What is Chaos? An Introduction for Beginners

In summary, chaos refers to systems that exhibit large response to small changes in initial conditions, creating complex behavior and patterns through simple rules. It is a broad concept with different definitions and applications in various fields of study. Chaotic systems do not reach equilibrium and can undergo reversals, making their behavior difficult to predict. Turbulent flow and weather patterns are examples of chaotic systems. Chaos theory also deals with finding order in seemingly chaotic situations through the use of mathematical principles.
  • #1
What is chaos?

Hello everybody.

I just wanted to know if anyone could explain to me what is chaos all about in verrrrry simple terms, because I found a book on it in the library and I wanted to know if anyone could please give me a simple introduction to what it is as I find the book quite confusing. (sorry if I seem a bit dim)

Thank you!
 
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  • #2
A system exhibits chaos if it has arbitrarily large response to a fixed small change in any of its initial conditions.
 
  • #3
the study of simple or large systems..."creating" complex behaviour or patterns through a simple set of rules.
 
  • #4
Any system with some sort of positive feedback will exhibit chaos.

It is probably worth noting that there are many definitions of chaos floating around, two of which have already been mentioned.

Claude.
 
  • #5
My life... :frown:
 
  • #6
Claude Bile said:
Any system with some sort of positive feedback will exhibit chaos.

It is probably worth noting that there are many definitions of chaos floating around, two of which have already been mentioned.
James Gleik's book on chaos covers so much ground it's hard to extrapolate a definition that covers everything that's been put into the category.

The thing that seemed important to me was that chaotic systems are those that don't ever achieve equilibrium and which also undergo reversals, as in the Earth's magnetic field flips.
 
  • #7
krab said:
A system exhibits chaos if it has arbitrarily large response to a fixed small change in any of its initial conditions.

Yes, but this is the definition for deterministic chaos.

Non-deterministic system chaotic evolution is observed even for fixed initial conditions, due to Poincaré resonances (i.e. non-determinacion in the evolution of the mechanical system).
 
  • #8
Danger said:
My life... :frown:
-_-; sorry (laughing)
 
  • #9
krab said:
A system exhibits chaos if it has arbitrarily large response to a fixed small change in any of its initial conditions.
Would you consider unstable -systems (you know, like closed loop systems)to be chaotic ? Personally, i would say know because there is still some kind of structural behaviour in the closed loop's output (for example : a sinusoidal output function, expressing the fact that the output just oscillates between two fixed values)

marlon
 
  • #10
krab said:
A system exhibits chaos if it has arbitrarily large response to a fixed small change in any of its initial conditions.

marlon said:
Would you consider unstable -systems (you know, like closed loop systems)to be chaotic ? Personally, i would say know because there is still some kind of structural behaviour in the closed loop's output (for example : a sinusoidal output function, expressing the fact that the output just oscillates between two fixed values)

marlon

Deterministic chaos is related to the "imprecision" of initial conditions and effect in diferent final states, and applies to systems far from equilbrium. Chaotics systems are generally "perpetuum" if externals conditions let it.

An instable system, in general, is not chaotic, because a small change on initial conditions by a perturbation does not provide always a radical modification of final states. For example if you can change the perturbation of an instable pendulum and obtain always the same final state, whereas a small different perturbations in a chaotic molecular system always provide diferent final states due to exponential increase of initial difference.
 
  • #11
Danger said:
My life... :frown:
:rofl: :rofl: :rofl: :rofl: :rofl: Sorry, Can't stop! :rofl: :rofl: :rofl: :rofl: :rofl: :rofl: :rofl: :approve: :approve: :biggrin: :biggrin:
 
  • #12
Juan R. said:
Deterministic chaos is related to the "imprecision" of initial conditions and effect in diferent final states, and applies to systems far from equilbrium. Chaotics systems are generally "perpetuum" if externals conditions let it.

An instable system, in general, is not chaotic, because a small change on initial conditions by a perturbation does not provide always a radical modification of final states. For example if you can change the perturbation of an instable pendulum and obtain always the same final state, whereas a small different perturbations in a chaotic molecular system always provide diferent final states due to exponential increase of initial difference.

