Certain iteration methods for the determination of periodic solutions of nonautonomous systems of differential equations

1985 ◽  
Vol 36 (3) ◽  
pp. 310-316 ◽  
Author(s):  
A. M. Samoilenko ◽  
K. Kenzhebaev ◽  
V. N. Laptinskii
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Nonautonomous Systems ◽  
Systems Of Differential Equations ◽  
Iteration Methods

scholarly journals On the periodic solutions of linear homogenous systems of differential equations

1982 ◽  
Vol 5 (2) ◽  
pp. 305-309
Author(s):  
A. K. Bose
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Fundamental Matrix ◽  
Solution Space ◽  
Order System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Scalar Matrix ◽  
Systems Of Differential Equations ◽  
Necessary And Sufficient

Given a fundamental matrixϕ(x)of ann-th order system of linear homogeneous differential equationsY′=A(x)Y, a necessary and sufficient condition for the existence of ak-dimensional(k≤n)periodic sub-space (of periodT) of the solution space of the above system is obtained in terms of the rank of the scalar matrixϕ(t)−ϕ(0).

scholarly journals V. On periodic solutions of Hamiltonian systems differential equations

1928 ◽  
Vol 227 (647-658) ◽  
pp. 137-221 ◽  
Keyword(s):  
Hamiltonian System ◽  
Differential Equations ◽  
Ab Initio ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Celestial Mechanics ◽  
Divergent Series ◽  
Hamiltonian Form ◽  
Systems Of Differential Equations ◽  
Independent Variable

In the following pages it is proposed to develop ab initio a theory of periodic solutions of Hamiltonian systems of differential equations. Such solutions are of theoretical importance for the following reason: that whereas the attempt to obtain, for a real Hamiltonian system, solutions valid for all real values of the independent variable leads in general to divergent series, for certain solutions which are formally periodic the series can be proved convergent. In the words of Poincare, “ce qui nous rend ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule breche paroù nous puissions essayer de pénétrer dans une place jusqu'ici reputee inabordable. The existing theory of periodic solutions of differential equations was developed by Poincare mainly with reference to the equations of Celestial Mechanics. With a suitable choice of co-ordinates these are of the Hamiltonian form.

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