703 research outputs found
A practical approach to computing Lyapunov exponents of renewal and delay equations
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists of the reformulation of the delay equation as an abstract differential equation, the reduction of the latter to a system of ordinary differential equations via pseudospectral collocation and the application of the standard discrete QR method. The effectiveness of the method is shown experimentally and a MATLAB implementation is provided
Piecewise orthogonal collocation for computing periodic solutions of renewal equations
We extend the use of piecewise orthogonal collocation to computing periodic
solutions of renewal equations, which are particularly important in modeling
population dynamics. We prove convergence through a rigorous error analysis.
Finally, we show some numerical experiments confirming the theoretical results,
and a couple of applications in view of bifurcation analysis.Comment: 32 pages, 6 figure
Collocation methods for complex delay models of structured populations
openDottorato di ricerca in Informatica e scienze matematiche e fisicheopenAndo', Alessi
Computation of Mixed Type Functional Differential Boundary Value Problems
This is the published version, also available here: http://dx.doi.org/10.1137/040603425.We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the modeling of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such problems including discrete Nagumo equations, an Ising model, and Frenkel--Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function
Spectrum-based stability analysis and stabilization of a class of time-periodic time delay systems
We develop an eigenvalue-based approach for the stability assessment and
stabilization of linear systems with multiple delays and periodic coefficient
matrices. Delays and period are assumed commensurate numbers, such that the
Floquet multipliers can be characterized as eigenvalues of the monodromy
operator and by the solutions of a finite-dimensional non-linear eigenvalue
problem, where the evaluation of the characteristic matrix involves solving an
initial value problem. We demonstrate that such a dual interpretation can be
exploited in a two-stage approach for computing dominant Floquet multipliers,
where global approximation is combined with local corrections. Correspondingly,
we also propose two novel characterizations of left eigenvectors. Finally, from
the nonlinear eigenvalue problem formulation, we derive computationally
tractable expressions for derivatives of Floquet multipliers with respect to
parameters, which are beneficial in the context of stability optimization.
Several numerical examples show the efficacy and applicability of the presented
results
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
Optimization along Families of Periodic and Quasiperiodic Orbits in Dynamical Systems with Delay
This paper generalizes a previously-conceived, continuation-based
optimization technique for scalar objective functions on constraint manifolds
to cases of periodic and quasiperiodic solutions of delay-differential
equations. A Lagrange formalism is used to construct adjoint conditions that
are linear and homogenous in the unknown Lagrange multipliers. As a
consequence, it is shown how critical points on the constraint manifold can be
found through several stages of continuation along a sequence of connected
one-dimensional manifolds of solutions to increasing subsets of the necessary
optimality conditions. Due to the presence of delayed and advanced arguments in
the original and adjoint differential equations, care must be taken to
determine the degree of smoothness of the Lagrange multipliers with respect to
time. Such considerations naturally lead to a formulation in terms of
multi-segment boundary-value problems (BVPs), including the possibility that
the number of segments may change, or that their order may permute, during
continuation. The methodology is illustrated using the software package coco on
periodic orbits of both linear and nonlinear delay-differential equations,
keeping in mind that closed-form solutions are not typically available even in
the linear case. Finally, we demonstrate optimization on a family of
quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with
delay and parametric forcing. The quasiperiodic case is a further original
contribution to the literature on optimization constrained by partial
differential BVPs.Comment: preprint, 17 pages, 9 figure
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