Thomas' Cyclically Symmetric Attractor

• Thomas’ Cyclically Symmetric Attractor is a nonlinear dynamical model defined by cyclic permutation invariance among variables, exemplifying key behaviors in feedback circuits.
• The attractor’s cyclic symmetry governs a rich bifurcation landscape, transitioning from stable fixed points to chaotic, hyperchaotic, and multistable regimes.
• Extensions of the model include coupled oscillators and swarmalators, linking its dynamics to pattern formation, wall-crossing phenomena, and advanced mathematical invariants.

Thomas’ Cyclically Symmetric Attractor is a paradigmatic structure arising in nonlinear dynamical systems whose equations admit cyclic symmetry among their variables. Originally introduced by René Thomas in the context of biological feedback circuits and later generalized in various mathematical and physical models, such attractors capture a diverse spectrum of complex dynamical phenomena—including chaos, hyperchaos, multistability, and self-organization—while being deeply shaped by the underlying symmetry. The defining feature is invariance under cyclic permutation of the system’s dynamical variables, which not only governs spectra of bifurcations and transitions to chaos, but also has broad implications for mathematical objects such as Donaldson–Thomas invariants, wall-crossing phenomena, and pattern formation in active matter systems.

1. Mathematical Formulation and Symmetry Principles

The archetype of Thomas’ cyclically symmetric attractor is encapsulated by systems of coupled ordinary differential equations (ODEs) with the form:

$$\begin{align*} \dot{x}_1 &= -b x_1 + f(x_2) \ \dot{x}_2 &= -b x_2 + f(x_3) \ &\vdots \ \dot{x}_n &= -b x_n + f(x_1) \end{align*}$$

where $b > 0$ is a dissipation parameter and $f(u)$ is a nonlinear function—commonly $f(u) = \sin(u)$—with indices taken modulo $n$ to enforce cyclic symmetry (Basios et al., 2018, Ho, 2019, Sorin et al., 18 Aug 2024). In the canonical three-dimensional case:

$$\dot{x} = -b x + \sin(y), \quad \dot{y} = -b y + \sin(z), \quad \dot{z} = -b z + \sin(x)$$

The key property is invariance under the cyclic permutation $(x, y, z) \mapsto (y, z, x)$. This symmetry is reflected in the Jacobian structure, bifurcation scenarios, and invariant manifolds of the system. Generally, higher-dimensional versions preserve these features, with each variable driven by the "next" in the cycle (Ho, 2019).

The dynamical consequences of this symmetry are profound. The bifurcation structure includes pitchfork and saddle-node bifurcations that always result in the simultaneous emergence of symmetrically related fixed points or limit cycles, and Hopf bifurcations occurring at values that maintain the same invariance (Sorin et al., 18 Aug 2024). For example, as the dissipation $b$ is decreased, the system moves from a single stable fixed point (the origin) into regimes with multiple coexisting, symmetry-related attractors and, ultimately, into chaotic or hyperchaotic dynamics, still constrained by cyclic permutation symmetry.

2. Bifurcation Landscape and Attractor Dynamics

As established in (Sorin et al., 18 Aug 2024), the primary dynamical regimes are organized through the control parameter $b$:

• Stable fixed point for $b > 1$: Only the origin is stable, with global attraction due to strong damping.
• Pitchfork bifurcation near $b = 1$: The origin loses stability and two new fixed points arise, symmetric under the cyclic group—an explicit marker of symmetry breaking in the attractor landscape.
• Double saddle-node bifurcations: As $b$ decreases further, additional pairs of fixed points are generated in a periodic cascade, controlled by maxima of the function $\sin(x)/x$.
• Infinite sequence of Hopf bifurcations: Each new fixed point can give rise to limit cycles via Hopf processes, with bifurcation points governed by the condition $b = -\cos(x^*)/2$ for a fixed point at $x^*$.
• Chaotic and hyperchaotic regimes: For lower $b$, multiple coexisting attractors emerge—including those related by cyclic permutation—resulting in symmetry-invariant but dynamically rich and possibly fractal attractor sets with positive Lyapunov exponents and dimensions approaching the system dimension as $b \to 0$.

The attractor structure thus typically consists of several symmetric orbits in phase space, connected by the cyclic transformations, with transitions to more intricate patterns through further loss of regularity as dissipation weakens (Sorin et al., 18 Aug 2024).

