Chaos visualization: how attractors of dynamical systems are represented

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There is a fable among scientists about a non-trivial way to make your report interesting and exciting. During the speech, you need to choose the most perplexed, the most lost listener in the hall, and tell him personally, so much so as to light a spark of interest in his eyes.

There is also a well-known aphorism attributed to physicist Richard Feynman: "If you are a scientist, a quantum physicist, and cannot explain in a nutshell to a five-year-old what you are doing, you are a charlatan."

Explaining the structure of complex things is a great skill, but there are stories about which even the most skillful speaker will break the tongue. The theory of dynamical systems is an area where, without visualization, one feels like a blind gardener surrounded by thorny, thorny plants.

Complex non-periodic modes of behavior of dynamical systems can be described by non-periodic trajectories - the so-called strange attractors with a fractal structure. Today we will show how the behavior of strange and some other attractors is visualized.

Great attractor

If you stop the first person who comes across on the street, shine a flashlight in his face and ask what he knows about attractors, then most likely we will not hear anything about the Great Attractor, which attracts hundreds of thousands of galaxies in the depths of space in order to restart the Matrix one day.

In fact, cosmological attractors are areas of the gravitational anomaly, apparently caused by special galactic clusters, and not directly related to the topic of the article.

Of course, it should be noted that the theory of dynamical systems is particularly well suited for determining the possible asymptotic states of various cosmological models. And the video is interesting - take a look.

Lorenz attractor

One of the most famous attractors is the Lorenz attractor, which became famous due to the massive distribution of the term "butterfly effect". Besides the fact that when visualizing an attractor its shape resembles a butterfly, it is a set of chaotic solutions of the Lorentz system.

Demonstration of chaotic systems like the Lorenz attractor (you can do it yourself in C ++).

The essence of Edward Lorentz's solutions in a nonlinear system of ordinary differential equations can be conveyed as follows: in any physical system, in the absence of perfect knowledge of the initial conditions, we are not able to fully predict its future. Physical systems can be completely unpredictable even in the absence of quantum effects.

Hidden attractor

An attractor is called hidden if its area of ​​attraction does not intersect with a certain open neighborhood of equilibrium points. Otherwise, it is called a self-excited attractor.

The classification of attractors (hidden or self-excited) appeared only in 2009 - after a hidden attractor was discovered in Chua's simplest electrical circuit with one nonlinear resistor, demonstrating modes of chaotic oscillations.

Multiscroll attractor

This is a whole family of multicomponent attractors, including the modified Chua hidden chaotic attractor.

Nonchaotic attractor

In addition to "ordinary" chaotic attractors, there are periodic, quasiperiodic, and also strange non-chaotic attractors.

One of the main criteria by which an attractor can be classified as non-chaotic is the calculation of Lyapunov exponents . In this type of attractors for the system, the Lyapunov exponents are not positive.

Hyperchaotic attractor

Hyperchaotic attractor is a visualization of Safieddine Bouali's differential equations. Hyperchaotic attractors exist only in dynamical systems whose phase space dimension is more than or equal to four. Hyperchaotic attractor models can be used in real-world applications related to secure communication and encryption.

Limit Cycle

A continuous dynamic system with an isolated orbit, implying self-sustaining oscillations (for example, pendulum clock oscillation or heartbeat while resting).

Rössler attractor

Chaotic attractor of the Rössler system of differential equations. In 1976, physician Otto Rössler presented a three-dimensional model of the dynamics of chemical reactions proceeding in a certain mixture with stirring. The Rössler attractor is characterized by a fractal structure in the phase plane.

On the Rössler attractor, the trajectories do not intersect themselves. The surfaces that form the strange attractor are divided into separate layers, creating an infinite number of surfaces, each of which is extremely close to the neighboring one. It can be assumed that the tape that forms the base of the attractor is similar to a multilayer Mobius strip.

Spiral attractor

The Spiral attractor is an attractor that made it possible to study the life of the amoeba Dictyostelium discoideum. When nutritional resources are depleted, amoebas secrete cyclic adenosine monophosphate (cAMP), signaling molecules that attract neighboring cells to a central location. Hungry mixamyoba (unicellular stage of development of Dictyostelium), obeying the signals, creep to the center, which was formed as a result of “gluing” the first mixamyobas that happened to be nearby. Connecting with the help of cell adhesion molecules, they form an aggregate of several tens of thousands of cells. Actually, this process is presented in the video.

Tinkerbell attractor

The Tinkerbell chart is a discrete-time dynamical system that exhibits chaotic behavior in two-dimensional space. Tinkerbell's shape can be modified to create other chaotic attractors in secure communications systems that exploit communications chaos .

Thomas' cyclically symmetric attractor

The three-dimensional attractor, proposed by bioinformatist Rene Thomas, can be viewed as the trajectory of a damping particle moving in a three-dimensional lattice of forces.

Ikeda attractor

A fractal set to which the orbit of any point on the plane is attracted if we continue to iterate a certain map from the plane to itself.


We have considered only a few known types of attractors. In total, you can find references to hundreds of different attractors.

It should be noted that this is a very young field of science, and the search, which began with the idea of ​​moving away from mathematical abstraction towards the practical "creation" of chaos, continues to this day.

One thing is invariable: our interest with the power of the Great Attractor is attracted by systems that are extremely sensitive to small deviations in the description of the initial state. We do not come across these systems out of idle curiosity - we live among them and thanks to them.

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