Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2001.05427

On the Independence of Fundamental Decompositions of Power-Law Kinetic Systems

Bryan S. Hernandez
View original source ↗Read source document ↗Export metadata ↓

Abstract

The fundamental decomposition of a chemical reaction network (CRN) is induced by partitioning the reaction set into "fundamental classes". It was the basis of the Higher Deficiency Algorithm for mass action systems of Ji and Feinberg, and the Multistationarity Algorithm for power-law kinetic systems of Hernandez et al. In addition to our previous work, we provide important properties of the independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and the incidence-independence (i.e., the image of the network's incidence map is the direct sum of the incidence maps' images of the subnetworks) of these decompositions. Feinberg established the essential relationship between independent decompositions and the set of positive equilibria of a network, which we call the Feinberg Decomposition Theorem (FDT). Moreover, Farinas et al. recently documented its version for incidence-independence. Fundamental decomposition divides the network into subnetworks of deficiency either 0 or 1 only. Hence, available results for lower deficiency networks, such as the Deficiency Zero Theorem (DZT), can be used. These justify the study of independent fundamental decompositions. A MATLAB program which (i) computes the subnetworks of a CRN under the fundamental decomposition and (ii) is useful for determining whether the decomposition is independent and incidence-independent is also created. Finally, we provide the following solution for determining multistationarity of CRNs with the following steps: (1) the use of the program, (2) the application of available results for CRNs with deficiency 0 or 1 (e.g., DZT), and (3) the use of FDT. We illustrate the solution by showing that the generalization of a subnetwork of Schmitz's carbon cycle model by Hernandez et al., endowed with mass action kinetics, has no capacity for multistationarity.

Explore related subjects

math.DS

Keep this discovery

BibTeXRIS

Bryan S. Hernandez. 2020-01-14. On the Independence of Fundamental Decompositions of Power-Law Kinetic Systems. https://arxiv.org/abs/2001.05427

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