Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems

Hassard, Brian

doi:10.1006/jdeq.1996.3127

Cited by 55 publications

(41 citation statements)

References 3 publications

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“…Hence, to analyse the stability of a steady-state solution, one must determine reliably all roots satisfying Re(e(λ)) r, for a given r < 0 close to zero. Analytical conditions for stability can be found in Stépán [70] and Hassard [42]. These conditions are deduced by using the argument principle of complex analysis, and they give a practical method for determining stability.…”

Section: Stability Of Steady-state Solutionsmentioning

confidence: 99%

“…The symmetry about x = 0 is exploited by considering only the interval [0, 0.5]. Splitting (41) into real and imaginary part and discretizing (42) in space with a second order central difference formula with constant stepsize ∆x = 0.5/128. (41)- (42) with α = 3, φ = 0, T = 1000, d = 1.68 × 10 −2 and τ = 1000, obtained by continuation with DDE-Biftool, with the feedback strength η as the parameter.…”

Section: Dde-pde Model Of a Laser With Optical Feedbackmentioning

confidence: 99%

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Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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Mathematical modeling with delay differential equations (DDEs) is widely used in various application areas of science and engineering (e.g., in semiconductor lasers with delayed feedback, high-speed machining, communication networks, and control systems) and in the life sciences (e.g., in population dynamics, epidemiology, immunology, and physiology). Delay equations have an infinite-dimensional state space because their solution is unique only when an initial function is specified on a time interval of length equal to the largest delay. Consequently, analytical calculations are more difficult than for ordinary differential equations andnumerical methods are generally the only way to achieve a complete analysis, prediction and control of systems with time delays.Delay differential equations are a special type of functional differential equation (FDE). In FDEs the time evolution of the state variable can depend on the past in an arbitrary way as long as the dependence is a bounded function of the past. However, DDEs impose a constraint on this dependence, namely that the evolution depends only on certain past values of the state at discrete times. (We do not consider here the case of distributed delay.) The delays can be constant or state dependent. The equations can also involve delayed values of the derivative of the state, which leads to equations of neutral type.In this chapter we mainly discuss the simplest case, namely a finite number of constant delays. Specifically, we consider a nonlinear system of DDEs with constant delays τ j 0, j = 1, . . . , m, of the formwhere x(t) ∈ R n , and f : R (m+1)n+p → R n is a nonlinear smooth function depending on a number of (time-independent) parameters η ∈ R p . We assume that the delays are in increasing order and denote the maximal delay by

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Section: Stability Of Steady-state Solutionsmentioning

confidence: 99%

Section: Dde-pde Model Of a Laser With Optical Feedbackmentioning

confidence: 99%

Continuation and Bifurcation Analysis of Delay Differential Equations

Roose

Szalai

2007

Understanding Complex Systems

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“…The book of Kolmanovskii and Nosov [21] summarizes the main theorems on stability of DDEs, and it contains several examples as well. A more sophisticated method was developed by StÃ epÃ an [2] (generalized also by Hassard [22]) applicable even for the combination of several discrete and continuous time delays.…”

Section: Introductionmentioning

confidence: 99%

Semi‐discretization method for delayed systems

Insperger

Stépàn

2002

Numerical Meth Engineering

592

282

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SUMMARYThe paper presents an e cient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past e ect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high-dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay.

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“…In [1,[11][12][13], attention has been paid to the linear stability analysis of the trivial steady state of delayed oscillators of type (1). While the investigation of stability of nontrivial solutions in (1) has received little attention from analytical view point, the influence of high-frequency excitation on nontrivial steady state has not been tackled.…”

Section: Introductionmentioning

confidence: 99%

Control of Bistability in a Delayed Duffing Oscillator

Hamdi

Belhaq

2012

Advances in Acoustics and Vibration

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The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding.

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Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems

Cited by 55 publications

References 3 publications

Continuation and Bifurcation Analysis of Delay Differential Equations

Continuation and Bifurcation Analysis of Delay Differential Equations

Semi‐discretization method for delayed systems

Control of Bistability in a Delayed Duffing Oscillator

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