30,133 research outputs found

    Fast generation of stability charts for time-delay systems using continuation of characteristic roots

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    Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.Comment: 12 pages, 6 figure

    Asymptotic properties of the spectrum of neutral delay differential equations

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    Spectral properties and transition to instability in neutral delay differential equations are investigated in the limit of large delay. An approximation of the upper boundary of stability is found and compared to an analytically derived exact stability boundary. The approximate and exact stability borders agree quite well for the large time delay, and the inclusion of a time-delayed velocity feedback improves this agreement for small delays. Theoretical results are complemented by a numerically computed spectrum of the corresponding characteristic equations.Comment: 14 pages, 6 figure

    Brownian Molecules Formed by Delayed Harmonic Interactions

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    A time-delayed response of individual living organisms to information exchanged within flocks or swarms leads to the emergence of complex collective behaviors. A recent experimental setup by (Khadka et al 2018 Nat. Commun. 9 3864), employing synthetic microswimmers, allows to emulate and study such behavior in a controlled way, in the lab. Motivated by these experiments, we study a system of N Brownian particles interacting via a retarded harmonic interaction. For N3N \leq 3 , we characterize its collective behavior analytically, by solving the pertinent stochastic delay-differential equations, and for N>3N>3 by Brownian dynamics simulations. The particles form molecule-like non-equilibrium structures which become unstable with increasing number of particles, delay time, and interaction strength. We evaluate the entropy and information fluxes maintaining these structures and, to quantitatively characterize their stability, develop an approximate time-dependent transition-state theory to characterize transitions between different isomers of the molecules. For completeness, we include a comprehensive discussion of the analytical solution procedure for systems of linear stochastic delay differential equations in finite dimension, and new results for covariance and time-correlation matrices.Comment: 36 pages, 26 figures, current version: further improvements and one correctio

    Canard explosion in delayed equations with multiple timescales

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    We analyze canard explosions in delayed differential equations with a one-dimensional slow manifold. This study is applied to explore the dynamics of the van der Pol slow-fast system with delayed self-coupling. In the absence of delays, this system provides a canonical example of a canard explosion. We show that as the delay is increased a family of `classical' canard explosions ends as a Bogdanov-Takens bifurcation occurs at the folds points of the S-shaped critical manifold.Comment: arXiv admin note: substantial text overlap with arXiv:1404.584

    Congestion at airports: the economics of airport expansions

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    Congestion and subsequent delays have been prevalent in many U.S. airports in recent years. A common response to congestion, championed by many community leaders, is to expand capacity by constructing new runways and terminals. Airport expansions are costly, complex, and controversial. We begin by using basic economic theory to analyze congestion at those airports that are part of an air transportation system. Next, we describe how benefit-cost analysis is used to assess the desirability of airport expansions. Many of the key points are illustrated in the context of Lambert–St. Louis International Airport. We also examine two especially controversial aspects of expansions—the displacement of people and businesses and the effects of airport noise. Finally, we discuss congestion-based pricing of landing fees as an alternative to airport expansions.Airports ; Economic development
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