Halvorsen System strange attractor digital artwork by Riccardo Zerbinati explores nonlinear motion through parametric geometry and light.

06 / Strange Attractors · Halvorsen System

Halvorsen System

An exploration of chaotic motion where defined rules give rise to an open-ended visual form.

The Halvorsen System is a three-dimensional nonlinear dynamical system studied for chaotic behavior. This artwork takes that mathematical name as a starting point for an authored visual interpretation. It does not attempt to reproduce an experiment; instead, it considers how the trajectory of a defined system might be experienced when translated from abstract motion into a composed image.

The work sits between a trace and an object. Its mathematical origin suggests movement through space, while the act of rendering gives the form a particular presence through viewpoint, material and light. The viewer can follow the idea of a trajectory without needing to read the image as a technical diagram.

Digital artworkRhinoceros · 3ds MaxCreated without generative AI

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Halvorsen System equations

Each equation repeats the same pattern with the variables cycled. Squared terms bend the path while the linear terms couple its coordinates.

dxdt=−ax − by − bz − y²

dydt=−ay − bz − bx − z²

dzdt=−az − bx − by − x²

Here, t is time and the variables are coordinates in the system’s state space. Letters such as a, b and c denote fixed coefficients. These are reference equations for the named model or family, not a recovered record of the parameters, initial conditions or integration settings used in this particular image. Chaotic behaviour depends on those choices.

Formula reference: View the documented system and source.

From a trajectory to Halvorsen System art

Riccardo develops the rail in Grasshopper for Rhinoceros using Chimpanzee. The system evolves over time, producing successive three-dimensional points. Connecting those samples creates a path. A cross-section is then swept along that rail, giving the line thickness and a surface. This is how a time-dependent mathematical trajectory becomes visible as sculptural geometry.

The sweep adds a visual layer to the mathematics: its width and orientation affect how the form catches light. Riccardo then uses 3ds Max to develop the material, camera and illumination. The finished work is a digital render of this geometry, rather than a photograph of a fabricated object. No generative AI is used to create the artwork.

To read the image, follow a single edge as it moves through light and shadow. Notice where the projected path overlaps, opens or disappears behind another surface. The rail comes from a rule evolving through time; the artist’s composition lets us experience that motion as a still image. The aim is to reveal the hidden beauty of chaos theory.

A closer look
Mathematics → Geometry → Light

From system to image

A strange attractor is a bounded structure that can emerge as a dynamical system evolves. Its motion follows defined rules, yet sensitivity to starting conditions can make a long path difficult to predict. This contrast between structure and uncertainty is one of the ideas explored across the Strange Attractors series.

For Halvorsen System, the named system is a starting point for visual research. Riccardo Zerbinati uses a parametric workflow in Rhinoceros to explore mathematical traces as three-dimensional geometry, then shapes materials, composition and lighting in 3ds Max. The finished image is an authored interpretation of mathematical form, not a scientific plot or an AI-generated image.

Study 06 belongs to a collection of forty vertical digital artworks. View it alongside the other studies to compare how distinct systems can lead to different visual readings.

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