A differential equation with state-dependent delay from cell population biology

Getto, Philipp; Waurick, Marcus

doi:10.1016/j.jde.2015.12.038

Cited by 25 publications

(24 citation statements)

References 22 publications

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“…This can be argued successfully, for example, in machining when the tool has nearly infinite stiffness perpendicular to the cutting direction [75], or in laser dynamics where light travels over a fixed distance [40]. On the other hand, in many contexts, including in biological systems and in control problems [9,10,11,21,36,38,68,82], the delays one encounters are not actually constant. In particular, they may depend on the state in a significant way, that is, change dynamically during the time-evolution of the system.…”

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confidence: 99%

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Calleja¹,

Humphries²,

Krauskopf³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation, and the resulting code HHnfDDE is supplied as a supplement to this article. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincaré section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds (when there is a single unstable Floquet multiplier). This allows us to study transitions through resonance tongues and the breakup of a 1 : 4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.Key words. State-dependent delay differential equations, bifurcation analysis, invariant tori, resonance tongues, Hopf-Hopf bifurcation, normal form computation AMS subject classifications. 34K60, 34K18, 37G05, 37M201. Introduction. Time delays arise naturally in numerous areas of application as an unavoidable phenomenon, for example, in balancing and control [8,19,35,39,64,65,66,67], machining [36], laser physics [40,46,54], agent dynamics [52,53,70,73], neuroscience and biology [1,18,20,42,79], and climate modelling [13,41,48]. Important sources of delays are communication times between components of a system, maturation and reaction times, and the processing time of information received. When they are sufficiently large compared to the relevant internal time scales of the system under consideration, then the delays must be incorporated into its mathematical description. This leads to mathematical models in the form of delay differential equations (DDEs). In many situations the relevant delays can be considered to be fixed; examples are the travel time of light between components of a laser syst...

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confidence: 99%

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Calleja¹,

Humphries²,

Krauskopf³

2017

SIAM J. Appl. Dyn. Syst.

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“…The statement of Theorem 3.1 consists of three parts and we show them in a series of propositions. But before doing so, we prove the following auxiliary result which is of similar type as Theorem 1.6 in Getto & Waurick [4]: Proposition 3.3. Given f as in Theorem 1.1, there is some b > 0 such that if ϕ ∈ X f and if the associated solution x ϕ : [−h, t + (ϕ)) → R n of Eq.…”

Section: Local Center and Center-unstable Manifoldsmentioning

confidence: 86%

Local stability analysis of differential equations with state-dependent delay

Stumpf¹

2015

DCDS-A

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In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.

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“…(2011). Here we consider a slightly adapted SD-DDE formulation derived by Getto and Waurick (2016) (see again Sect. 2).…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

et al. 2019

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We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.

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A differential equation with state-dependent delay from cell population biology

Cited by 25 publications

References 22 publications

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

Local stability analysis of differential equations with state-dependent delay

Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical methods

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