1,721,029 research outputs found
Solution operator approximation for characteristic roots of delay differential equations
No full textIn this paper a new method for the numerical computation of characteristic roots for linear autonomous systems of Delay Differential Equations (DDEs) is proposed. The new approach enlarges the class of methods recently developed (see [SIAM J. Numer. Anal. 40 (2002) 629; D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: experimental comparison, in: BIOCOMP2002: Topics in Biomathematics and Related Computational Problems at the Beginning of the Third Millennium, Vietri, Italy, 2002, Sci. Math. Jpn. 58 (2) pp. 377–388; D. Breda, The infinitesimal generator approach for the computation of characteristic roots for delay differential equations using BDF methods, Research Report RR2/2002, Department of Mathematics and Computer Science, Università di Udine, Italy, 2002; IMA J. Numer. Anal. 24 (2004) 1; SIAM J. Sci. Comput. (2004), in press]) and in particular it is based on a Runge–Kutta (RK) time discretization of the solution operator associated with the system. Hence this paper revisits the Linear Multistep (LMS) approach presented in [SIAM J. Numer. Anal. 40 (2002) 629] for the multiple discrete delay case and moreover extends it to the distributed delay case. We prove that the method converges with the same order as the underlying RK scheme and illustrate this with some numerical tests that are also used to compare the method with other existing techniques
Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents
No full textA general class of linear and nonautonomous delay differen-
tial equations with initial data in a separable Hilbert space is treated.
The classic questions of existence, uniqueness, and regularity of solutions
are addressed. Moreover, the semigroup approach typically adopted
in the autonomous case for continuous initial functions is extended,
and thus the existence of an equivalent abstract ordinary formulation
is shown to hold. Finally, the existence of infinitely many Lyapunov
exponents for the associated evolution is proven and their meaning is
discussed
Methods for numerical computation of characteristic roots for delay differential equations: experimental comparison
No full textThis paper is a collection of tests about the numerical computation of characteristic roots for linear delay differential equations (DDEs) with multiple discrete and distributed delays. Two different approaches are tested, based on the discretization of the infinitesimal generator of the solution operators semigroup associated to the DDE and of the solution operator itself. These approaches are implemented using different numerical techniques such as Runge-Kutta (RK), linear multistep (LMS) and spectral methods
On roots and charts of delay equations with complex coefficients
Full text linkThis 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
On characteristic roots and stability charts of delay differential equations
No full textThis work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed on the robust analysis of the position of the roots in the complex plane with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the relevant stability boundaries and charts. While the real case is almost standard (and known), the investigation of the complex case is not as immediate. Hence, a preliminary shift of the coefficients is proposed, which reduces the number of free parameters. This allows to extend the techniques used for the real case, also allowing for useful graphical visualization of the relevant stability charts. The present research is motivated on the basis of studying the stability of systems with delay
Discrete or distributed delay? Effects on stability of population growth
Full text linkThe 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
Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations
No full textAn SEIR epidemic model with constant latency time and infectious period
No full textWe present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison principle, we provide global attractivity results for both the equilibria, i.e. the disease-free equilibrium E0 and the positive equilibrium E+, which exists iff the basic reproduction number R0 is larger than one. If R0>1 we also provide a permanence result for the model solutions. Finally we prove that the two delays are harmless in the sense that, by the analysis of the characteristic equations, which result to be polynomial trascendental equations with polynomial coefficients dependent upon both delays, we confirm all the standard properties of an epidemic model: E0 is locally asymptotically stable for R01, while if R0>1 then E+ is always asymptotically stable
Approximating Lyapunov exponents and Sacker-Sell spectrum for retarded functional differential equations
No full textWe consider Lyapunov exponents and Sacker–Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the eduction to finite dimension of the evolution family associated to the system,
to which a classic discrete QR method is then applied. The discretization of the evolution family is accomplished by a combination of collocation and generalized Fourier projection. A rigorous error analysis is developed to bound the difference between the
computed stability spectra and the exact stability spectra. The efficacy of the results is illustrated with some numerical examples
Efficient Computation of Stability Charts for Linear Time Delay Systems
No full textA new efficient algorithm for the computation of the stability chart of linear time delay systems is proposed and tested on several examples. The stability chart is obtained by investigating the 2d-parameter space by a first coarse square grid which is then adaptively refined by triangulation to match the desired tolerance. This leads to a considerable reduction in computational cost with respect to investigate a uniform fine square grid. Stability of each point is determined by approximating the rightmost characteristic root real part via a numerical scheme recently developed by the authors and based on pseudospectral differencing methods. A Matlab code is included in appendix
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