SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

# Invited Talk

### A novel method to compute Lyapunov exponents of switched linear systems (I)

M. Zennaro and N. Guglielmi

Abstract
For a given finite set of matrices $\mathcal{G} = \{ C_1,\ldots,C_m\}$, where $C_i \in \mathbb{R}^{d,d}$, $i=1,\ldots,m$, we consider the switched linear system of ODEs $\dot{x}(t) = C( u(t)) x(t), x(0) = x_0$, where $x(t) \in \mathbb{R}^d$, $u: \mathbb{R} \longrightarrow \{1,\ldots,m\}$ is a piecewise constant control function and $C(i)=C_i$. We propose a novel method for the approximation of the upper Lyapunov exponent and, under suitable assumptions, of the lower Lyapunov exponent which is based on the computation of the joint spectral radius and lower spectral radius of a sequence of discretized systems obtained by forcing the switching instants to be multiple of $\Delta^{(k)} t$, where $\Delta^{(k)} t \to 0$ as $k \to \infty$. In the first talk we shall give general ideas to bound the Lyapunov exponents. In the second talk we shall provide details about the approximation of the joint (lower) spectral radius.

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