Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena like gravity, electricity, or chemical reactions, Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. These phenomena are often described by fractal mathematics, which captures the infinite complexity of nature. Many natural objects exhibit fractal properties, including landscapes, clouds, trees, organs, rivers etc, and many of the systems in which we live exhibit complex, chaotic behavior. Recognizing the chaotic, fractal nature of our world can give us new insight, power, and wisdom. For example, by understanding the complex, chaotic dynamics of the atmosphere, a balloon pilot can “steer” a balloon to a desired location. By understanding that our ecosystems, our social systems, and our economic systems are interconnected, we can hope to avoid actions which may end up being detrimental to our long-term well-being.

**Principles of Chaos**

- The Butterfly Effect: This effect grants the power to cause a hurricane in China to a butterfly flapping its wings in New Mexico. It may take a very long time, but the connection is real. If the butterfly had not flapped its wings at just the right point in space/time, the hurricane would not have happened. A more rigorous way to express this is that small changes in the initial conditions lead to drastic changes in the results. Our lives are an ongoing demonstration of this principle. Who knows what the long-term effects of teaching millions of kids about chaos and fractals will be?

- Unpredictability: Because we can never know all the initial conditions of a complex system in sufficient (i.e. perfect) detail, we cannot hope to predict the ultimate fate of a complex system. Even slight errors in measuring the state of a system will be amplified dramatically, rendering any prediction useless. Since it is impossible to measure the effects of all the butterflies (etc) in the World, accurate long-range weather prediction will always remain impossible.

- Order / Disorder Chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

- Mixing: Turbulence ensures that two adjacent points in a complex system will eventually end up in very different positions after some time has elapsed. Examples: Two neighboring water molecules may end up in different parts of the ocean or even in different oceans. A group of helium balloons that launch together will eventually land in drastically different places. Mixing is thorough because turbulence occurs at all scales. It is also nonlinear: fluids cannot be unmixed.

- Feedback: Systems often become chaotic when there is feedback present. A good example is the behavior of the stock market. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing it to rise or fall chaotically.

- Fractals: A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos. Geometrically, they exist in between our familiar dimensions. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc.

*“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.”*

*-Albert Einstein*

Dear Fractal Folk

Thank you for sharing the wonderful world of fractals! I have visited your site many times since finding it about a year ago and am encouraging my kids to take an interest. Thank you. Louise, Australia

Chaos is one of Physics Foibles. As Godel revealed – maybe we can’t prove everything!! long term weather prediction is trumped by Chaos Theory.

Thank you for your useful writing.It helped me to write my article.Somayeh,Iran,Khomeini shahr

i think each and erything in the universe inter related and interconnected.

A puzzle for “chaos thinking” is that the universe is NOT chaotic in the sense of actual butterfly wings having an effect even one meter away, never mind Texas. The world of reality is not an approximation. By contrast, the world of chaos theory is full of approximations, because one CANNOT specify initial conditions to an infinite degree.

The “butterfly effect” does not apply to the macroscopic real world. The “butterfly effect” is a mathematical construct, or perhaps better, an artifact of “digitization for computation.” Beautiful, but not real.

I have my objections to what meteorologicalengineer wrote. He is wrong on what mathematically “initial conditions” are and as a result he claims that chaos deals with a lot of approximations. That is entirely untrue because chaos is remarkably precise in terms of chaotic behavior. Turbulences, triggers, fractal expansions, borders of chaos possess the quality of exactness. The key feature of chaotic behavior is order and precision. It is also untrue that “the butterfly effect” is a construct or an artifact nonexistent in reality. I can present loads of heuristic evidence from the financial charts like indices, stocks, bonds and currencies falsifying meateorologicalengineer’s conjecture.

It also forces me to ask him, do you really think that mathematics together with its constructs and artifacts is simply manmade? The theory of chaos denies it.

I do agree with the engineer with the world of chaos theory is full of approximations and Paul for chaotic behavior. Imagine if, the butterfly can affect in some way the world, how would effect: the direct and horrible acts like; war, terrorism, war and violent movie or war-game and more and more… which in every second has a direct influence in our world and life, as Calvino says, rains in our mind.