3. Random Walks, Brownian Motion, and High-Dimensional Extensions

In the limit $b \to 0$, Thomas’ systems exhibit a transition to conservative dynamics. Here, the phase space flow becomes volume-preserving and the attractor structure is dramatically altered (Ho, 2019):

• Infinite set of neutrally stable fixed points at $(x^*, y^*, z^*) = (\pm\pi n, \pm\pi m, \pm\pi k)$, $n, m, k \in \mathbb{Z}$.
• Labyrinth chaos and chaotic walks: Typical solutions no longer settle onto attractors in the classical sense but exhibit random-walk behavior across the lattice of fixed points. This is evidenced by the trajectory statistics, with the difference function $D(t) = \sum_i [x_i(t) - x_i(0)]^2$ approximated by

$D(t) = 2E(1 - e^{-t/\tau})$

for total energy $E$ and diffusion time $\tau$.

In high-dimensional generalizations, the system's state is constrained onto the surface of a hypersphere, and error growth between nearby solutions transitions from initial exponential divergence (chaotic) to linear growth and then to saturation, bounded by the system’s conserved energy. This mirrors the error dynamics seen in rotating turbulence and provides a model for understanding limitations on predictability in complex flows (Ho, 2019).

4. Coupled Systems, Chimera States, and Pattern Formation

Thomas’ cyclically symmetric attractor also serves as a building block for spatially extended and coupled systems. When individual Thomas oscillators are linearly or nonlinearly coupled, the following phenomena are observed (Basios et al., 2018, Vijayan et al., 2020):

• Synchronization regimes: Depending on coupling strength and type (linear vs. sinusoidal), systems can display complete synchronization, lag/anti-lag synchronization, and space-lag (chiral/eddy) synchronization, with distinct phase relations and coordinated or swirling motion.
• Transient chaos: Coupled systems often pass through prolonged transiently chaotic epochs before relaxing onto their eventual attractors (either periodic, quasi-periodic, or chaotic).
• Chiral phenomena: The presence of strong nonlinearities or anisotropies in coupling generates attractors that lack mirror symmetry (chirality) in their collective trajectories, reminiscent of structures in fluid dynamics.
• Chimera states: In networks, especially rings with nonlocal coupling, coherent (locked) and incoherent (drifting) domains can spontaneously form, paralleling chimera states known from the Kuramoto model and other studies of coupled oscillators.

These results reinforce the robustness of cyclic symmetry: regardless of the complexity generated by coupling or spatial structure, the organization of the system remains linked to the cyclic group actions.

5. Modelling Active Matter and Swarmalators

The Thomas oscillator can be further endowed with internal phase dynamics, resulting in hybrid models known as "swarmalators"—entities exhibiting both self-oscillation and spatial self-organization (Vijayan et al., 2022). Explicitly, the motion of particle $i$ with position $\mathbf{r}_i$ and phase $\Theta_i$ is governed by:

$$\begin{align*} \dot{\mathbf{r}}_i &= \mathbf{f}(\mathbf{r}_i) + \frac{1}{N}\sum_{j\neq i} \left\{ \frac{\mathbf{r}_j - \mathbf{r}_i}{|\mathbf{r}_j - \mathbf{r}_i|} [A + J\cos(\Theta_j - \Theta_i)] - \frac{B(\mathbf{r}_j - \mathbf{r}_i)}{|\mathbf{r}_j - \mathbf{r}_i|^3} \right\} \ \dot{\Theta}_i &= \frac{K}{N}\sum_{j\neq i} \frac{\sin(\Theta_j - \Theta_i)}{|\mathbf{r}_j - \mathbf{r}_i|^2} \end{align*}$$

Here, particle dynamics combine cyclically symmetric self-oscillation (the Thomas core) with spatial aggregation and phase-coupled migration. These models display:

• Pattern formation: Hexagonal lattices, active phase waves, cluster synchronization, and fully turbulent (active turbulence) regimes depending on coupling and damping parameters.
• Self-organization in active matter: The system is a prototypical model for collective behavior in biological systems (e.g., bacterial colonies, micro-swimmers) and novel materials, with transitions between ordered, semi-ordered, and strongly chaotic collective states directly controlled by modification of cyclic symmetry and coupling terms (Vijayan et al., 2022).

