Luca Dieci

Lyapunov Spectral Intervals: Theory and Computation

Different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell [J. Differential Equations, 7 (1978), pp. 320--358] and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the so-called continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence how to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and Sacker--Sell spectra and present some numerical results.

Numerical calculation of invariant tori

The problem of computing a smooth invariant manifold for a finite-dimensional dynamical system is considered. In this paper, it is assumed that the manifold can be parameterized over a torus in terms of a subset of the system variables. The approach used here then involves solving a system of partial differential equations subject to periodic boundary conditions.The resulting numerical approach is analyzed and contrasted with some previously tested ones, and several methods of implementation are considered. Numerical results are given for the forced van der Pol equation and for a system of two linearly coupled oscillators.

Computing Connecting Orbits via an Improved Algorithm for Continuing Invariant Subspaces

A successive continuation method for locating connecting orbits in parametrized systems of autonomous ODEs was considered in [Numer. Algorithms, 14 (1997), pp. 103--124]. In this paper we present an improved algorithm for locating and continuing connecting orbits, which includes a new algorithm for the continuation of invariant subspaces. The latter algorithm is of independent interest and can be used in contexts different than the present one.

Computation of invariant tori by the method of characteristics

In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.

On the Error in QR Integration

An important change of variables for a linear time varying system

Block M -matrices and computation of invariant tori

\dot x=A(t)x, t\ge 0

Jacobian Free Computation of Lyapunov Exponents

, is that induced by the QR-factorization of the underlying fundamental matrix solution:

Numerical Techniques for Approximating Lyapunov Exponents and Their Implementation

X=QR

Lyapunov-type numbers and torus breakdown: numerical aspects and a case study

, with

Some stability aspects of schemes for the adaptive integration of stiff initial value problems

Q