Forced Periodic Solutions of Systems of Differential Equations

1953 ◽  
Vol 57 (2) ◽  
pp. 314 ◽  
Author(s):  
H. A. Antosiewicz
1982 ◽  
Vol 5 (2) ◽  
pp. 305-309
Author(s):  
A. K. Bose

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).


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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