Planar chemical reaction systems with algebraic and non-algebraic limit cycles

TL;DR

The paper investigates Hilbert-type questions for planar ODEs arising from two-species chemical reaction networks, defining analogues $S(n)$, $M(n)$, $W(n)$ and their algebraic variants, and connecting limit-cycle counts to network structure. It develops a framework linking algebraic limit cycles to algebraic curves $h(x,y)=0$ via cofactors, transversality, and perturbation theory, enabling robust persistence of multiple algebraic limit cycles under controlled perturbations. Leveraging Harnack's curve theorem, it demonstrates constructions with up to four ovals (and thus four stable algebraic limit cycles) and provides a general method to realize $N$ algebraic limit cycles in reversible networks, yielding bounds like $W^a(4N+2) \ge N$. The results offer explicit reversible-CRN realizations, analytic criteria for robustness, and concrete examples (including quartic curves) that illustrate how algebraic limit cycles can be embedded and counted within chemical reaction networks, with implications for higher-dimensional and stochastic settings.

Abstract

The Hilbert number $H(n)$ is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most $n \in {\mathbb N}$. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where $n$ is equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number $H(n)$ for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to the $n$-th order; (ii) systems with up to $n$-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve $h(x,y)=0$ of degree $n_h \in {\mathbb N}$ and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most $n=2\,n_h+1.$ Considering $n_h \ge 4,$ the algebraic curve $h(x,y)=0$ can contain multiple closed components with the maximum number of ovals given by Harnack's curve theorem as $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for $n_h=4.$ Algebraic curve $h(x,y)=0$ with $n_h=4$ and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.

Figures (8)

• Figure 1: Schematics of planar E-graphs associated with (a) Lotka-Volterra chemical system $(\ref{['LotkaVolterraCRN']})$; (b) chemical system $(\ref{['illustrativenetwork1']})$; (c) chemical system with six reactions given by $(\ref{['illustrativenetwork1']})$ and $(\ref{['illustrativenetwork2']})$.
• Figure 2: (a) The quartic algebraic curve $h(x,y)=0$ given by $(\ref{['threeovalcurve']})$ has three (blue) ovals in the positive quadrant. The red line is given by $7x-y=0$. The red line does not intersect the (blue) ovals of $h(x,y)=0$ in the positive quadrant. We use log scale on the $x$-axis and $y$-axis. (b) The chemical reaction network corresponding to the blue edges of the planar E-graph realizes the 'unperturbed' ODE system $(\ref{['WR_network_small_x']})$--$(\ref{['WR_network_small_y']})$, while the $\varepsilon$-perturbations are shown by the red edges. (c) The weakly reversible network that consists of the blue edges (with some modified rate constants) together with the red edge provides a weakly reversible realization of the perturbed system $(\ref{['WR_network_small_perturbed_x']})$--$(\ref{['WR_network_small_perturbed_y']})$.
• Figure 3: (a) Planar E-graph of a reversible chemical reaction network corresponding to the ODE system $(\ref{['unit_square_system_x']})$--$(\ref{['unit_square_system_y']})$ as its reaction rate equations with all reaction rate constants equal to $1.$ (b) A geometric representation of the monomials of the polynomial $h(x,y)$ given by $(\ref{['h_x_y_9']})$. The blue points represent the monomials with positive coefficients and the red point represents its negative monomial $-9xy$. (c) The dynamical system obtained by multiplying the network shown in (a) by the factor $h(x,y)$ shown in (b)$($which gives rise to the equations $(\ref{['min_reversible_system_x']})$--$(\ref{['min_reversible_system_y']}))$ can be realized by this reversible 'full-grid' network.
• Figure 4: (a) The algebraic curve $h(x,y)=0$ given by $(\ref{['h_x_y_9']})$ is plotted as the blue line, together with algebraic curves $h_i(x,y)=0$ given by $(\ref{['h_i']})$ for $\delta_i = 2$ (red line), $\delta_i=3$ (green line) and $\delta_i = 4$ (magenta line). (b) The negative coefficients of product $h_0$ given by $(\ref{['h_0']})$ correspond to $($a subset of$)$ monomials that are represented by the red points, while the coefficients of the monomials that are represented by the blue points are all positive. (c) A reversible chemical reaction network which can be modelled by the reaction rate equations written in the form of the ODE system $(\ref{['reversible_system_perturbed_prod_hi_x']})$--$(\ref{['reversible_system_perturbed_prod_hi_y']})$, which has $N$ algebraic limit cycles.
• Figure 5: (a) The phase plane of the ODE system $(\ref{['WR_network_small_x']})$--$(\ref{['WR_network_small_y']})$, i.e. the ODE system $(\ref{['WR_network_small_perturbed_x']})$--$(\ref{['WR_network_small_perturbed_y']})$ for $\varepsilon=0$. We plot the algebraic curve $h(x,y)\equiv x^2 + x y^2 + y - 4xy = 0$ (black dashed line) together with some illustrative trajectories starting at the boundary of the visualized square. All trajectories converge to a stable steady state on the curve $h(x,y)=0.$ (b) The phase plane of the ODE system $(\ref{['odex43b']})$--$(\ref{['odey43b']})$, i.e. the ODE system $(\ref{['WR_network_small_perturbed_x']})$--$(\ref{['WR_network_small_perturbed_y']})$ for $\varepsilon=1$. The algebraic curve $h(x,y)=0$ becomes a stable limit cycle for $\varepsilon>0$.

Theorems & Definitions (37)

• Definition 1
• Lemma 1
• proof
• Definition 2
• Lemma 2
• proof
• Definition 3
• Lemma 3
• proof
• Definition 4