Do you think turbulent flow is chaotic?
 
  • #13
Clausius2 said:
Do you think turbulent flow is chaotic?

Yes, i think so. Two elements of volume initially very closely between them can evolutionate in time to very different positions.
 
  • #14
Juan R. said:
Yes, i think so. Two elements of volume initially very closely between them can evolutionate in time to very different positions.

Then, why the final solution of N-S equations for turbulent flow is unique for a set of initial and boundary conditions given?
 
  • #15
Clausius2 said:
Then, why the final solution of N-S equations for turbulent flow is unique for a set of initial and boundary conditions given?

Hum!

1) NS are not dynamic equations.

2) NS are already based in chaos: in molecular chaos.

3) I am not sure of application of NS to full chaotic regimes. In completely developed chaos with long-range correlation the local approach of usual NS is no longer valid. What then? modified large correlation NS equations? Perhaps but i am not sure that version of NS would correctly work: rational extended one? q-like ones? :confused:

4) Your statement about uniqueness of flow is only valid for the conditional average limit, which is not very valid for realistic studies, for example in the "nano "regime.

Any case, that do you attempt to say me?
 
  • #16
So through all this chaos what is chaos supposed to described in physics?

~Kitty
 
  • #17
LedZep_Kamal said:
Hello everybody.

I just wanted to know if anyone could explain to me what is chaos all about in verrrrry simple terms, because I found a book on it in the library and I wanted to know if anyone could please give me a simple introduction to what it is as I find the book quite confusing. (sorry if I seem a bit dim)

Thank you!

Chaos deals with the existence of order in situations of seeming disorder. For example, scientists have discovered that such seemingly unrelated sitautions as the stock market and fluctuations of populations of various biological organisms may be described by certain mathematical series or numbers.

Some situations like climate/weather involve the complex interactions of various cycles such as the El Nino/La Nina cycles in the Pacific and the North Atlantic Oscillation as well as changes in solar energy output.

You may have read of the Butterfly Effect, a butterfly fluttering in Beijing may cause a thunderstorm in Kansas a week later if reinforced by other events.
An old poem: "For want of a [horseshoe] nail, the horse was lost. For want of a horse, the rider was lost. For want of a rider the battle was lost."

A more modern example might be a situation in which you need to travel to a city some distance away. You think you have plenty of time, but as you start to lock your door you drop your keys. The few seconds it takes to pick up the keys mean you miss a green light. The 30 seconds you wait for the light to change mean you reach a railroad crossing 5 seconds after a long slow freight starts crossing it instead of 25 seconds before. The 3 minutes you wait for the train mean you reach a point on the highway 1 minute after a traffic blocking wreck occurs instead of 2 minutes before. The 30 minutes you wait in traffic mean you get to the airport too late to pass through the security check.
 
  • #18
misskitty said:
So through all this chaos what is chaos supposed to described in physics?

~Kitty

Chaos theory was developed in part to deal with weather/climate. It may also deal with the flowing of fluids.
 
  • #19
Chaos cannot be used in periodic environments such as ocean current analysis. For non periodic systems like Atmospheric properties and fluid flow, chaos works nicely.

Regards,

Nenad
 

What is chaos?

Chaos refers to the mathematical concept of a system that is highly sensitive to initial conditions, meaning that small differences in the starting conditions can lead to vastly different outcomes.

Why is chaos important?

Chaos theory has a wide range of applications in science, including weather forecasting, ecology, and economics. It also has implications for understanding complex systems and patterns in nature.

What are some examples of chaos in the natural world?

Examples include the weather, the movement of planets, and population dynamics in animal populations. Chaos can also be found in systems such as the stock market and even in our own brains.

How is chaos different from randomness?

While chaos can appear random, it is actually deterministic, meaning that the behavior of the system can be predicted using mathematical equations. Randomness, on the other hand, is truly unpredictable.

Can chaos be controlled or predicted?

While chaos cannot be controlled, it can be predicted to some extent using mathematical models and simulations. However, even small changes in initial conditions can lead to vastly different outcomes, making accurate long-term predictions difficult.

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