Learn how natural systems exhibit characteristics of chaos theory

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.

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.

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

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.

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.

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

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

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.

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.

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.

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

The science of symmetry | The Royal Society

Nature is full of symmetries, from spiral galaxies to florets of broccoli. So why do complex creatures break the pattern?This short video discusses why symmetry came to dominate life on Earth, and how many complex life forms, such as humans, transitioned from radial to lateral symmetry.

Chaos gardening involves scattering seeds without any planning

The trend is intended to produce colorful, dense gardens by mixing various seeds. While encouraging experimentation and lower stakes, the success of these gardens depends on choosing plants with similar needs and incorporating native, nectar-rich flowers.

Challenges for synthetic biology

Can engineering approaches tame the complexity and unpredictability of living systems? An exploration of five challenges for the field and how they might be resolved.

Traffic flow can be improved through research into card shuffling

Card-shuffling mathematics has been applied to solving problems in biology to estimate the evolutionary distance between organisms and in computer science to determine the optimal distribution of files and folders in databases.

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 how cells respond to signals from their environment

In "Dropping Signals," users can drag stimuli, such as hormones and light, into various cellular environments to observe cellular responses. These include changing their shape, enabling the transport of chemicals into or out of the cell via the cell membrane, or dividing to compensate for surrounding damaged cells.

Circadian rhythm disruption can influence various mental health disorders

This 2020 research overview surveys the relationship between disrupted circadian clocks and disorders including schizophrenia, anxiety, and bipolar disorder. It incorporates findings and therapeutic approaches drawing from human night shift studies and jet lag studies alongside animal research.

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.

Even in nature reserves, climate change is driving rapid ecological collapse

Studies show a dramatic decline in insect populations, resulting in cascading global biodiversity losses across non-insect species. The balances in tropical ecosystems are the most affected by climate change.

See how populations across the world could change in the 21st century

This tool lets you model the populations of individual countries by adjusting key demographic inputs. See how changes in birth rates, mortality, and migration affect the size and structure of future populations.

Watch how viruses like Ebola can spill over from animals to humans—and vice-versa.

Social media can reinforce echo chambers that amplify pre-existing sentiments

A 2023 Nature study showed that exposure to diverse content left participants with unchanged core beliefs, suggesting social media alone cannot create hateful viewpoints. However, such platforms allow rhetoric to spread widely, potentially exacerbating geopolitical tensions.

Dawkins believes cultural memes have mutated into internet memes

The world's creations fall into two categories: those that look designed but whose appearance or function came about due to random mutation and natural selection, such as birds, and those whose designs and mutations are knowingly crafted, such as airplanes.

Watch how viruses like Ebola can spill over from animals to humans—and vice-versa

More than 70% of all new infectious diseases originate in animals. Referred to as zoonotic diseases, these are diseases that originate in animals and occasionally spill over or jump from animals to humans. Bats, mosquitoes, and primates have all been linked to spillover events. As human populations push further into previously uninhabited wilderness, animal-human contact increases, and disease spillover is inevitable.

Too little stress can be as problematic as too much stress

Hormesis—mild or moderate stress followed by recovery—helps make us more resilient. A stress expert speaking during this roughly 30-minute CNN special explains that our bodies depend on periodic mental and physical challenges to activate our innate defense mechanisms, spurring cellular repair and regeneration.

The ingredients that make up severe weather

A seven-minute video illustrating how a storm is made. These factors—moisture, instability, lift, and shear—are explained through a series of data sets and creative, descriptive diagrams.

Learn how your body is, scientifically, like a firework with its own slow, controlled burn.

Watch how algorithms can be designed to train themselves and self-improve without human input, to the point where the optimized code is too complex for programmers to understand.

Stars appear to twinkle because turbulence in the motion of molecules in Earth's atmosphere causes light to bend and wiggle rapidly on its way to the surface—a phenomenon called scintillation.

Studies suggest when you sleep is as important as how much you get, and syncing with your circadian rhythms may lead to less fatigue and improved long-term health outcomes

Indefinite cloning of some endangered species may not sustain their longevity because cloning success rates decrease as genetic mutations accumulate.

Radioactive elements can introduce software errors because the ionizing radiation they produce can detach electrons from atoms, and computer code relies on the distribution of electrons.

Crows are believed to have intelligence on par with a seven-year-old child and demonstrate complex behaviors, from problem-solving with tools to recognizing human faces.

Visualize how CRISPR-Cas9—a type of immune system used by single-celled organisms to attack invasive viruses—targets and makes precise changes to a gene's DNA.

As of 2025, the most cited papers in science include a review of graphene, methods to improve image recognition, streamlining electron density calculations, and optimizing protein studies.

Some renewable energy sources may weaken the resiliency of electric grids by removing outage buffers provided by coal- and natural gas-fired plants, which provide the system with more time to find a new power source if a plant goes down.

According to the physics of black holes, our 4D reality may be a hologram—a projection of information coded on a 2D surface on the edge of the universe.

How Amazon operates a 'flywheel' business model, a self-reinforcing cycle that drives continuous growth

Scientists testing how climate change affects plants found, in a years-long simulation study, that poison ivy grew 70% faster with the resulting extra carbon dioxide.

Explore an interactive map built from more than 100,000 survey responses on where people find joy in their lives

Fear or disgust related to clusters of holes and cracks, called trypophobia, may stem from evolutionary processes to help us avoid pathogens.

Some Pokémon have been named after real and fictional organisms, while some real-world species have been named after Pokémon, such as Chilicola charizard and Binburrum zapdos.

Exploring the Larry Bird punch that almost changed March Madness

The hormonal disorder PCOS was renamed PMOS in 2026, a move designed to improve care and research by shifting the focus to the hormonal causes of this condition that affects 1 in 8 women.

A 2019 Facebook event called on attendees to "Storm Area 51, They Can't Stop All of Us," resulting in countless memes, the creation of a music festival, and eventually the arrests of those who did try to approach the area.

Memecoins are a type of cryptocurrency—a form of digital currency with a wide variety of uses—created around popular 'memes,' or viral internet jokes.

View the ten deadliest tornadoes in US history.

The United States is one of only around two dozen countries with a fully federal system of government, in which power is divided between national and state governments.