1,721,023 research outputs found
Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach
No full textDelay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the evolution family associated to linear problems. The focus is on local stability of either equilibria and periodic orbits as well as on generic nonautonomous systems, for either delay differential and renewal equations
Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach
No full textDelay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the infinitesimal generator of the semigroup of solution operators associated to the system. The focus is on both local stability of equilibria and general bifurcation analysis of nonlinear problems, for either delay differential and renewal equations
Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations
Full text linkWe 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
Piecewise orthogonal collocation for computing periodic solutions of coupled delay equations
No full textWe extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and provide a theoretical estimate of the error. We conclude with some numerical experiments to further support the theoretical results
Preface to Stability of linear delay differential equations: A numerical approach with MATLAB
No full textStability of Linear Continuous-Time Systems with Stochastically Switching Delays
Full text linkNecessary and sufficient conditions for the stability of linear continuous-time systems with stochastically switching delays are presented in this paper. It is assumed that the delay random paths are piece-wise constant functions of time where a finite number of values may be taken by the delay. The stability is assessed in terms of the second moment of the state vector of the system. The solution operators of individual linear systems with constant de- lays, chosen from the set of all possible delay values, are extended to form new augmented operators. Then for proper formulation of the second moment in continuous time, tensor products of the augmented solution operators are used. Finally the finite-dimensional versions of the stability conditions, that can be obtained using various time discretization techniques, are presented. Some examples are provided that demonstrate how the stability conditions can be used to assess the stability of linear systems with stochastic delay
Numerical approximation of characteristic values of Partial Retarded Functional Differential Equations
No full textThe stability of an equilibrium point of a dynamical system is determinedby the position in the complex plane of the so-called characteristic values of the linearizationaround the equilibrium. This paper presents an approach for the computationof characteristic values of partial differential equations of evolution involving timedelay, which is based on a pseudospectral method coupled with a spectral method.The convergence of the computed characteristic values is of infinite order with respectto the pseudospectral discretization and of finite order with respect to the spectralone. However, for one dimensional reaction diffusion equations, the finite order of thespectral discretization is proved to be so high that the convergence turns out to be asfast as one of infinite order
How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology
Full text linkA prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents' decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results
An adaptive algorithm for efficient computation of level curves of surfaces
No full textAn adaptive algorithm for efficient computation of level curves of surface
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