118,694 research outputs found
SVD algorithms to approximate spectra of dynamical systems
No full textIn this work we consider algorithms based on the singular value decomposition (SVD) to approximate Lyapunov and exponential dichotomy spectra of dynamical systems. We review existing contributions, and propose new algorithms of the continuous SVD method. We present implementation details for the continuous SVD method, and illustrate on several examples the behavior of continuous (and also discrete) SVD method. This paper is the companion paper of [L. Dieci,C. Elia, The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects, J. Diff. Equat., in press]
Regularizing piecewise smooth differential systems: co-dimension 2 discontinuity surface
No full textIn this paper, we are concerned with numerical solution of piecewise smooth
initial value problems. Of specific interest is the case when the discontinuities occur on a
smooth manifold of co-dimension 2, intersection of two co-dimension 1 singularity surfaces,
and which is nodally attractive for nearby dynamics. In this case of a co-dimension 2 attracting
sliding surface, we will give some results relative to two prototypical time and space
regularizations. We will show that, unlike the case of co-dimension 1 discontinuity surface,
in the case of co-dimension 2 discontinuity surface the behavior of the regularized problems
is strikingly different. On the one hand, the time regularization approach will not select a
unique sliding mode on the discontinuity surface, thus maintaining the general ambiguity
of how to select a Filippov vector field in this case. On the other hand, the proposed space
regularization approach is not ambiguous, and there will always be a unique solution associated
to the regularized vector field, which will remain close to the original co-dimension 2
surface.We will further clarify the limiting behavior (as the regularization parameter goes to
0) of the proposed space regularization to the solution associated to the sliding vector field
of Dieci and Lopez (Numer Math 117:779–811, 2011). Numerical examples will be given
to illustrate the different cases and to provide some preliminary exploration in the case of co-dimension 3 discontinuity surface
Hermitian matrices of three parameters: Perturbing coalescing eigenvalues and a numerical method
No full textIn this work we consider Hermitian matrix-valued functions of 3 (real) parameters, and are interested in generic coalescing points of eigenvalues, conical intersections. Unlike our previous works [L. Dieci, A. Papini and A. Pugliese, Approximating coalescing points for eigenvalues of Hermitian matrices of three parameters, SIAM J. Matrix Anal. Appl., 2013] and [L. Dieci and A. Pugliese, Hermitian matrices depending on three parameters: Coalescing eigenvalues, Linear Algebra Appl., 2012], where we worked directly with the Hermitian problem and monitored variation of the geometric phase to detect conical intersections inside a sphere-like region, here we consider the following construction: (i) Associate to the given problem a real symmetric problem, twice the size, all of whose eigenvalues are now (at least) double, (ii) perturb this enlarged problem so that, generically, each pair of consecutive eigenvalues coalesce along curves, and only there, (iii) analyze the structure of these curves, and show that there is a small curve, nearly planar, enclosing the original conical intersection point. We will rigorously justify all of the above steps. Furthermore, we propose and implement an algorithm following the above approach, and illustrate its performance in locating conical intersections
A Survey of Numerical Methods for IVPs of ODEs with Discontinuous Right-Hand Side
No full textThis work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numerical
Analysis of the second author. We remember Donato as a generous teacher, always ready to discuss with
his students, able to give them profound and interesting suggestions.
Here, we present a survey of numerical methods for differential systems with discontinuous right
hand side. In particular, we will review methods where the discontinuities are detected by using an event
function (so-called event driven methods) and methods where the discontinuities are located by controlling
the local errors (so-called time-stepping methods). Particular attention will be devoted to discontinuous
systems of Filippov’s type where sliding behavior on the discontinuity surface is allowed
A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis
No full textWe consider sliding motion, in the sense of Filippov, on a discontinuity surface σ of co-dimension 2. We characterize, and restrict to, the case of σ being attractive through sliding. In this situation, we show that a certain Filippov sliding vector field fF (suggested in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6], Dieci and Lopez, 2011 [10]) exists and is unique. We also propose a characterization of first order exit conditions, clarify its relation to generic co-dimension 1 losses of attractivity for σ, and examine what happens to the dynamics on σ for the aforementioned vector field fF. Examples illustrate our results
Hermitian matrices depending on three parameters: Coalescing eigenvalues.
No full textWe consider Hermitian matrix valued functions depending on three parameters
that vary in a bounded surface of . We study how to detect when
such functions have coalescing eigenvalues inside this surface. Our criterion to
locate these singularities is based on a construction suggested by Stone in [20]. For
generic coalescings, any such singularity is related to a particular accumulation
of a certain phase, or lack thereof, as we cover the surface
The singular value decomposition to approximate spectra of dynamical systems. Theoretical aspects
No full textIn this paper we consider the singular value decomposition (SVD) of a fundamental matrix solution in
order to approximate the Lyapunov and exponential dichotomy spectra of a given system. One of our main
results is to prove that SVD techniques are sound approaches for systems with stable and distinct Lyapunov
exponents. We also show how the information which emerges with the SVD techniques can be used to
obtain information on the growth directions associated to given spectral intervals
Minimum variation solutions for sliding vector fields on the intersection of two surfaces inR3
No full textIn this work, we consider model problems of piecewise smooth systems inR3, for which we propose minimum variation approaches to find a Filippov sliding vector field on the intersection Σ of two discontinuity surfaces. Our idea is to look at the minimum variation solution in theH1-norm, among either all admissible sets of coefficients for a Filippov vector field, or among all Filippov vector fields. We compare the resulting solutions to other possible Filippov sliding vector fields (including the bilinear and moments solutions). We further show that-in the absence of equilibria-also these other techniques select a minimum variation solution, for an appropriately weightedH1-norm, and we relate this weight to the change of time variable giving orbital equivalence among the different vector fields. Finally, we give details of how to build a minimum variation solution for a general piecewise smooth system inR3
Erratum: A comparison of Filippov sliding vector fields in codimension 2 (Journal of Computational and Applied Mathematics (2014) 262 ((161-179))
No full textCorrigendum to ‘‘A comparison of Filippov sliding vector
fields in codimension 2’’ [J. Comput. Appl. Math. 262 (2014)
161–179
A comparison of Filippov sliding vector fields in codimension 2
No full textWe consider several possibilities on how to select a Filippov sliding vector field on a codimension 2 singularity surface Σ, intersection of two codimension 1 surfaces. We discuss and compare several, old and new, approaches, under the assumption that Σ is nodally attractive. Of specific interest is the selection of a smoothly varying Filippov sliding vector field. As a result of our analysis and experiments, the best candidates of the many possibilities explored are those based on the so-called barycentric coordinates. In the present context, one of these possibilities appear to be new. © 2013 Elsevier B.V. All rights reserved
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