Explore interactive simulations of strange attractors, the graphs that help visualize chaos

View the Lorenz attractor, a graphical solution to chaotic equations that resembles a butterfly

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

Learn how natural systems exhibit characteristics of chaos theory

In biology, some mathematical models of ecosystem populations and heart irregularities show strong sensitivity to initial conditions, which scientists can simulate to determine quantitative tipping points at which these systems become unstable. Within the social sciences, crowd behavior and the virality of social media posts have shown how human behavior fits within the framework of chaos theory.

A mathematical and visual analysis distinguishing chaos from randomness

Random systems lack any underlying pattern or causal relationships, while chaos is mathematically deterministic because the behavior follows established laws of nature. In 3D plots, chaotic systems produce outputs that approximate a mathematical function, while random points appear as statistical noise.

Visualize chaos with the orbit of three or more objects bound by gravity

In this simulation, users can simulate randomized and preset orbital configurations for up to 10 celestial bodies to observe the system's evolution in an interactive 3D space. The same configurations can produce different orbital behavior—a feature of chaos—by making small changes to an object's mass under advanced settings.

An analysis of the logistic map, which shows how chaos can emerge from simple equations

When modeling systems such as the predator-prey cycle of an organism's population in an ecosystem, the logistic map function shows the population normally settles into an equilibrium value. However, at high reproduction and starvation rates, multiple such values arise, making it difficult to accurately model long-term population growth in some environments.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

The origin of chaos theory, colloquially known as the 'butterfly effect'

The theory explains that systems following well-understood physical laws can still produce unpredictable outcomes due to small variations in their initial conditions. Originally identified in weather data, elements of chaos have been observed in stock market fluctuations, cardiac arrhythmias, and the evolution of online social movements.

The life of Edward Lorenz, who originally used a seagull in his 'butterfly effect' metaphor

By accidentally finding that rounding one variable to its third decimal place and re-running a weather simulation produced a drastically different two-month forecast, the American mathematician established the field of chaos theory. His paper—"Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?"—introduced the well-known metaphor for the behavior of chaotic systems. (Some readers may experience a paywall.)

The story of two female programmers running the simulations behind chaos theory's discovery

Margaret Hamilton and Ellen Fetter worked on MIT's 800-pound Royal McBee LGP-30 computer, which was fed binary code on paper tape to perform computations. Among the computations performed were those involving equations describing the motion of gas in a circulating fluid for Edward Lorenz, widely considered the father of chaos theory.

Chaos theory—informally known as the butterfly effect—is the reason why weather forecasts are presented with probabilities rather than certainties.

The 'butterfly effect' is the informal name for chaos theory

In mathematics, chaos theory involves complex systems that are highly sensitive to their initial conditions. Originally discovered when analyzing weather data, slight variations in the data input into such systems create drastically different results, despite following known physical laws.

How chaos helps explain why small market corrections can produce major economic instability

Oxford University complex systems professor Doyne Farmer discusses how predictions from many economic models are overly weighted toward simple relationships, such as supply and demand, and external shocks, such as the COVID-19 pandemic. However, real-world macroeconomic systems involve public and private actors interacting, producing feedback loops and cascading effects that resemble the oscillatory behavior seen in chaotic systems.

An interview with James Yorke, the man who coined the mathematical term 'chaos'

Along with co-author Tien-Yien Li, the former University of Maryland mathematics department chair published the paper, "Period Three Implies Chaos" in 1975. "Chaos" then became the terminological placeholder for systems with sensitive dependence on initial conditions, which had been well known to mathematicians for over a decade.

Understanding turbulence, which is characterized by growing chaos at multiple scales

As energy is lost in fluids undergoing turbulent, non-linear flow, particles will collectively form eddies, or whirlpools, of various sizes that mix together. The energy cascade of larger eddies becoming smaller ones before dissipating completely varies based on the initial conditions of the fluid, a characteristic feature of chaotic systems.

Researchers identify a hidden instability pattern in simple ocean wave models

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

Mathematics proves turbulence's chaos creates super diffusion

Turbulent flow is responsible for the rapid scattering of dandelion seeds, volcanic ash, and untied balloons. Its swirling behavior accelerates particle scattering faster than other mixing methods, as particles acquire energy from the interactions of eddies.

