Explore the Dynamic Gumowski-Mira Attractor
How does a simple system transform into a myriad of shapes?
If you set a point in motion following a fixed rule, you might assume: 'Surely, it will eventually trace out a fixed path, right?'
Sometimes, that is indeed the case. It might repeat a simple movement—looping around, again and again.
However, if we slightly tweak the parameters controlling its motion, things begin to change; a once-regular path might evolve into something far more complex. Adjust it a little more, and branches may appear; continue changing it, and the system might even descend into chaos.
All the while, the moving point leaves behind a variety of wondrous shapes on the canvas: sometimes resembling jellyfish, starfish, flowers, or butterflies—or even tiny life forms viewed under a microscope.
This is the dynamic Gumowski-Mira attractor.
The most fascinating part: it doesn't have just one form
If you are encountering the Gumowski-Mira attractor for the first time, you might ask: 'What does it actually look like?'
In truth, there is no simple answer to that question, because it has no single, fixed appearance.
Changing the parameters can yield completely different structures. Some parameters produce regular paths, others create periodic structures, and some generate intricate patterns. Certain combinations of parameters can even drive the system into a state of chaos.
That makes it perfect for a 'blind box' style of exploration—a chance to see where a specific set of numbers will take the system.
Starting with a Single Point
Like many attractors, we can begin with something very simple: a single point.
It has a position defined by x and y—that is, its horizontal and vertical coordinates.
Then, mathematical rules instruct it: “Based on your current position, calculate the next step.” The point moves. Upon reaching the new position, it calculates again, moves again, and the process repeats.
Current position → Calculate next step → Move → New position → Calculate again → Move again → Repeat many, many times
At first, the image might show only a few scattered points. But as the calculations continue, more of the trajectory appears. Eventually, a complete structure gradually emerges.
It’s Not Just a Random Pattern
Seeing these complex patterns, you might wonder: “Are they randomly generated?”
They aren't. There are clear rules governing every step. If the initial position, parameters, and calculation rules remain the same, the system will evolve in the exact same way.
So, rather than simply scattering points at random, it involves repeatedly performing calculations based on specific rules.
This is one of the most fascinating aspects of attractors: they may look chaotic and random, yet they arise from very precise, well-defined rules.
So, what exactly is the Gumowski-Mira attractor calculating?
If we look at the mathematical formulas, they look roughly like this:
xn+1 = yn + αx(1 - σyn2)yn + f(xn)
yn+1 = -xn + f(xn+1)
f(x) = μx + 2(1 - μ)x2 / (1 + x2)
Together, they determine how the system moves next. You don't actually need to memorize these formulas; you just need to understand the concept: current position + parameters → next position, and then new position → calculate the next step—repeating the process over and over.
Three parameters, like three control knobs
Imagine α, σ, and μ as three knobs; turning one changes the way the system moves. Since this movement repeats continuously, the resulting pattern changes as well.
μ: Think of this as the twisting strength; it has the greatest impact, and even a slight change can transform the pattern from a "flower" into a "vortex."
α: Think of this as the feedback strength; it has a moderate impact, primarily affecting the pattern's tightness and drift speed.
σ: The feedback coefficient; it makes subtle adjustments to the feedback strength of α. It has the weakest impact and acts as a "fine-tuning" parameter.
To use an analogy: μ (twisting strength) is like the "distortion knob" on a dial, determining the overall shape of the pattern; α (feedback strength) is like "tension," determining whether the pattern is loose or tight; and σ (feedback coefficient) is like a "fine-tuning screw" that works in tandem with α.
But the truly interesting part isn't just that the patterns look prettier
If it were merely a case of "adjust parameters → get a pretty picture," then it would indeed resemble the Clifford attractor or the De Jong attractor.
What makes the Gumowski-Mira attractor special is that changing the parameters can alter the behavior of the entire system.
In other words, what changes isn't just what the pattern looks like, but how the point actually moves.
Imagine a system moving in a certain way: it might take a step or two and then return to its original position. This is a form of periodic motion—like A → B → A → B → A → B...—which appears very regular.
Now, suppose we tweak a parameter slightly. A single cycle might turn into two cycles. Change it a bit more, and four cycles might appear. As we continue to vary the parameters, the system can become increasingly complex. We call this process of evolving from simple cycles to more complex ones period-doubling.
If we keep changing the parameters, the system might enter a highly complex state known as chaos. This doesn't mean the program has suddenly become random; rather, the underlying rules remain, but the long-term motion becomes incredibly intricate. The system still follows the formula step-by-step, yet it is difficult to intuitively predict where it will end up next—or far into the future. That is precisely what makes chaotic systems so fascinating.
With ordinary mathematical visualizations, you might simply think, "I don't know what the next image will look like." With the Gumowski-Mira attractor, however, the situation goes a step further: "I don't even know what kind of motion the system will exhibit after I change the parameters." You might get regular trajectories, periodic structures, or complex chaotic patterns. Ultimately, what you see isn't just different images, but different mathematical behaviors.
