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Multidelay Differential Equations: A Taylor Expansion Approach
It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.
Authors:
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Award ID(s):
Publication Date:
NSF-PAR ID:
10335429
Journal Name:
International Journal of Bifurcation and Chaos
Volume:
32
Issue:
03
ISSN:
0218-1274
2. We study the relationship between the input phase delays and the output mode orders when using a pixel-array structure fed by multiple single-mode waveguides for tunable orbital-angular-momentum (OAM) beam generation. As an emitter of a free-space OAM beam, the designed structure introduces a transformation function that shapes and coherently combines multiple (e.g., four) equal-amplitude inputs, with the$k$th input carrying a phase delay of$(k−<#comment/>1)Δ<#comment/>φ<#comment/>$. The simulation results show that (1) the generated OAM order ℓ is dependent on the relative phase delay$Δ<#comment/>φ<#comment/>$; (2) the transformation function can be tailored by engineering the structure to support different tunable ranges (e.g., $l={−<#comment/>1},{−<#comment/>1,+1},{−<#comment/>1,0,+1}$, or${−<#comment/>2,−<#comment/>1,+1,+2}$); and (3) multiple independent coaxial OAM beams can be generated by simultaneously feeding the structure with multiple independent beams, such that each beam has its own$Δ<#comment/>φ<#comment/>$value for the four inputs. Moreover, there is a trade-off between the tunable range and the mode purity, bandwidth, and crosstalk, such that the increase of the tunable range leads to (a) decreased mode purity (from 91% to 75% formore »), (b) decreased 3 dB bandwidth of emission efficiency (from 285 nm for$l={−<#comment/>1}$to 122 nm for$l={−<#comment/>2,−<#comment/>1,+1,+2}$), and (c) increased crosstalk within the C-band (from$−<#comment/>23.7$to$−<#comment/>13.2dB$when the tunable range increases from 2 to 4).
4. ABSTRACT We present a blind time-delay cosmographic analysis for the lens system DES J0408−5354. This system is extraordinary for the presence of two sets of multiple images at different redshifts, which provide the opportunity to obtain more information at the cost of increased modelling complexity with respect to previously analysed systems. We perform detailed modelling of the mass distribution for this lens system using three band Hubble Space Telescope imaging. We combine the measured time delays, line-of-sight central velocity dispersion of the deflector, and statistically constrained external convergence with our lens models to estimate two cosmological distances. We measure the ‘effective’ time-delay distance corresponding to the redshifts of the deflector and the lensed quasar $D_{\Delta t}^{\rm eff}=$$3382_{-115}^{+146}$ Mpc and the angular diameter distance to the deflector Dd = $1711_{-280}^{+376}$ Mpc, with covariance between the two distances. From these constraints on the cosmological distances, we infer the Hubble constant H0= $74.2_{-3.0}^{+2.7}$ km s−1 Mpc−1 assuming a flat ΛCDM cosmology and a uniform prior for Ωm as $\Omega _{\rm m} \sim \mathcal {U}(0.05, 0.5)$. This measurement gives the most precise constraint on H0 to date from a single lens. Our measurement is consistent with that obtained from the previous sample of six lenses analysed by the H0more »