I am by no means an expert in the field of nonlinear dynamics, but I’ve taken a few college courses on it and I’ve read a few pop-science books (Chaos by James Gleick is great, as is Does God Play Dice? by Ian Stewart). So, I consider myself knowledgeable enough on the subject to clarify a few points. The beauty of so-called “chaos math” is that we can simulate impossibly complex systems (i.e. everyone’s favorite, weather) with incredibly simple models. Even more importantly, the behavior of those simple systems is complex and erratic. I forget which of the two books I mentioned before the quote is in, but one of my favorite quotes about models used in nonlinear dynamics is paraphrased as “there is no point in having a map of a city which is as large and complex as the city itself”. What this means is that mathematicians want to model complex activity with as simple of a model as possible. This is the inherent beauty of chaos.

Now, on a more personal aside, I have a pet peeve against the phrase “butterfly effect”. It’s misleading, in my opinion. It gives the layperson a nice perspective, but the way it’s portrayed (“a butterfly flapping its wings causes a hurricane across the globe”) implies a direct causality. The effect is not that straightforward, which I think is the point the meteorological engineer is trying to make.

Thank you for this wonderful, yet simplistic explanation of chaos theory! I use this theory (and consequently this page) to inform and empower friends and strangers alike. Too many believe they are too small to make a difference in most if not all of the aspects of modern civilization, but wisdom has shown us for thousands of years that it’s not so. Thank you simplifying such a monumental concept and keeping the mathematics and science intact. You do the alchemists of old great justice!

My perspective, obviously, is a stark contrast to the calculating outlook some in math and science tend to have. I am horrible at math/physics, but I have a tremendous respect and admiration for their tremendous power. So here’s the artist/spiritualist perspective: Chaos theory existed long before the great minds of physics and mathematics named it such. In fact, the Chinese word for “chaos” contains the root word meaning “opportunity”. Dating some philosophies and ideas of chaos theory around the 14th century B.C. Although the connection may seem like grasping straws to some, the study of alchemy (which evidence suggests occured as early as ancient Sumeria) directly correlates with some of the ideas of chaos theory in very transcendent ways. Though equations may sometimes seem to disprove or augment our core ideas of what is natural or “real”, more often than not it is information we lack that renders our understanding, which is received through lifetimes of apt research and relentless study. Perhaps we do not understand reality to the extent in which science leads us to believe. History is full of prideful human error made by many ingenious minds, swept away by the vastness of their own intelligence. To make my point, many scholars, philosophers, psychologists, physicists etc. have dedicated a portion, if not the entirety of their lives piecing together the mysteries of global consciousness and interconnectedness as well as the intimate relationship between order and disorder. Carl Jung spent a great deal of time on the subject of enlightenment by means of chaos. The butterfly effect is indeed indirect when we investigate the lineage of cause and effect, however the point is not HOW influential the butterfly is but rather that it has any influence at all. The idea of a universal consciousness is becoming more feaseable with the help of science and mathematics, that we somehow affected the butterfly, who affected the hurricane, which affected us (to grossly oversimplify), a never ending fractal cycle of cause and effect. Lately, those seasoned in hard fact finding have had to play a bit of catch up with metaphysics, as larger numbers of people have spiritual or unexplainable experiences in their lives that have nothing to do with religion and actually, more to do with the ideas of chaos theory.

I just say yes to all because that is chaos in its self. I believe chaos is universal while being precise and linear it is non linear and unpredictable which by definition causes chaos in its self and I call this true chaos and chaos is everywhere. This maybe a simple idea compared to all other ideas but it is how I believe chaos is.

