| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | S | u | b | j | e | c | t | : | | C | h | a | o | s | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
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| | Q3a: | What is chaos? |
| | A3a: | Chaos is apparently unpredictable behavior arising in a deterministic |
| system because of great sensitivity to initial conditions. Chaos arises in a |
| dynamical system if two arbitrarily close starting points diverge |
| exponentially, so that their future behavior is eventually unpredictable. |
| Weather is considered chaotic since arbitrarily small variations in initial |
| conditions can result in radically different weather later. This may limit the |
| possibilities of long-term weather forecasting. (The canonical example is the |
| possibility of a butterfly's sneeze affecting the weather enough to cause a |
| hurricane weeks later.) |
| Devaney defines a function as chaotic if it has sensitive dependence on |
| initial conditions, it is topologically transitive, and periodic points are |
| dense. In other words, it is unpredictable, indecomposable, and yet contains |
| regularity. |
| Allgood and Yorke define chaos as a trajectory that is exponentially |
| unstable and neither periodic or asymptotically periodic. That is, it |
| oscillates irregularly without settling down. |
| sci.fractals may not be the best place for chaos/non-linear dynamics |
| questions, sci.nonlinear newsgroup should be much better. |
| |
| | Q3b: | Are fractals and chaos synonymous ? |
| | A3b: | No. Many people do confuse the two domains because books or papers |
| about chaos speak of the two concepts or are illustrated with fractals. |
| Fractals and deterministic chaos are mathematical tools to modelise different |
| kinds of natural phenomena or objects. The keywords in chaos are |
| impredictability, sensitivity to initial conditions in spite of the |
| deterministic set of equations describing the phenomenon. |
| On the other hand, the keywords to fractals are self-similarity, invariance |
| of scale. Many fractals are in no way chaotic (Sirpinski triangle, Koch |
| curve...). |
| However, starting from very differents point of view, the two domains have |
| many things in common : many chaotic phenomena exhibit fractals structures (in |
| their strange attractors for example... fractal structure is also obvious in |
| chaotics phenomena due to successive bifurcations ; see for example the |
| logistic equation | Q9 | ) |
| The following resources may be helpful to understand chaos: |
| |
| sci.nonlinear FAQ (UK) |
| http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html |
| |
| sci.nonlinear FAQ (US) |
| http://amath.colorado.edu/appm/faculty/jdm/faq.html |
| |
| Exploring Chaos and Fractals |
| http://www.lib.rmit.edu.au/fractals/exploring.html |
| |
| Chaos and Complexity Homepage (M. Bourdour) |
| http://www.cc.duth.gr/~mboudour/nonlin.html |
| |
| The Institute for Nonlinear Science |
| http://inls.ucsd.edu/ |