Cinzia Elia

On the Equivalence between the Sigmoidal Approach and Utkin's Approach for Piecewise-Linear Models of Gene Regulatory Networks ∗

This paper is concerned with piecewise-linear dynamical systems modeling a simple class of gene regulatory networks. One of the main issues when dealing with these problems is that the vector field is not defined on the discontinuity hyperplanes. Two different methods are usually employed in the literature to overcome this issue: Filippovs convexification approach and the steep sigmoidal approach. A particular selection of Filippovs vector field, namely Utkins vector field, will be of interest to us. Our purpose is twofold: show that Utkins vector field is well defined on the intersection

SVD algorithms to approximate spectra of dynamical systems

\Sigma

Sharp sufficient attractivity conditions for sliding on a co-dimension 2 discontinuity surface

of two discontinuity hyperplanes (under assumptions of attractivity) and prove that, for

Computation of few Lyapunov exponents by geodesic based algorithms

\Sigma

Uniqueness of Filippov Sliding Vector Field on the Intersection of Two Surfaces in \mathbb {R}^3 and Implications for Stability of Periodic Orbits

nodally attractive and attractive with three surfaces, Utkins approach and the steep sigmoidal approach are equivalent, i.e., the corresponding solutions on

Exponential monotonicity of quadratic forms in ODEs and preserving methods

\Sigma

Symplectic Method Based on the Matrix Variational Equation for Hamiltonian System

are the same. This allows us to study the piecewise dynamical system, and hence the gene regulatory network it models, with no ambiguity.

Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics

In 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].

A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis☆

We consider Filippov sliding motion on a co-dimension 2 discontinuity surface. We give conditions under which Σ is attractive through sliding which are sharper than those given in a previous paper of ours. Under these sharper conditions, we show that the sliding vector field considered in the same paper is still uniquely defined and varies smoothly in x?Σ. A numerical example illustrates our results.

Exponential dichotomy on the real line: SVD and QR methods☆

In this paper, we apply numerical methods based on the embedded geodesics for computing few Lyapunov exponents of a finite dimensional dynamical system. These new numerical algorithms are designed using geometric structure of the Stiefel manifold and they preserve orthogonality to machine accuracy. Numerical tests are also provided in order to show the features of our methods.