I’m a high school student looking at chaos theory for a Math project, so I may not be correct about my idea of what the chaos theory is but please try to understand as I have not received any “advance” education as I am only a high school student. Now I have visited many websites and read many of the comments on their and here. Everyone seems to hate the Butterfly fly effect as they say it is “irrational” or “misleading”. The fact that a butterfly flapping its wings in China can cause a hurricane in another far away place, I can understand, that many would find it to be ambiguous. The butterfly effect is indeed indirect when we investigate the lineage of cause and effect, however the point is not HOW influential the butterfly is but rather that it has any influence at all. The idea of a universal consciousness is becoming more feasible with the help of science and mathematics, that we somehow affected the butterfly, who affected the hurricane, which affected us (to grossly oversimplify), a never ending fractal cycle of cause and effect.

I have coined a term for this topic…Computational Density…..it states the deeper the zoom, the greater the number of iterations needed for viewing and the greater an objects robustness. This will lead to a blending of fractals, chaos, celluar automata, and the simulation hypothesis. I will be brief. Consider the Mandlebrot set, infinitely many copies, smaller and smaller and smaller, each one a complete set unto itself. Computational density states that a trillion iteration M set is in fact more “powerful” than the 1st zoom level.

Now, allow me to take out my galaxy scale microscope. On this slide I have the milkyway. Let’s take a look at the universal spin fractal. 1st zoom the galaxy….1,000 zooms the solar system…..10,000 the earth…..100,000 zooms a hurricane……1,000,000 zooms a tornado….1,000,000,000,000 your DNA. According to Computational density in a simulated universe….you are much harder to render than the galaxy as a whole.

Now, the interesting question is what does one do with all that power. Can the one make a butterfly that spawns up the tree of scale. Can one use simple tools, taking advantage of the Joseph effect, using complexity and iteration to participate more fully. Much love, Quartaro Industries

Without life, it would be possible to predict the fate of the universe at any given time, given enough initial conditions and powerful enough supercomputer

I’ve been a fan of Chaos Theory and fractals ever since I picked up the book Chaos (Gleick). However, I’m a bit confused (I am in no way a math whiz or scientist of any kind), doesn’t the theory basically state that even in chaotic systems patterns are identifiable, one just has to look close enough or far enough away to identify them?

thanks!! it’s a wonderful site!! It helped me so much!!fantastic!! North Korea jiyoon

It sure appears that way, but as we move on, chaotic systems of today will be very predictable things as our data gets more robust and we are clear about what problems need accuracy (in initial conditions) with the accuracy we can provide.

As I read all the content and comments in this article, I realize that: “Chaos theory is our key to understand every system in the universe no matter how complex it is.” ~Sir Doone, Philippines

I agree with Molson (above). Chaotic systems may very well be deterministic, but not modelisable due to unstable coefficients, feedbacks, and phase changes. However,the patterns thrown-off therein are identifiable; that may be more useful than attempting to predict in non ergodicity. Great site, bravo.

I don’t get it. In no way does a butterfly in China cause a hurricane in the Carribean. The physics concept of entropy explains the ability to mix gases but not unmix them. Entropy (randomness) in the Universe is always increasing. Gaseous expansion of a pure gas from a cylinder to a bigger cylinder can be done without the loss of energy but cannot be compressed back to the original state without the expenditure of energy. Theoretically, if the Universe is finite (albeit large) there will be an eventual run down of intrinsic energy to a steady state of equal energy throughout. Local zones in which there is an increase in energy such as a water hammer rely on more energy being expended by the surrounding environment. This is classic entropy. In a sense, these local zones of hyper energy will occur in ever decreasing amplitudes “forever” asymptotically approaching the eventual uniform mean. Predicting magnitude and location of these local zones of hyper energy. Maybe that is what chaos theory is all about.

i believe that everything in the universe is somehow interconnected. i mean, it makes sense how a butterfly could have so much force. even the smallest of animals can have so much power. by the way, this website explained everything so clearly to me. i have a school project, and i only went to this website, then i had all the information i needed.

Very interesting reading the comments posted and this very well put together site. The comments ranging from Mathematical sciences to Eastern Philosophy show that theories of Chaos have been studied for years. Energy can not be created nor destroyed, some people will even argue this fact, however the idea of a butterflies wings flapping causing exponential expressions until it causes a tropical storm may seem hard to imagine. Let’s try an easier example, I apologize in advance for the morbidity of the topic, but negative effects are easier to notice and more obvious for Chaos Theory. (See mythology for Chaos) Anyway, imagine the equation for nuclear reactions, let’s say an explosion, the amount of elemental particulates increases after the intial explosion, however the reaction continues, with the same formula’s until things are back in balance with our current environments of land, water, air to the molecular level sometimes within thousands of miles from the atomic explosion. You would think that as the environment reaches homeostasis, the Chaos Therory ceases to multiply, in fact it has and will continue until every exponential possibility has expired. (Never)

Very intrigued by this “chaos theory”. However as Stephen has so graciously pointed out. The concept of a butterfly flapping its wings, thus causing a hurricane to occur in a country miles away seems a bit too far-fetched in my opinion. Now I do not entirely dismiss the butterfly effect as simple hogwash. I thoroughly believe that with time and effort one would be able to calculate the effects of butterflies around the world to prove whether or not that a simple butterfly flapping it’s wings can cause such strife and chaos in a country miles away. Simply fascinating. How even the most slightest of events can cause such chaos in other parts of the world. Even still, why does the whole idea of the butterfly effect apply only to butterflies. Is there not any other creature out there that can cause chaos in a country miles away? Similarly, like a butterfly, a lion roaring in sub-Saharan Africa, can cause a tsunami to occur in Indonesia. Or a Gorilla pounding its chest in a New York zoo, could cause an earthquake to occur somewhere in Spain? Can someone enlighten me further on this theory? Of what I say could be possible?

Furthermore, if what Emily said was true, and that universes are in some way interconnected with everything. Think of the possibilities!

The n-body problem (gravity) is chaotic (might want to update your intro). Julia set fractals and the like are predictable in the same context as gravity, through iterative and rounded calculation. There is a difference between systems such as weather and stock markets which have many unknown variables, and systems like the purely mathematical n-body problem and the Mandelbrot set, with which we know all variables, but for which solutions require exhaustive calculation. These type of chaotic systems are chaotic in the sense that given some set of parameters, there exists no known “shortcut” equation to calculate the final result. Instead, the calculation expands into a massive Taylor series, which in practice we don’t/can’t expand fully and must truncate. Those n-body simulations of galaxies you see on youtube use non-relativistic gravity by the way, the relativist n-body problem is a whole other cat.

Note that the butterfly does not power or directly create the tornado. The Butterfly effect does not convey the notion – as is often misconstrued – that the flap of the butterfly’s wings causes the tornado. The flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn’t. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different

Paul Koszarny says:

April 12, 2013 at 1:28 pm

“I have my objections to what meteorologicalengineer wrote. He is wrong on what mathematically “initial conditions” are and as a result he claims that chaos deals with a lot of approximations. That is entirely untrue because chaos is remarkably precise in terms of chaotic behavior. Turbulences, triggers, fractal expansions, borders of chaos possess the quality of exactness…”

I have heard regarding fractals a demand that physical phenomena are not fractal unless they conform to mathematical exactness. What often is forgotten in such demands is that mathematics merely describe physical phenomena. They are not in themselves a physical phenomenon nor do they necessarily explain physical phenomena. Yet as a mode of description, the fractal concept describes many physical phenomena far better than do other alternatives. One should challenge people who make such demands for perfection to show truly physically perfect equivalents of Euclidean forms such as rectangles of circles. At some scale of measure imperfections will appear that depart from the theoretical. Nevertheless, the theoretical equations of Euclidian rectangles and circles describe these imperfect forms better than any other alternatives. The frustrating problem is that narrow-minded critics do not apply the standard they demand in fractals to the Euclidian physical forms with which they are familiar.

Here’s a passing thought I had about Chaos Theory:

There is order in chaos.

There is no chaos in the theory of chaos. Chaos is a human construct to explain the difference between control and not controlled.

It is the substance between the here and now and then and there. There is no dis order in the universe for that would mean the universe has a plan of what should be and should not be.

