Computing Convex Hulls of Trajectories

TL;DR

This work studies the convex hulls of trajectories of polynomial dynamical systems and presents numerical algorithms for identifying patches that furnishes a key step in computing attainable regions of chemical reaction networks.

Abstract

We study the convex hulls of trajectories of polynomial dynamical systems. Such trajectories include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We present numerical algorithms for identifying these patches. An implementation based on the software Bensolve Tools is given. This furnishes a key step in computing attainable regions of chemical reaction networks.

Figures (10)

• Figure 1: The Hamiltonian vector field defined by the Trott curve and two of its trajectories.
• Figure 2: A pair of ellipses encloses the Trott curve and bounds the attainable region.
• Figure 3: A Hausdorff convergent sequence of facets $F_\varepsilon$ of $A_\varepsilon$ need not converge to a face $F$ of $C$. The face $F$ on the left contains a curve point $y \in \mathcal{C}$ which is not extremal in $C$. The endpoint $F$ of the curve $\mathcal{C}$ on the right is an exposed face of $C$ but it is not uniquely exposed. There is no sequence of facets $F_\varepsilon$ of $A_\varepsilon$ that Hausdorff converges to that face $F$.
• Figure 4: A sample of points (left) from a space curve and its convex hull (right).
• Figure 5: Two $2$-patches (left) and three $1$-patches (right) in the boundary of a $4$-dimensional convex body. It is the convex hull of a trigonometric curve of degree six. The picture shows the graph $G$, with five connected components $G_i$, found by Algorithm \ref{['alg:boundary_examining_3']}.

Theorems & Definitions (33)

• Corollary 2.2
• Example 2.3: $n=2$
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof