A novel strange attractor featuring a logic-saturation term (\(\tanh\)).
\(\frac{dx}{dt} = \sigma(y - x) + \mu \cdot \tanh(z)\)
\(\frac{dy}{dt} = \rho x - y - xz\)
\(\frac{dz}{dt} = xy - \beta z\)
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The Tantara Attractor's phase space trajectory.
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."
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.
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:
Standard chaotic systems often treat variables as abstract quantities. The Tantara system assigns them semantic meaning:
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.
The Tantara Attractor is defined by the following system of three coupled, autonomous, non-linear ordinary differential equations.
To observe the unique Tantara topology, the following parameter values are established:
| Parameter | Symbol | Value | Description |
|---|---|---|---|
| Interaction Rate | \(\sigma\) | 10.0 | The coupling strength between Tension and Creativity. |
| System Energy | \(\rho\) | 28.0 | The Rayleigh number equivalent; drives the chaos. |
| Dissipation | \(\beta\) | 8/3 | The rate at which Logic naturally decays. |
| The Shift Factor | \(\mu\) | 15.0 | The Novel Term. Controls the magnitude of the logic saturation effect. |
The Tantara Attractor belongs to the family of Lorenz-like systems but is distinguished by symmetry breaking via saturation.
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).
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.
Visually, the Tantara Attractor resembles a "Tilted Butterfly."
For researchers, artists, or developers wishing to simulate the Tantara Attractor, snippets are provided below for common environments.
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)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);
}For high-performance rendering in web or game engines.
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);
}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);
}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}
]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 (\(x\)).
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Burton Rast is a designer, a photographer, and a public speaker who loves to make things.