1,720,995 research outputs found
Singular Values of Two-Parameter Matrices: An Algorithm To Accurately Find Their Intersections
No full textConsider the Singular Value Decomposition (SVD) of a two-parameter function
, , where is simply connected and compact,
with boundary . No matter how differentiable the function is (even
analytic), in general the singular values lose all smoothness at points where they coalesce. In thiswork, we propose and implement algorithms which locate points in
where the singular values coalesce. Our algorithms are based on the
interplay between coalescing singular values in , and the periodicity
of the SVD-factors as one completes a loop along
Hermitian matrices of three parameters: Perturbing coalescing eigenvalues and a numerical method
Full text linkIn 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
Decay behaviour of functions of skew-symmetric matrices
No full textIn this paper we study the decay behaviour for
the entries of functions of skew symmetric matrices arising
in the applications. Thanks to an algorithm proposed in [4]
for computing the exponential of tridiagonal skew symmet-
ric matrices we will be able to study the decay behaviour of
more general functions of matrices and to define a banded
version of such function of matrices. Numerical tests show-
ing this kind of behaviour are reported for several kind of
functions
Cusp bifurcations: Numerical detection via two-parameter continuation and computer-assisted proofs of existence
No full textThis paper introduces a novel computer-assisted method for detecting and constructively proving the existence of cusp bifurcations in differential equations. The approach begins with a two-parameter continuation along which a tool based on the theory of Poincaré index is employed to identify the presence of a cusp bifurcation. Using the approximate cusp location, Newton’s method is then applied to a given augmented system (the cusp map), yielding a more precise numerical approximation of the cusp. Through a successful application of a Newton-Kantorovich type theorem, we establish the existence of a non-degenerate zero of the cusp map in the vicinity of the numerical approximation. Employing a Gershgorin circles argument, we then prove that exactly one eigenvalue of the Jacobian matrix at the cusp candidate has zero real part, thus rigorously confirming the presence of a cusp bifurcation. Finally, by incorporating explicit control over the cusp’s location, a rigorous enclosure for the normal form coefficient is obtained, providing the explicit dynamics on the center manifold at the cusp. We show the effectiveness of this method by applying it to four distinct models
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
Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters
No full textOn an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion
No full textIn this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues λ1=0<λ2<⋯<λN and null-vector Image 1. Then, we show how this result can be used to guarantee –if possible– that a synchronous orbit of a connected tridiagonal network associated to the matrix L above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for L
Two-Parameter SVD: Coalescing Singular Values and Periodicity
No full textWe consider matrix valued functions of two parameters in a simply connected
region . We propose a new criterion to detect when such functions
have coalescing singular values. For {\it generic\/} coalescings,
the singular values come together in a ``double cone''-like intersection.
We relate the existence of any such singularity to the periodic structure of the
orthogonal factors in the singular value decomposition of the one-parameter
matrix function obtained restricting to closed loops in .
Our theoretical result is very amenable to approximate numerically the
location of the singularities
Blow-up profile for solutions of a fourth order nonlinear equation
Full text linkIt is well known that the nontrivial solutions of the equation
u′′′′(r)+κu′′(r)+f(u(r))=0u′′′′(r)+κu′′(r)+f(u(r))=0
blow up in finite time under suitable hypotheses on the initial data, κκ and ff. These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher–Kolmogorov equation or Swift–Hohenberg equation.
In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation
Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions
Full text linkIn this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation
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