A Novel Strange Attractor of Logic and Tension

1. Abstract

The Tantara Attractor is a newly defined strange attractor designed to model how creative tension, logic, and paradigm shifts interact. Inspired by the investment thesis of Tara Tan (Strange Ventures), the system modifies the classical Lorenz equations by introducing a transcendental "saturation" term, the hyperbolic tangent (\(\tanh\)).

This modification breaks the inherent symmetry of the standard chaotic fold, resulting in a topologically distinct, asymmetric "tilted butterfly" shape. The system is both a valid mathematical object and a computational metaphor for "creativity grounded in logic."

2. Inspiration & Philosophy

The design of the Tantara Attractor is grounded in the "Strange" philosophy: the belief that true creativity is not random, but emerges from structured, self-reflexivity and cognitive tension.

2.1 The "Tan" Sigmoid

A core poetic element of the system is the choice of the non-linear operator. We utilize the Hyperbolic Tangent (\(\tanh\)), a function fundamental to two fields:

1. Neural Networks: It is a primary activation function, representing the firing thresholds of artificial neurons, a nod to the "human-machine intersection."
2. Nomenclature: The function's name, \(\tanh\), establishes a direct symbolic link to the system's inspiration.

2.2 The Narrative of Equations

Standard chaotic systems often treat variables as abstract quantities. The Tantara system assigns them semantic meaning:

• \(x\) (Tension): The contrarian energy; the drive to diverge from the norm.
• \(y\) (Creativity): The generative output; the "art" produced by the system.
• \(z\) (Logic): The structural foundation; the technical truth or "world model."

In this model, Logic (\(z\)) does not merely dampen the system. Through the novel \(\tanh\) term, fully saturated Logic exerts a constant, unyielding pressure on Tension (\(x\)), forcing a paradigm shift in the system's trajectory.

3. Mathematical Definition

The Tantara Attractor is defined by the following system of three coupled, autonomous, non-linear ordinary differential equations.

3.1 Parameters (The "Strange" Regime)

To observe the unique Tantara topology, the following parameter values are established:

4. Novelty & Analysis

4.1 Classification: The Asymmetric Lorenz Variant

The Tantara Attractor belongs to the family of Lorenz-like systems but is distinguished by symmetry breaking via saturation.

• Classical Lorenz: The standard Lorenz system is invariant under the transformation \((x, y) \to (-x, -y)\). This produces a perfectly symmetrical "butterfly" where both wings are mirror images.
• Tantara Mechanism: The term \(+ \mu \tanh(z)\) destroys this symmetry. The sophistication of this modification lies in the bounded influence of the transcendental term. Unlike polynomial modifications (e.g. Chen or Lü systems) where feedback grows indefinitely with state variables, the \(\tanh(z)\) term enforces a "soft saturation." This creates a dual-regime system:
  1. Linear Regime \((z \approx 0)\): The system behaves like a locally twisted Lorenz.
  2. Saturated Regime \((|z| \gg 1)\): The feedback to tension becomes a constant drift \((\pm \mu)\). This structure forces the trajectory to fold not just due to rotation, but due to a hard asymptotic limit on the "logic" feedback, effectively simulating a cognitive capacity constraint.

4.2 Behavior

• Boundedness: The system is dissipative. The divergence of the flow is \(\nabla \cdot F = -(\sigma + 1 + \beta)\), which is negative for all positive parameter values. This guarantees that phase space volume contracts, confining trajectories to a bounded strange attractor. By analogy with the classical Lorenz system (whose Kaplan-Yorke dimension is approximately 2.06), the Tantara attractor is expected to have a fractal dimension slightly greater than 2.
• Sensitivity: The system exhibits high sensitivity to initial conditions (The Butterfly Effect). However, the \(\tanh\) term introduces regions of "soft stability" where high Logic (\(z\)) temporarily linearizes the feedback to Tension, creating a unique "fast-slow" dynamic not present in pure polynomial systems.

4.3 Stability & Jacobian Analysis

To analyze the local stability of the system, we compute the Jacobian Matrix \(J\). The eigenvalues of this matrix at equilibrium points reveal the nature of nearby trajectories.

Jacobian Matrix \(J(x,y,z)\):

Dissipation Criterion:

The trace of the Jacobian equals the divergence of the flow, which determines whether the system is dissipative (contracting phase space volume):

Since this is strictly negative for all positive parameter values, the system is globally dissipative: all trajectories are eventually confined to a set of measure zero (the attractor).

4.4 Equilibrium Points

The equilibrium points of the system occur where all derivatives vanish simultaneously. Setting \(\dot{x} = \dot{y} = \dot{z} = 0\):

The Origin \((0, 0, 0)\):

The origin is always an equilibrium point. Evaluating the Jacobian at \((0,0,0)\) with \(\operatorname{sech}^2(0) = 1\):

The eigenvalues are approximately \(\lambda_1 \approx 11.83\), \(\lambda_2 \approx -22.83\), and \(\lambda_3 \approx -2.67\). The presence of one positive eigenvalue confirms the origin is an unstable saddle point: trajectories are repelled along one direction while attracted along others, forcing them onto the strange attractor.

Non-Trivial Equilibria:

Unlike the classical Lorenz system, which has symmetric equilibria at \((\pm\sqrt{\beta(\rho-1)}, \pm\sqrt{\beta(\rho-1)}, \rho-1)\), the Tantara system's \(\tanh\) term breaks this symmetry. The non-trivial equilibria satisfy the transcendental system:

These must be solved numerically and yield asymmetric equilibrium points, another manifestation of the symmetry breaking that gives the Tantara its distinctive "tilted" topology.

