Where are nonlinear dynamics and chaos theory used?

[MathWarrior]

Basically where are nonlinear dynamics and chaos theory used in the real world? Like if someone studies it what type of areas might they find it being useful for? The only example I can seem to think of is stuff like weather/fluids/air resistance/physics.

What might be some other more applications of it, or where have you seen it come up?

 

[epenguin]

AFAIK when chaos became fashionable and the field exploded in the 1970's its value firstly appeared to be in pure understanding or in poofs of impotence, e.g. impossibility of long-range weather forecasting. Only in 1990 did Ott, Yorke and Grebogi point out in a seminal paper that precisely the butterfly effect allows control of chaotic systems with minimal interventions ((so-called 'non-invasive control'). This should allow to extend the operating ranges and performances of devices. Various strategies for control of chaos were developed known as the OGY method, occasional proportional feedback, delayed feedback, adaptive strategy, targeting, selective filtering.

I am completely out of touch and ignorant of what has happened but 15 years ago I heard talk of:

secure communications - understand using a chaotic signal as carrier modulated by the message signal to produced a still chaotic signal from which anyone not possessing the large number which is the 'initial value' of the chaotic carrier cannot filter it out;

lean burners - where the most efficient operation is in or near the chaotic regime. The trajectories of the system make excursions that put the flame out, so one wants controls that maintain the chaos but with occasional controls that avoid the flameout, and in general combustion devices (therefore engines) and all applications where turbulence is important;

lasers - some potentially most useful ones had chaotic outputs.

These are all large-scale potential applications, I do not know if anything has been realized outside the laboratory.

Longer or more speculative possibilities I heard of were:
better control of industrial cutting tools and their problem of 'chatter' (another potentially large-scale industrial benefit), more intelligent climate control, neuropsychiatry, prostheses, cardiac and other medicine, controlled thermonuclear fusion.

 

[HallsofIvy]

Turbulent flow of fluids can also be dealt with in terms of "chaotic behavior".

 

[rahul.6sept]

The concept of non-linear dynamics is in use in MEMS in fabrication of resonators.

 

[yus310]

Look at Lorentz effect=awesome, weather related

heart beat=disease

structural mechanics

electromagnteism

 

[rahul.6sept]

hi Yus310,

Can u elaborate ur reply please ??

 

[jackmell]

Basically where are nonlinear dynamics and chaos theory used in the real world?

Most of the world is non-linear. I remember using some in school a book entitled, "The Equations of Mathematical Physics". Everything in there was non-linear! So I suppose you could take a look at that book to see what's up. I mean even the pendulum is non-linear. Imagine that, just some blob going back and forward. Economies and societies are non-linear, evolution is non-linear, the human brain is non-linear, and cosmology too.

 

[yus310]

hi Yus310,

Can u elaborate ur reply please ??

First of all the question of interest is too broad. Hence my reply.

Second, there are many applications of

http://en.wikipedia.org/wiki/Van_der_Pol_oscillator

are utilized in studying neurons and irregular heart beats.

The Butterfly effect, is a thoroughly interesting theory and one that has been studied in non-linear e.g.

http://en.wikipedia.org/wiki/Butterfly_effect

I call it the Lorenz effect, but that something you should explore yourself.

Non-linear science is something you should explore. By the fact, that you ask what is the applications, indicates that you yet to have an appreciation for this beautiful science. Applications are not of importance, until an appreciation can be established.

 

[jackmell]

I believe all the secrets of the Universe can be found in differential equations and in particular non-linear differential equations. After some time studying them, it will all begin to make sense and when you see things in life, not just science things, that puzzle most people and often causes them grief because they dont' understand, you'll say, "yeah, I know just why that's happening, and that knowledge comforts me and I am at peace with that understanding."

To me that is the best application of non-linear dynamics. :)