ISSN:

1937-1632

eISSN:

1937-1179

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## Discrete & Continuous Dynamical Systems - S

January 2020 , Volume 13 , Issue 1

Issue on delay differential equations

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**Abstract:**

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation.

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**Abstract:**

The long-time behavior of positive solutions of a differential equation with state-dependent delay

*+*[Abstract](1368)

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**Abstract:**

We consider a real-valued differential equation

with strictly monotonic negative feedback and state-dependent delay, that has a nontrivial periodic solution

*+*[Abstract](1414)

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**Abstract:**

The paper is concerned with a linear neutral differential equation

where

*+*[Abstract](1487)

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**Abstract:**

We present generalised Lyapunov-Razumikhin techniques for establishing global asymptotic stability of steady-state solutions of scalar delay differential equations. When global asymptotic stability cannot be established, the technique can be used to derive bounds on the persistent dynamics. The method is applicable to constant and variable delay problems, and we illustrate the method by applying it to the state-dependent delay differential equation known as the sawtooth equation, to find parameter regions for which the steady-state solution is globally asymptotically stable. We also establish bounds on the periodic orbits that arise when the steady-state is unstable. This technique can be readily extended to apply to other scalar delay differential equations with negative feedback.

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