Chemical systems with chaos

TL;DR

This work addresses the gap between chaotic dynamics in low-dimensional polynomial dynamical systems and their realization via chemical reaction networks by introducing quasi-chemical maps (QCMs) that transform DSs into chemically feasible CDSs of the same dimension with fewer nonlinear terms. It proves that, under robustness assumptions, $N$-dimensional quadratic DSs with a single quadratic monomial can be mapped to quadratic CDSs preserving chaos, and that quadratic DSs with two quadratics can map to cubic CDSs with a single cubic, enabling rich chaos in $3$-D CDSs. The authors construct explicit, comparatively simple CDS examples exhibiting one-wing chaos, two-wing chaos, and hidden chaos, each with a minimal number of highest-degree monomials and a compact CRN, illustrating chemically plausible routes to chaos. They also develop refinement techniques (splittings) to minimize nonlinear terms and discuss robustness and scaling considerations, providing concrete open questions about the ultimate limits of CDS simplicity and bounded chaotic behavior.

Abstract

Three-dimensional polynomial dynamical systems (DSs) can display chaos with various properties already in the quadratic case with only one or two quadratic monomials. In particular, one-wing chaos is reported in quadratic DSs with only one quadratic monomial, while two-wing and hidden chaos in quadratic DSs with only two quadratic monomials. However, none of the reported DSs can be realized with chemical reactions. To bridge this gap, in this paper, we investigate chaos in chemical dynamical systems (CDSs) - a subset of polynomial DSs that can model the dynamics of mass-action chemical reaction networks. To this end, we develop a fundamental theory for mapping polynomial DSs into CDSs of the same dimension and with a reduced number of non-linear terms. Applying this theory, we show that, under suitable robustness assumptions, quadratic CDSs, and cubic CDSs with only one cubic, can display a rich set of chaotic solutions already in three dimensions. Furthermore, we construct some relatively simple three-dimensional examples, including a quadratic CDS with one-wing chaos and three quadratics, a cubic CDS with two-wing chaos and one cubic, and a quadratic CDS with hidden chaos and five quadratics.

Figures (5)

• Figure 1: Three-dimensional CDSs with chaos. Panels (a)--(c) display solutions of the quadratic CDS $(\ref{['eq:CDS_1']})$ with exactly $3$ quadratic monomials, cubic CDS $(\ref{['eq:CDS_2']})$ with exactly $1$ cubic monomial, and the quadratic CDS $(\ref{['eq:CDS_3']})$ with exactly $5$ quadratic monomials, respectively. The parameters and initial conditions are chosen as in Figures \ref{['fig:CDS_1']}--\ref{['fig:CDS_3']}, respectively.
• Figure 2: Quadratic $(11,5)$ chemical Rössler system. Panels (a) and (b) display respectively the $(y,z)$- and $(t,y)$-space for the reflected Rössler system $(\ref{['eq:Rossler_reflected']})$ with initial condition $(x_0,y_0,z_0) = (5,-5,5)$; panel (c) shows time-evolution of the three LCEs for this solution, along with the values at $t = 10^6$ as dashed black lines. Analogous plots are shown in panels (d)--(f) for the perturbed Rössler system $(\ref{['eq:Rossler_reflected_perturbed']})$, and in (g)--(i) for its translated and rescaled version $(\ref{['eq:CDS_Rossler']})$ with the translated and rescaled initial condition $x_0 = \varepsilon \mu (5 + (\varepsilon^2 \mu)^{-1})$, $y_0 = -5 + (\varepsilon \mu)^{-1}$, $z_0 = 5 \varepsilon (5 - \varepsilon)^{-1} (5 + \mu^{-1})$. The parameters are fixed to $(a,b,c) = (\varepsilon^{-2},\varepsilon^{-1},1)$ and $(\varepsilon,\mu) = (10^{-3}, 10^{-2})$.
• Figure 3: Quadratic $(10,3)$ CDS with one-wing chaos. Panels (a)--(b) display respectively the $(x,y)$- and $(t,x)$-space for DS $(\ref{['eq:DS_1']})$ with initial condition $(x_0,y_0,z_0) = (0.5,0,0)$; the two equilibria are shown as black dots in (a). Panel (c) shows the three finite-time LCEs, along with the values at $t = 10^6$ shown as the dashed lines. Panels (d)--(e) show analogous plots for CDS $(\ref{['eq:CDS_1']})$ with $(\varepsilon,\mu) = (10^{-2},10^{-2})$ and the translated and rescaled initial condition $x_0 = |\varepsilon^{-2} + 2.7 \varepsilon^{-1} - 2 \mu^{-1}|^{-1} (0.5 + \mu^{-1})$, $y_0 = (10/27) ((\varepsilon^2 \mu)^{-1} + 2.7 (\varepsilon \mu)^{-1})$, $z_0 = (\varepsilon \mu)^{-1}$. Panel (f) shows the LCEs for the perturbed DS $(\ref{['eq:1_perturbed']})$ with $a = 1$ and $c = \varepsilon^{-1}$, which are in the long-run identical to the long-time LCEs of $(\ref{['eq:CDS_1']})$.
• Figure 4: Cubic $(11,5,1)$ CDS with two-wing chaos. Panels (a)--(b) display respectively the $(x,y)$- and $(t,y)$-space for DS $(\ref{['eq:DS_2']})$ with initial condition $(x_0,y_0,z_0) = (0,0,-1)$; the two equilibria are shown as black dots in panel (a). Panel (c) shows the three finite-time LCEs, along with the values at $t = 10^6$ shown as the dashed lines. Panels (d)--(e) show analogous plots for CDS $(\ref{['eq:CDS_2']})$ with $(\varepsilon,\mu) = (10^{-3},10^{-2})$ and the translated and rescaled initial condition $x_0 = 2 (\varepsilon \mu)^{-1}$, $y_0 = \varepsilon \mu^{-1}$, $z_0 = 2 \mu^{-1} (-1 + \mu^{-1})$. Panel (f) shows the LCEs for the perturbed DS $(\ref{['eq:2_perturbed']})$ with $b = 1$, which are in the long-run identical to the long-time LCEs of $(\ref{['eq:CDS_2']})$.
• Figure 5: Quadratic $(11,5)$ CDS with hidden chaos. Panels (a)--(b) respectively display the $(x,y)$- and $(t,y)$-space for DS $(\ref{['eq:DS_3']})$, with initial condition $(x_0,y_0,z_0) = (-5,0,7.5)$; the stable equilibrium is shown as a black dot in (a). Panel (c) shows the three finite-time LCEs, along with the values at $t = 10^6$ shown as the dashed lines. Panels (d)--(e) show analogous plots for CDS $(\ref{['eq:CDS_3']})$ with $(\varepsilon,\mu) = (10^{-5}, 10^{-5})$ and the translated and rescaled initial condition $x_0 = (3.1 + 0.285 \mu)^{-1} (-5 + (\varepsilon \mu)^{-1})$, $y_0 = 2 \mu^{-1}$, $z_0 = 0.6 \mu^{-1} (3.1 + 0.285 \mu)^{-1} (7.5 + 2 \mu^{-1})$. Panel (f) shows the LCEs for the perturbed DS $(\ref{['eq:3_perturbed']})$ with $b = c = 2$, which are in the long-run identical to the long-time LCEs of $(\ref{['eq:CDS_3']})$.

Theorems & Definitions (49)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Example 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Theorem 3.1
• proof