scholarly journals The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems

2018 ◽  
Vol 20 (3) ◽  
pp. 295-303
Author(s):  
Vladimir I. Nikonov
Keyword(s):  
Differential Operator ◽  
Lie Algebras ◽  
Stability Problem ◽  
Linear Differential Operator ◽  
Original System ◽  
First Integrals ◽  
Partial Stability ◽  
Stability Of Dynamical Systems ◽  
Linear Differential ◽  
The One

The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

scholarly journals The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems

2018 ◽  
Vol 20 (3) ◽  
pp. 295-303
Author(s):  
V. I. Nikonov
Keyword(s):  
Differential Operator ◽  
Lie Algebras ◽  
Stability Problem ◽  
Linear Differential Operator ◽  
Original System ◽  
First Integrals ◽  
Partial Stability ◽  
Stability Of Dynamical Systems ◽  
Linear Differential ◽  
The One

The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

Method of Forced Monotonicity for Conjugate type Boundary Value Problems for Ordinary Differential Equations

1988 ◽  
Vol 31 (1) ◽  
pp. 79-84
Author(s):  
P. W. Eloe ◽  
P. L. Saintignon
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Linear Differential ◽  
Type Boundary ◽  
Sign Conditions ◽  
Conjugate Type

AbstractLet I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

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