6. Cyclic Symmetry in Donaldson–Thomas Invariants and Wall-Crossing

The notion of cyclically symmetric attractors finds a deep mathematical counterpart in the paper of Donaldson–Thomas (DT) invariants, wall-crossing, and related structures (1303.3253, Reineke, 4 Oct 2024). In these contexts:

• Attractor flows on (complex) moduli spaces are modeled as gradient flows with cyclic symmetry in the underlying charge or cohomology lattice (for example, on the holomorphic volume forms of a Calabi–Yau threefold). An attractor flow with charge vector $\gamma$ and central charge $Z(\gamma) = \int_{\gamma} \Omega^{3,0}$ selects a unique attractor point invariant under cyclic shifts, providing initial data for wall-crossing recursion and wall-crossing formulas.
• Cyclic ordering and wall-crossing: The attractor’s cyclic symmetry is essential in determining wall-crossing behavior for DT-invariants, especially in non-compact Calabi–Yau and spectral curve settings, where a wheel-like (cyclic) gluing of local models encodes the global wall-crossing structure.
• Symmetric quivers and cyclic invariance: In the representation theory of quivers, DT-invariants of symmetric quivers can be written in product forms

$$P_A(x) = \prod_{d \in \mathbb{N}^n} \prod_{k \in \mathbb{Z}} (1 - q^{k/2} x^d)^{(-1)^{k-1}c_{d, k}}$$

which remain invariant under cyclic permutation of the factors. The invariants themselves can be interpreted as "attractor coefficients," counting cyclically invariant objects—whether as graded pieces in a Cohomological Hall algebra or as combinatorial break divisors on symmetric graphs (Reineke, 4 Oct 2024).

This interplay underscores a unifying principle: cyclic symmetry organizes not only dynamical attractors in ODEs but also the deeper algebraic and geometrical data in moduli problems, stability structures, and wall-crossing for DT-invariants.

7. Applications, Generalizations, and Broader Significance

Thomas’ cyclically symmetric attractor has found application and theoretical resonance in multiple domains:

• Dissipative and conservative systems: Serving as a testbed for bifurcation and chaos theory, error growth analysis, and delineation of predictability horizons in high-dimensional flows (e.g., turbulence foregrounds) (Ho, 2019).
• Active fluids and materials science: As a prototypical model for pattern formation and dynamic self-organization in active matter, and for exploring routes to “swarming” and “active turbulence” (Vijayan et al., 2022).
• Networks and synchronization: Underpinning studies of coupled oscillators, emergence of chimera states, and scenarios where synchronization is governed by local and global symmetry constraints (Basios et al., 2018, Vijayan et al., 2020).
• Algebraic geometry and mathematical physics: Providing a structural parallel in the algebraic decomposition and wall-crossing theory of DT-invariants, bonded by cyclic symmetry and combinatorial invariance (1303.3253, Reineke, 4 Oct 2024).
• Design and realization of attractors with stylized symmetry: Classification and construction of attractors with prescribed cyclic or Platonic symmetry, leveraging combinatorial frameworks such as Sturm permutations and Thom–Smale complexes (1708.00690).

In all these instances, the enforcement or emergence of cyclic symmetry both constrains and enriches the spectrum of possible behaviors—embedding deep connections between nonlinear dynamics, symmetry groups, complex geometry, and collective pattern formation.

---

Table: Key Regimes and Phenomena in the Thomas System

Regime	Control Parameter(s)	Attractor Structure
Stable fixed point	$b > 1$	Single point (origin), globally attracting
Pitchfork / saddle-node bifurcation	$b \lesssim 1$	Multiple fixed points, cyclically arranged
Hopf bifurcations / limit cycles	Decreasing $b$	Cyclically symmetric limit cycles
Chaotic / hyperchaotic regime	$b \to 0$	Multiple coexisting chaotic attractors
Conservative / Brownian-like motion	$b = 0$	Infinite neutrally stable fixed points,
		chaotic walks (Labyrinth chaos)

---

In summary, Thomas’ Cyclically Symmetric Attractor is a central object in the paper of nonlinear systems with cyclic symmetry, offering a model of remarkable dynamical richness and serving as a nexus linking feedback circuits, pattern formation, chaotic turbulence, and wall-crossing invariants. Its mathematical structure, shaped by cyclic invariance, continues to inform theoretical advances and experimental modeling across disciplines.