Use of forms of high order in discussions of the stability of systems of differential equations with delay

By defining a Lyapunov functional, we investigate the stability and boundedness of solutions to nonlinear third order differential equation with constant delay, r : x'''(t) + g(x(t), x'(t))x''(t) + f (x(t - r), x'(t - r)) + h(x(t - r)) = p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)), when p(t, x(t), x'(t), x(t - r), x'(t - r), x''(t)) = 0 and ? 0, respectively. Our results achieve a stability result which exists in the relevant literature of ordinary nonlinear third order differential equations without delay to the above functional differential equation for the stability and boundedness of solutions. An example is introduced to illustrate the importance of the results obtained.

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Word series high-order averaging of highly oscillatory differential equations with delay

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2019.2.00042 ◽

2019 ◽

Vol 4 (2) ◽

pp. 445-454 ◽

Cited By ~ 1

Author(s):

J. M. Sanz-Serna ◽

Beibei Zhu

Keyword(s):

Differential Equations ◽

Delay Differential Equations ◽

High Order ◽

Constant Delay ◽

Integer Multiple ◽

Averaged Equations ◽

Highly Oscillatory ◽

Highly Oscillatory Differential Equations ◽

Delay Differential ◽

Equations With Delay

AbstractWe show that, when the delay is an integer multiple of the forcing period, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word series techniques to obtain high-order averaged equations for differential equations without delay.

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Deterministic chaos in pendulum systems with delay

Applied Mathematics and Nonlinear Sciences ◽

10.2478/amns.2019.1.00001 ◽

2019 ◽

Vol 4 (1) ◽

pp. 1-8 ◽

Cited By ~ 22

Author(s):

Aleksandr Shvets ◽

Alexander Makaseyev

Keyword(s):

Mathematical Model ◽

Steady State ◽

Differential Equations ◽

Deterministic Chaos ◽

Phase Portraits ◽

Characteristic Exponents ◽

Systems Of Differential Equations ◽

Lyapunov Characteristic Exponents ◽

The Mathematical Model ◽

Equations With Delay

AbstractDynamic system "pendulum - source of limited excitation" with taking into account the various factors of delay is considered. Mathematical model of the system is a system of ordinary differential equations with delay. Three approaches are suggested that allow to reduce the mathematical model of the system to systems of differential equations, into which various factors of delay enter as some parameters. Genesis of deterministic chaos is studied in detail. Maps of dynamic regimes, phase-portraits of attractors of systems, phase-parametric characteristics and Lyapunov characteristic exponents are constructed and analyzed. The scenarios of transition from steady-state regular regimes to chaotic ones are identified. It is shown, that in some cases the delay is the main reason of origination of chaos in the system "pendulum - source of limited excitation".

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