On roots and charts of delay equations with complex coefficients

This work is devoted to the analytic study of the characteristic roots of scalar autonomous Delay Differential Equations (DDEs) with complex coefficients. The focus is placed on the robust analysis of the position of the roots in C with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the relevant stability boundaries and charts. The investigation benefits from a preliminary shift of the coefficients which reduces the number of free parameters allowing for useful graphical visualizations. The present research is motivated on the base of studying the stability of systems of DDEs

Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations

2noWe analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math., 133 (2016), pp. 525-555], [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2771-2793], and [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794-2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable to such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical since it is directly linked to the course of time. Finally, we prove that the finite element method is convergent, while we limit ourselves to commenting on the infeasibility of this approach as far as the spectral element method is concerned.openopenANDO A.; BREDA D.Ando, A.; Breda, D

Discrete or distributed delay? Effects on stability of population growth

The growth of a population subject to maturation delay is modeled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and sufficient conditions are provided by analyzing the relevant characteristic equations. It is shown that for any choice of parameter values for which the discrete delay model presents stability switches there exists a maximum delay variance beyond which no switch occurs for the continuous delay model: the delay variance has a stabilizing effect. Moreover, it is illustrated how, in the presence of switches, the unstable delay domain is as larger as lower is the ratio between the juveniles and the adults mortality rates

Pseudospectral reduction to compute Lyapunov exponents of delay differential equations

A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included

On discretizing the semigroup of solution operators for linear time invariant - time delay systems

in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(-\u3c4,0;C) and generalized Fourier projection when the state space is \u3c7 = C 7 L2(-\u3c4,0;C). Full discretization details for constructing the approximation matrices are given. Moreover, concise, yet fundamental, convergence results are discussed, with particular attention to their similarities and differences as well as pros and cons with regards to solution approximation and asymptotic stability detection

Numerical continuation and delay equations: A novel approach for complex models of structured populations

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of {it external} problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this {it internal} continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold

Numerical computation of characteristic multipliers for linear time periodic coefficients delay differential equations

In this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to the system of DDEs, i.e. the solution operator over the period of the coefficients, we discretize it by using pseudospectral differencing techniques based on collocation and approximate the dominant multipliers by the eigenvalues of the resulting matrix. The use of pseudospectral methods has already been proposed in the context of simpler DDEs. Here we fully generalize the method to the class of linear time periodic coefficients DDEs with arbitrary period and multiple discrete and distributed delays. The scheme is shown to have spectral accuracy by means of several numerical examples

Delay equations and characteristic roots: stability and more from a single curve

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubt- edly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equilibria inherit the stability properties of the corresponding linearizations, the study of which relies on associated characteristic equations. The transcendence of the latter, due to the presence of the delay, leads to infinitely-many roots in the com- plex plane. Simple algebraic manipulations show, first, that all such roots belong to the intersection of two curves. Second, only one of these curves is crucial for stability, and relevant sufficient and/or necessary criteria can be easily derived from its analysis. Other aspects can be investigated under this framework and a link to the theory of modulus semigroups and monotone semiflows is also discussed

An Invitation to Stochastic Differential Equations in Healthcare

An important problem in finance is the evaluation of the value in the future of assets (e.g., shares in company, currencies, derivatives, patents). The change of the values can be modeled with differential equations. Roughly speaking, a typical differential equation in finance has two components, one deterministic (e.g., rate of interest of bank accounts) and one stochastic (e.g., values of stocks) that is often related to the notion of Brownian motions. The solution of such a differential equation needs the evaluation of Riemann–Stieltjes’s integrals for the deterministic part and Ito’s integrals for the stochastic part. For A few types of such differential equations, it is possible to determine an exact solution, e.g., a geometric Brownian motion. On the other side for almost all stochastic differential equations we can only provide approximations of a solution. We present some numerical methods for solving stochastic differential equations