View 3D renderings of atomic orbitals created with quantum mechanics

Scientists use the Schrödinger equation to determine how a system governed by a wavefunction—an equation that predicts the probability of certain outcomes—evolves with time. Computers can animate atomic structure and behavior by depicting the wavefunction as a fluid that guides a particle, like water moving a speck of dust.

The most bizarre numbers in the universe

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

Experimenting with space-time

Einstein's theory of general relativity—that mass literally bends space and time—can be difficult for the human brain to imagine. This interactive resource lets you create celestial objects, imagine their effect on space-time, and see how they interact with each other (best viewed on desktop).

Explore an optical table to create quantum physics experiments

This interactive teaches users to create quantum experiments through lessons resembling video game levels. Users can also use the virtual lab to visualize landmark quantum experiments like entanglement, teleportation, cryptography, and quantum computing, or create their own experiments with drag-and-drop elements.

Neural networks mimic the structure and behavior of real physical systems

Early generative AI systems were inspired by scientific models such as the Ising model, which explains how atoms interact with one another and align in magnets. As these networks have expanded, their behavior has more closely resembled that of quantum fields, enabling them to be used in exploring complex particle physics interactions.

Explore an interactive map of the unified theories of everything

This interactive concisely describes many highly complex theories ranging from quantum gravity and black holes to dark matter and energy. Supplementary resources, key questions, and possible solutions accompany each description. As of 2026, no developed theory provides an accepted explanation of all observed phenomena in the universe at all scales.

Build atomic nuclei to explore their instability and resulting radioactive decay

In this simulation, users can add protons and neutrons to create an atomic nucleus, view its half-life and decay options, and identify patterns in how specific combinations of particles create extremely stable nuclei, from which the nuclear shell model of stability emerged.

View the transition from laminar to turbulent flow

Using a dye released in the water, streamlines, which are parallel to one another and consistent at low flow velocities, become visible. As the water moves more quickly, viscosity cannot suppress the velocity change, and streamlines become chaotic, creating turbulence.

Quantum computers can simulate nature more effectively than classical computers

Ordinary computers struggle to simulate quantum fields due to entanglement, but systems built from those elements can model the complexity. Systems using particle in more than two states—qudits—have successfully modeled 2D electromagnetic fields and done so more efficiently.

Visualize how data becomes organized through a variety of sorting algorithms

This interactive shows how different algorithms shift elements in a randomized set to order them from smallest to largest. By comparing the different procedures and their corresponding functions, users can explore how runtimes vary across algorithms that complete the same task.

Explore a simulation of ideal gas behavior, which can be explained by kinetic molecular theory

The theory describes particles as moving in random directions, with a distribution of velocities determined by the gas's temperature. The particles collide elastically like billiard balls, are negligibly small, and do not exert forces on one another. The gas laws can be derived from these principles using statistics or observed via this interactive.

Computational and biological systems share fundamental principles, like randomness

Proposed by Alan Turing and John von Neumann, the concept originates from how DNA and computers run by executing encoded instructions in parallel. The advent of AI and the temperature parameter has seen neural networks mimic the randomness of protein encoding.

How a competition honoring a king's birthday uncovered history's first chaotic system

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

Climate models use complex equations and supercomputers to predict future climate

The equations apply the laws of thermodynamics, fluid motion, and radiative transfer to a 3D grid along Earth's surface. After solving them for each cell in the grid over a time interval, the results carry over to neighboring cells, and the process repeats into the future.

Explore a simplified breakdown of the Einstein field equations

Often depicted as one formula, Einstein's equation represents a suite of 16 nonlinear partial differential equations, which can be simplified to six unique ones, that relate the geometry of spacetime to the presence of mass and energy.

Explore a visual and auditory guide to wave properties and waveforms to learn how every sound imaginable can be created by combining multiple vibrations of air.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

When experiencing turbulence, fluid elements undergo stretching and folding

Simulations show patches of fluid particles, such as splashes of milk, bend under turbulent flow, creating curvature that grows linearly then exponentially. This accelerated deformation enhances folding and mixing faster than other methods, such as stirring or beating.

