Flip and Neimark–Sacker bifurcation in a differential equation with piecewise constant arguments model

2017 ◽  
Vol 23 (4) ◽  
pp. 763-778 ◽  
Author(s):  
S. Kartal
Keyword(s):  
Differential Equation ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant

EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

2021 ◽  
Vol 10 (9) ◽  
pp. 3113-3128
Author(s):  
M.I. Muminov ◽  
Z.Z. Jumaev
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Linear Equations ◽  
Periodic Functions ◽  
Piecewise Constant Arguments ◽  
Piecewise Constant ◽  
Second Order Differential Equations ◽  
Greatest Integer Function ◽  
Soluble Problem ◽  
Existence Conditions

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

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