PERTURBATION OF LINEAR DIFFERENTIAL EQUATIONS BY A HALF-LINEAR TERM DEPENDING ON A SMALL PARAMETER

1981 ◽  
pp. 59-63
Author(s):  
I. Bihari
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Linear Differential Equations ◽  
Linear Term ◽  
Linear Differential

scholarly journals Reducibility by Moments of Linear Impulse Markov Dynamical Systems with Almost Constant Coefficients

2016 ◽  
Vol 70 (1) ◽  
pp. 34-40
Author(s):  
Jevgeņijs Carkovs ◽  
Aija Pola ◽  
Kārlis Šadurskis
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Markov Process ◽  
Differential Equations ◽  
Small Parameter ◽  
Linear Differential Equations ◽  
Phase Trajectories ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Coordinate Changes

Abstract This paper deals with linear impulse dynamical systems on ℝd whose parameters depend on an ergodic piece-wise constant Markov process with values from some phase space 𝕐 and on a small parameter ɛ. Trajectories of Markov process x(t,y)∈ ℝd satisfy a system of linear differential equations with close to constant coefficients on its continuity intervals, while its phase coordinate changes discontinuously when Markov process switching occur. Jump sizes depend linearly on the phase coordinate and are proportional to the small parameter ɛ. We propose a method and an algorithm for choosing the base 𝔹(t,y) of the space ℝd that provides approximation of average phase trajectories E{x(t,y)} by a solution of a system of linear differential equations with constant coefficients.

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