3D Quadratic CDSs: Minimal Chemical Chaos

• The paper demonstrates that 3D quadratic CDSs are minimal mass-action models that inherently exhibit chaos through precise chemical and structural constraints.
• These systems require at least six monomials and negative cross-quadratic terms, ensuring the inevitability of chaotic dynamics under strict chemical rules.
• Enumeration identifies twenty minimal chaotic examples with distinct reaction counts and attractor types, validated by numerical Lyapunov exponent analysis.

Three-dimensional quadratic chemical dynamical systems (CDSs) form a mathematically and computationally minimal class of mass-action kinetic models exhibiting chaos, with a direct correspondence to small chemical reaction networks (CRNs). These systems—governed by ordinary differential equations (ODEs) in three state variables (concentrations of chemical species)—are among the simplest continuous-time, polynomial dynamical systems in which robust deterministic chaos is rigorously and computationally observed. Recent results establish sharp bounds on their structural complexity and enumerate explicit minimal examples, revealing that chaotic behavior is both structurally inevitable and relatively ubiquitous within this model class (Plesa et al., 6 Jan 2026, Plesa, 19 Nov 2025).

1. Mass-action Form and Chemical Constraints

A three-dimensional quadratic CDS is an ODE system of the form:

$$\begin{aligned} \dot{x} &= f_1(x, y, z), \ \dot{y} &= f_2(x, y, z), \ \dot{z} &= f_3(x, y, z), \end{aligned}$$

where each $f_i$ is a quadratic polynomial in %%%%2%%%%. The system is "chemical" if it respects the non-negativity of concentrations: for any $x_i=0$ and all other $x_j\geq0$, every monomial $m(x, y, z)$ appearing in $f_i$ is nonnegative. Equivalently, negative monomial coefficients are only allowed for terms containing $x_i$ itself, ensuring that mass is never driven negative from the boundary of the positive orthant. Each such monomial corresponds canonically to a chemical reaction under mass-action kinetics (Plesa et al., 6 Jan 2026).

The general normal form is:

$$f_i(x, y, z) = \alpha_{i,0} + \sum_{j=1}^3 \alpha_{i,j} x_j + \sum_{j\le k} \beta_{i,jk} x_j x_k,$$

with chemical sign and support restrictions on the $\beta_{i,jk}$ coefficients.

2. Structural Bounds and Necessary Conditions for Chaos

Two key theorems rigorously constrain the algebraic form of chaotic three-dimensional quadratic CDSs:

• Sign Theorem: In any $N$-dimensional polynomial CDS with a compact invariant set in $\mathbb{R}_+^N$, each equation must have at least one positive and one negative monomial; the total number of monomials must be at least $2N$. For $N=3$, this gives a minimum of six monomials (Plesa et al., 6 Jan 2026).
• Nonlinearity Theorem: Any chaotic 3D CDS must contain at least one negative nonlinear (usually quadratic) monomial in at least two variables, e.g., $-x y$ or $-y z$. Without this, the Jacobian is Metzler and the system is cooperative, ruling out chaos via the Hirsch theorem (Plesa et al., 6 Jan 2026).

These theorems exclude the existence of chaotic 3D quadratic CDSs with, for instance, only positive quadratic terms, or systems lacking cross quadratic interactions.

3. Enumeration and Classification of Minimal Chaotic Examples

A comprehensive computational enumeration under the above constraints yielded twenty inequivalent minimal 3D quadratic CDSs. Each system is specified by its right-hand-side ODEs, together with a tuple $(a, b)$ of total and quadratic monomial counts, and $(c, d)$, the number of induced reactions and quadratic reactions in its corresponding CRN (Plesa et al., 6 Jan 2026):

System	CDS Monomials $(a,b)$	CRN Reactions $(c,d)$	Example ODEs ($\dot{x}$, $\dot{y}$, $\dot{z}$)
CS$_1$	(6,4)	(5,3)	$x^2 - 0.5xy$, $x^2 - y z$, $y - 0.9 z$
CS$_2$	(6,4)	(5,3)	$xy - 0.4 xz$, $-y + x^2$, $-2z + x y$
CS$_6$	(6,5)	(4,3)	$2y^2 - x y$, $x y - 0.5 y z$, $-z + x y$
CS$_9$	(7,2)	(5,2)	$-4 x + x y$, $6 y - 0.2 y z$, $4 x + 6 y - 2 z$

No chaotic 3D chemical system with fewer than six monomials or with fewer than four quadratic terms has been found (Plesa et al., 6 Jan 2026). The minimal number of reactions is four, with a case (CS$_6$) for which three are quadratic—establishing the lowest known reaction-level complexity for chaos in mass-action CRNs.

4. Canonical Dynamical Motifs

Across these minimal systems, structural recurrence is observed:

• Each possesses at least one negative cross-quadratic term (e.g., $-xy$ or $-yz$), matching the nonlinearity theorem.
• Competing quadratic terms, positive and negative, especially in $\dot{x}$ (e.g., $x^2$ versus $-x y$), are common.
• A single "stable" direction (e.g., linear $-\alpha z$ in $\dot{z}$) complements two coupled nonlinear equations.
• No chemical quadratic CDS with only two quadratic monomials is found to be chaotic, even though such minimality exists for unconstrained (non-chemical) quadratic ODEs (Plesa et al., 6 Jan 2026, Plesa, 19 Nov 2025).

5. Numerical Verification and Attractor Properties

Chaos in these systems is verified computationally via numerically computed largest Lyapunov characteristic exponents (LCEs): for each exhibited system, one LCE is positive (indicative of sensitive dependence), one is near zero, and one is negative. Integration is performed using the continuous QR algorithm to accumulate the growth rates of tangent vectors along the flow, confirming robust chaos (Plesa, 19 Nov 2025).

A subset of these systems presents special attractor types:

• "One-wing" attractors with only a single lobe (CS$_1$, see (Plesa, 19 Nov 2025)).
• Hidden chaotic attractors, which do not intersect neighborhoods of any stable equilibrium (e.g., five-quadratic system described in (Plesa, 19 Nov 2025)).

6. Minimality, Chemical Realizability, and Implementation

Previous quadratic chaotic systems not constrained to be chemical (Sprott P, Willamowski-Rössler) involve fewer quadratic terms but cannot be realized as CDSs under the mass-action restriction. By contrast, the systems described here are chemically valid and thus, in principle, directly implementable in biochemical settings or synthetic chemical reactors, with precise predictions on the required number of species and reactions for robust chaos.

The minimal CDSs elucidate the boundary between structural minimality and dynamical complexity in chemical network design—highlighting the role of negative cross-quadratic terms and mixed feedback loops as the indispensable archetypes of chaos in low-dimensional mass-action networks (Plesa et al., 6 Jan 2026).

7. Broader Context and Significance

The classification and explicit enumeration of minimal chaotic 3D quadratic CDSs establishes their ubiquity and sharp structural lower bounds for chaos in mass-action CRNs. These results also demonstrate that even at the lowest levels of complexity, deterministic chaotic dynamics are accessible in chemical contexts that strictly enforce positivity and reaction-compatibility, closing a theoretical gap left open by prior studies limited to general polynomial systems or to biological oscillations (Plesa et al., 6 Jan 2026, Plesa, 19 Nov 2025). The work provides templates for further analytical and synthetic work on minimal chemical computation, as well as a foundation for understanding the relationship between polynomial network architecture and dynamical behavior in natural and engineered chemical systems.