Core Concepts
This work presents a novel numerical method to compute the Lyapunov exponents of renewal equations, along with a rigorous convergence analysis of the approximations.
Abstract
The authors propose a numerical method for computing the Lyapunov exponents (LEs) of renewal equations (REs), which are delay equations of Volterra type. The method consists of:
Reformulating the RE as an abstract differential equation on a Hilbert state space.
Discretizing the associated evolution family using a pseudospectral collocation approach.
Adapting the discrete QR (DQR) method to the resulting finite-dimensional approximations of the evolution operators.
The key contributions are:
A more rigorous approach compared to previous work, treating the RE directly in the Hilbert space setting.
Proving the convergence of the discretized evolution operators and the approximated LEs.
Providing a MATLAB implementation of the method.
The authors first formulate the REs on an L2 state space and prove the existence and uniqueness of the associated initial value problem. They then define the family of evolution operators and prove their (eventual) compactness, which is necessary for the LE computation.
Next, the authors present the discretization of the evolution operators via pseudospectral collocation and show that the discretization is well-defined. They then prove the convergence of the discretized operators to the exact ones in the operator norm.
Finally, the authors define the LEs for REs, describe their approximation in both the infinite- and finite-dimensional settings, and present the proof of convergence of the approximated LEs to (part of) the exact LEs.
The proposed method is experimentally effective and its convergence is rigorously proved, in contrast with previous work. The authors also provide a MATLAB implementation of the method.