Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation

Optimization-based Framework for Stability and Robustness of Bipedal Walking Robots

As robots become more sophisticated and move out of the laboratory, they need to be able to reliably traverse difficult and rugged environments. Legged robots -- as inspired by nature -- are most suitable for navigating through terrain too rough or irregular for wheels. However, control design and stability analysis is inherently difficult since their dynamics are highly nonlinear, hybrid (mixing continuous dynamics with discrete impact events), and the target motion is a limit cycle (or more complex trajectory), rather than an equilibrium. For such walkers, stability and robustness analysis of even stable walking on flat ground is difficult. This thesis proposes new theoretical methods to analyse the stability and robustness of periodic walking motions. The methods are implemented as a series of pointwise linear matrix inequalities (LMI), enabling the use of convex optimization tools such as sum-of-squares programming in verifying the stability and robustness of the walker. To ensure computational tractability of the resulting optimization program, construction of a novel reduced coordinate system is proposed and implemented. To validate theoretic and algorithmic developments in this thesis, a custom-built “Compass gait” walking robot is used to demonstrate the efficacy of the proposed methods. The hardware setup, system identification and walking controller are discussed. Using the proposed analysis tools, the stability property of the hardware walker was successfully verified, which corroborated with the computational results

Asymptotic Poincaré maps along the edges of polytopes

For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope’s vertexes and edges. This piecewise linear flow is easy to compute even in higher dimensions, which allows the usage of numeric algorithms to find invariant dynamical structures such as periodic, homoclinic or heteroclinic orbits, which if robust persist as invariant dynamical structures of the original flow. We apply this method to prove the existence of chaotic behavior in some Hamiltonian replicator systems on the five dimensional simplex.info:eu-repo/semantics/publishedVersio

On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $f:K\to R^n$ on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within $L_\infty$ distance r from f for a given r>0. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps $f,f': K\to R^n$ with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.Comment: 20 pages main paper including bibliography, followed by 22 pages of Appendi