This is not a "painting", but a collection of trajectories
Suppose we only let the point walk: 10 steps, and you may not see anything. Let it walk 100 steps, and a little structure begins to appear. Let it walk 10,000 steps, and the pattern becomes more and more obvious. 100,000 steps, and the complex shape begins to appear completely. So, the pattern you see is actually the path this point has taken in the past. It is not a picture prepared in advance, but the traces left by the movement.
When a large number of trajectories are superimposed, the human brain will naturally discover familiar shapes. Some parameter combinations produce images that are reminiscent of: jellyfish, starfish, plankton, butterflies, flowers and even fruit sections.
So, what is an "attractor"?
You can simply understand an attractor as when a system moves for a long time, it will continue to return to or stay near a specific structure.
The point doesn't simply stop, it's still in motion.
However, it will not run across the entire plane indefinitely. In the long term, its trajectory will be limited to a specific area and structure.
Stacking a large number of trajectories together, we see theGumovsky-Mira attractor.
How does it differ from the De Jong attractor?
If you’ve already looked at the De Jong attractor, you might ask: "Don't they both involve changing parameters to create beautiful 2D patterns?"
There are indeed similarities, but there are also differences worth highlighting.
The De Jong attractor is more like a laboratory for mathematical textures defined by four numbers. You tweak the parameters to discover new clouds, ribbons, and abstract textures.
The Gumowski-Mira attractor is more like a mathematical system that changes its "behavioral patterns." When parameters shift, it’s not just the visual pattern that changes; the system itself may undergo a transition from periodicity → period-doubling → complex motion → chaos.
So, if the De Jong attractor is about "seeing what kind of image these numbers can draw," then the Gumowski-Mira attractor is more about "seeing how the system itself evolves when you change those numbers."
It is also different from the Lorenz attractor
The Lorenz attractor is best known for chaos and the butterfly effect; it originates from a continuous-time, three-dimensional system that ultimately forms the classic 'double-winged butterfly' shape.
In contrast, the Gumowski-Mira attractor is a two-dimensional discrete dynamical system; it calculates positions step-by-step—from the first position to the second, then the third, and so on—eventually forming a variety of 2D patterns.
So:
Lorenz attractor: Mathematical butterfly + the story of chaos.
Gumowski-Mira attractor: Parameter variation + transition from periodicity to chaos + diverse forms.
There is a real-world physics story behind it
The Gumowski-Mira attractor is not merely a formula created to generate beautiful patterns; it is linked to the research conducted by Gumowski and Mira at CERN. They studied the trajectories of accelerated particles, a process that led to the development of the Gumowski-Mira attractor.
In other words, the jellyfish, starfish, butterflies, and intricate textures we see on the screen today originated from a fundamental question: How do particles move within a complex dynamical system?
This represents a fascinating progression: a physics problem → a mathematical model → iterative calculation → complex trajectories → ultimately becoming the mathematical patterns we see today.
Explore a Gumowski-Mira world for yourself
You can start with a preset configuration.
Observe the motion without changing the parameters first. Then, alter α and regenerate. Next, change σ and observe the result. Finally, try adjusting μ to see if the system shifts into a completely different state.
The most interesting way to engage with the Gumowski-Mira attractor isn't asking, 'Which set of parameters is the correct one?' because there is no single right answer.
You can look for patterns that are the most regular, complex, symmetrical, organic, abstract, or even the strangest. You choose the parameters, and the system provides the response.
Three numbers, one ever-changing world
Let's return to where we started.
We have just a single point, three key parameters, and a set of mathematical rules.
Then, we let it calculate, move, and repeat endlessly.
Under certain conditions, the motion remains regular. Change the parameters, and period-doubling might occur. Vary them further, and the system could descend into chaos.
When we combine all the trajectories left behind by this motion, a pattern emerges—one that resembles life forms, plants, marine creatures, or even abstract art.
This is the most captivating aspect of the dynamic Gumowski-Mira attractor: you aren't just altering the appearance of an image; you are changing the behavior of the entire system.
Continue Exploring the World of Mathematics
The Gumowski-Mira attractor is a truly unique type of attractor.
If you enjoy adjusting parameters and exploring abstract textures, you might also like to try the De Jong attractor and the Clifford attractor.
If you are interested in observing strange structures in 3D space, you can explore the Aizawa attractor.
If chaos, weather patterns, and the butterfly effect fascinate you, take a look at the Lorenz attractor.
The Gumowski-Mira attractor has a special quality of its own: it reveals how a simple 2D system can evolve—as parameters change—from orderly motion into complexity or even chaos, creating an endless variety of mathematical forms. The next time you click the 'Next' button, you might not know exactly what you'll see, but one thing is certain: the shape wasn't pre-drawn. It is a form that emerges step-by-step, generated by mathematical rules.
You can also continue to enjoy more amazing mathematical patterns.