5. Visual Characteristics

Visually, the Tantara Attractor resembles a "Tilted Butterfly."

The three distinctive features of the Tantara: lobes that fold, a crossover that warps, and a spine that constrains.

1. The Fold: Like the Lorenz, it features two primary scrolling lobes.
2. The Warp: Unlike the Lorenz, the "crossover" point (where the trajectory switches wings) is offset from the origin.
3. The Spine: The Z-axis (Logic) acts as a rigid spine. As the trajectory climbs the Z-axis, the \(\tanh(z)\) term saturates to 1, applying a maximum constant force that "torques" the system back down, preventing infinite growth but ensuring constant motion.

6. Implementation Notes

For researchers, artists, or developers wishing to simulate the Tantara Attractor, snippets are provided below for common environments.

6.1 Python (Scientific Computing)

Ideal for plotting, analysis, and verifying Lyapunov exponents.

import numpy as np
from scipy.integrate import odeint

def tantara(state, t, sigma=10.0, rho=28.0, beta=8.0/3.0, mu=15.0):
    x, y, z = state
    dxdt = sigma * (y - x) + mu * np.tanh(z)
    dydt = rho * x - y - x * z
    dzdt = x * y - beta * z
    return [dxdt, dydt, dzdt]

# Usage
state0 = [1.0, 1.0, 1.0]
t = np.arange(0.0, 40.0, 0.01)
result = odeint(tantara, state0, t)

6.2 Processing / Java (Creative Coding)

Optimized for real-time generative art visualizations.

float x = 1.0, y = 1.0, z = 1.0;
float sigma = 10.0, rho = 28.0, beta = 8.0/3.0, mu = 15.0;

void updateTantara(float dt) {
  float dx = (sigma * (y - x) + mu * (float)Math.tanh(z)) * dt;
  float dy = (rho * x - y - x * z) * dt;
  float dz = (x * y - beta * z) * dt;

  x += dx;
  y += dy;
  z += dz;

  point(x * scale, y * scale, z * scale);
}

6.3 GLSL & WGSL (Shaders)

For high-performance rendering in web or game engines.

GLSL (WebGL)

vec3 tantara(vec3 p) {
    // Note: Use exact floating point division for beta
    float dx = 10.0 * (p.y - p.x) + 15.0 * tanh(p.z);
    float dy = 28.0 * p.x - p.y - p.x * p.z;
    float dz = p.x * p.y - (8.0 / 3.0) * p.z;
    return vec3(dx, dy, dz);
}

WGSL (WebGPU)

fn tantara(p: vec3<f32>) -> vec3<f32> {
    let sigma: f32 = 10.0;
    let rho: f32 = 28.0;
    let beta: f32 = 8.0 / 3.0;
    let mu: f32 = 15.0;

    let dx = sigma * (p.y - p.x) + mu * tanh(p.z);
    let dy = rho * p.x - p.y - p.x * p.z;
    let dz = p.x * p.y - beta * p.z;

    return vec3<f32>(dx, dy, dz);
}

6.4 Mathematica (Interactive Exploration)

For interactive parameter exploration and bifurcation analysis. The Manipulate interface allows real-time adjustment of all system parameters.

Manipulate[
 Module[{sol, x, y, z, t},
  sol = NDSolve[{
     x'[t] == sigma (y[t] - x[t]) + mu Tanh[z[t]],
     y'[t] == rho x[t] - y[t] - x[t] z[t],
     z'[t] == x[t] y[t] - beta z[t],
     x[0] == 1, y[0] == 1, z[0] == 1
     },
    {x, y, z}, {t, 0, T}, MaxSteps -> 100000];

  ParametricPlot3D[
   Evaluate[{x[t], y[t], z[t]} /. sol], {t, 0, T},
   PlotRange -> {{-30, 30}, {-30, 30}, {0, 60}},
   Boxed -> False, Axes -> False,
   PlotStyle -> {Directive[RGBColor[0, 0.8, 1], Thickness[0.003], Opacity[0.8]]},
   ImageSize -> {500, 450},
   Background -> Black,
   ViewPoint -> {2.4, 1.3, 1.0}
   ]
  ],
 {{sigma, 10, "Interaction (σ)"}, 1, 20, Appearance -> "Labeled"},
 {{rho, 28, "System Energy (ρ)"}, 10, 50, Appearance -> "Labeled"},
 {{beta, 8/3, "Dissipation (β)"}, 1, 5, Appearance -> "Labeled"},
 {{mu, 15, "Shift Factor (μ)"}, 0, 30, Appearance -> "Labeled"},
 {{T, 50, "Time"}, 10, 100, Appearance -> "Labeled"},
 SaveDefinitions -> True,
 TrackedSymbols :> {sigma, rho, beta, mu, T}
]

7. Conclusion

The Tantara Attractor was designed deliberately, not discovered accidentally. By injecting a neural network activation function into the heart of a chaotic workflow, it mathematically encodes the thesis that inspired it: that unexpected patterns emerge when human creativity (\(y\)) and technical logic (\(z\)) are allowed to collide, saturate, and shift the paradigm (\(x\)).

✶✶✶✶

About the Author

Burton Rast is a designer, a photographer, and a public speaker who loves to make things.

LinkedIn Instagram Contact Home