There is no plan to the universal mind. It is capable of creating whatever is wanted whenever it is wanted by whoever would want to do it.

The chaos you speak of is the human minds ability to substantiate and bring to the forefront an explanation of how things should be so and what should not be so.

This theory of chaos does not exist as such but will exist as long as the human construct deems it to be so.

I like what Marina has shared. It makes simple the human thought of chaos.

To me chaos theory explains the relationship that one particle, molecule, insect, animal, person, world, asteroid or any other simple thing broken down to the smallest or built into the largest object can and will have an effect on every other object.

To those confused about the “Butterfly Effect”, remember this was one mans attempt to explain what I have just stated above, but he used a metaphor. It can be that butterfly that moved the air or it could have been the child that scared the butterfly into flight or maybe the mother that sent the child out to play.

It is not just the wind the butterfly expelled with its wings it is the fact that if it had not moved it might not have affected the world at all.

Just my opinion. I am not a scholar, but I do try to think of myself as being able to grasp ideas that a lot of people may not be able to.

In point,

I have a though and please let me know if there is more on this.

Chaos theory as has been stated as “One Particle will affect another and so on for eternity with varying degrees. Thus the determination of an outcome is ever changing.” So if you add String theory into this concept and show that that particle is actually a string and therefore one wiggle of that string will cause another wiggle within that string as well as along other parts of that string. Would the two theories support each other.

Fascinating, I’m a nurse, not a scholar. I confess I lack a basic understanding but this gives me a small insight…and a start! Thank you

Great theory for expanding one’s thinking and musing. I am a Lutheran pastor and find that much of Chaos explains the church. The dynamics fit well with my process explanations of church life and the graphics I use. I find my thoughts going in directions that might provide all human organizations some needed insights on themselves.

“The Butterfly Effect” stated that a butterfly flapping its wings _CAUSES_ a hurricane far away in time and space is obviously credible only as a metaphor. If a flap could have _any_ noticeable effect on the weather,even near in time and space seems like an actual hypothesis. But there doesn’t seem to be a way to test it, cause we cannot repeat the exact conditions of an experiment, one time with the flap and one time without. The only way approach to testing the hypothesis is with a mathematical model — but a model is, by definition, a simplified representation of the real system. And if you are looking for an effect of a very small variation, using a _simplified_ representation to detect it is not very credible. As such, going back to origin of the “butterfly affect” (that a small change in the initial conditions in a mathematical model of the weather leads to large changes in the models predictions), I would say its more likely that the effect is an artifact of the need to simplify the system to be able to model it, rather than a real property of the weather system itself. This is what meteorologicalengineer was getting at in the post above.

Whether the effect is real in other real-world systems seems another question, as is whether there is anyway to test if its real.

I’ve never thought that the ‘butterfly effect’ was a very good example of chaos theory, the main reason being that it is so difficult to visualise. Here is a passing thought I had which although still related to weather conditions, I think is easier to picture.

Consider a child in for example southern Mali collecting water from a stream. He hears a call from his brother/sister ‘Hey come and look at this strange animal/insect I’ve found’. He runs across the sandy terrain to take a look and as he does so he kicks up several grains of sand. which get caught in a small thermal updraft. These few grains give critical mass to a dust cloud gathering high above which several hours later becomes a huge sand storm. The wind picks up and the storm proceeds northwards across the vast expanse of the Sahara gathering more sand as it does so, and onwards across the Mediterranean then through Southern Europe until it reaches the cold Atlantic airstream blowing Eastward through the English Channel. The storm finally dissipates but not before depositing several tons of sand on the English South coast. So the initial small effect of a child running across the sand in Mali has resulted in people waking up in South London to find their cars covered in sand. But of course you can take things much further back than that. What if the child’s brother/sister had shown indifference to his find and not bothered to call out? What if the creature had not appeared at that exact moment? What if it had been raining on that day? Any small change in these initial conditions could have resulted in an entirely different outcome.

LOVE THIS WEBSITE. New to chaos theory and you, Fractal Folk, really rock my chaotic socks off! Way to go! See you at Fractal Fest!