Inclusion conditions for Hurwitzian and Schur sets in

1976 ◽  
Vol 80 (1) ◽  
pp. 113-120 ◽  
Author(s):  
Russell A. Smith
Keyword(s):  
Differential Equations ◽  
Frequency Domain ◽  
Delay Differential Equations ◽  
Domain Type ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Vector Polynomial

AbstractA condition of frequency domain type on the vector polynomial G(z) is obtained which is both necessary and sufficient for the Hurwitzian set of G(z) to include the special Hurwitzian set , where E(z) = col (1, z, …, zn). This result is extended to Hurwitzian sets arising from scalar delay-differential equations. Analogous conditions are also given for each of the inclusions where denotes the Schur set of G(z).

Oscillation of the Riemann–Weber Version of Euler Differential Equations with Delay

2000 ◽  
Vol 7 (3) ◽  
pp. 577-584
Author(s):  
Jitsuro Sugie ◽  
Mitsuru Iwasaki
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Oscillation Of Solutions

Abstract Our concern is to consider delay differential equations of Euler type. Necessary and sufficient conditions for the oscillation of solutions are given. The results extend some famous facts about Euler differential equations without delay.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

scholarly journals Bifurcations in Delay Differential Equations: An Algorithmic Approach in Frequency Domain

2016 ◽  
Vol 26 (14) ◽  
pp. 1650238
Author(s):  
A. Bel ◽  
W. Reartes ◽  
A. Torresi
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Periodic Solutions ◽  
Frequency Domain ◽  
Delay Differential Equations ◽  
Arbitrary Order ◽  
Bifurcation Equation ◽  
Algorithmic Approach ◽  
Delay Differential

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.

Oscillatory and monotone solutions of first-order nonlinear delay differential equations

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Nino Partsvania ◽  
Zaza Sokhadze
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
First Order ◽  
Kneser Solution ◽  
Nonlinear Delay Differential Equations ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Monotone Solutions ◽  
Nonlinear Delay

AbstractFor first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.

ON THE STUDY OF BIFURCATIONS IN DELAY-DIFFERENTIAL EQUATIONS: A FREQUENCY-DOMAIN APPROACH

2012 ◽  
Vol 22 (06) ◽  
pp. 1250137 ◽  
Author(s):  
FRANCO S. GENTILE ◽  
JORGE L. MOIOLA ◽  
EDUARDO E. PAOLINI
Keyword(s):  
Differential Equations ◽  
Frequency Domain ◽  
Delay Differential Equations ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Delay Dependence ◽  
Collateral Findings ◽  
Delay Differential ◽  
Controllability And Observability ◽  
Frequency Domain Approach

An improved version of a frequency-domain approach to study bifurcations in delay-differential equations is presented. The proposed methodology provides information about the frequency, amplitude, and stability of the orbit emerging from Hopf bifurcation. We apply this method to different schemes of the delayed van der Pol oscillator. The time-delay dependence can appear intrinsically because of the system dynamics or can be intentionally introduced in a feedback loop. Also, a discussion about system controllability and observability is given for a proper and rigorous application of the frequency domain technique. Collateral findings involving some types of static bifurcations are included for completeness.

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