51--55

We prove the asymptotic global stability of the de Sitter solution in the Friedmann-Robertson-Walker conservative and dissipative cosmology. In the proof we construct a Lyapunov function in an exact form and establish its relationship with the first integral of dynamical system determining evolution of the flat Universe. Our result is that de-Sitter solution is asymptotically stable solution for general form of equation of state p = p(p, H), where dependence on the Hubble function H means that the effect of dissipation are included.

First integrals of evolution systems and nonlinear stability of stationary solutions for the ideal atmospheric, oceanic, hydrodynamical and plasma models

Gordin V.

Journal of Technical Physics

1998

Vol. 39, no 2

213-219

First integrals of the systems of nonlinear equations governing the behaviour of atmospheric, oceanic and MHD plasma models are determined. The Lyapunov stability conditions for the solutions under small initial disturbances are analyzed. \