Grasp the earth's orbit around the sun with this interactive

See why seasons exist with this interactive simulator. Toggle between different views of the earth in relation to the sun and watch as the earth's axis causes the sun to shine at different angles throughout the year, causing the weather to fluctuate and more. This simulator clarifies a process many take for granted.

Simulate the effects of changing supply and demand on the electric grid

In this application, users can modify the demand of fictional cities and the supply of various generators, including a nuclear power plant and a wind farm, which can be disconnected from the grid. Notice that current may flow into or out of external systems to compensate for available generation. See what it takes to cause a blackout in different cities or shut down the entire grid.

Build atoms to learn about atomic number, mass number, and charge notations

In this simulation, users can add protons, neutrons, and electrons to create atoms, view their stability and atomic configurations, and identify how the amounts of these subatomic particles are detailed around the abbreviation for the resulting element using AZE notation.

Explore a visualization of curved spacetime under general relativity

While depictions involving bends in elastic fabrics are common to illustrate gravity, they are circular representations that rely on gravity in the real world to depict gravity in the model. The river model instead shows how bends in spacetime convert speed through time into speed through space, producing acceleration.

Quantum computers harness weird physics to explore unsolvable problems

Classical computers manipulate bits as either 0 or 1, but quantum computers use qubits to hold combinations of both, enabling new algorithms. This new approach could one day help with breakthroughs in medicine, chemistry, and AI.

Visualize the components of fireworks

This article provides animated diagrams depicting how different parts of fireworks function. The elements work together through timed mechanisms to produce desired effects.

Turbulence has been called the most important unsolved problem in classical physics

Despite being the most common type of fluid flow in our world, scientists have struggled to model turbulent flow accurately. Though behaving unpredictably, understanding inertia and viscosity can help determine whether turbulence will occur.

The Navier-Stokes equations are fundamentally accurate but unsolved in 3D

The equations model fluids and have been shown to have solutions for all systems in two dimensions. Averaging and approximations let the equations be used in climate models, aerodynamics, and medicine, though their use in some systems produces nonsensical answers.

View a library of custom programs that can be loaded onto graphing calculators, including productivity tools, reference materials, games, and interactive applications, such as prom proposals.

Science is rarely a linear progression of understanding

Particularly at the frontiers, investigations conducted based on established knowledge can provide unexpected results. This opens the door to revising existing models, inspiring new lines of research, and ultimately leading to better future comprehension. For instance, experiments on the speed of light and the expansion of the universe yielded then-confusing data that led to revolutionary insights.

Explore the effects of tornadoes of different scales

In this simulation, users can adjust the tornado's diameter and rotation speed to see its impact as it passes through a series of trees and a one-story home.

Accurate climate models incorporate calculus to account for complex feedback loops

The simplest models assume uniformity across the Earth, with incoming energy from the sun being offset by the planet's albedo, or reflectiveness, which prevents the radiation from being fully absorbed. Incorporating elements such as atmospheric layers, latitude bands, geographic variability, and emissivity yields coupled differential equations quantifying Earth's interconnected systems.

A breakdown of algorithms, how they are constructed, and their applications

Algorithms are instructions for accomplishing a goal and typically take the form of a step-by-step procedure. Given the conditions, these procedures can take various paths, increasing an algorithm's complexity. While traditional computer algorithms are programmed to account for all possible scenarios within a given process, machine learning systems build their own rules from data-driven patterns.

Early 20th-century debates about horses with 'paranormal mathematical abilities' helped shape modern approaches to psychological research.

Mathematicians often solve problems using wordless mental processes

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

Studying 'sonic' black holes

If a fish screams as it falls down a cascading waterfall accelerating to ridiculous speeds, will his fish friends above hear him scream? This analogy, odd as it is, has become an essential tool in the astrophysicists kit for learning how enigmatic black holes operate. Lab experiments mirroring the so-called sonic black hole have achieved insights for their cosmic counterparts. Read how it works in this brief piece.