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 Post subject: Chaos - definition.Posted: Thu, 17 Nov 2011 22:14:09 UTC
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While reading the book I came across the concept of chaos. Is there any mathematical definition of chaos?

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 Post subject: Re: Chaos - definition.Posted: Thu, 17 Nov 2011 22:26:35 UTC
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 Post subject: Re: Chaos - definition.Posted: Thu, 17 Nov 2011 23:29:56 UTC
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I read it, but I still do not know how to strictly define chaos. I would like to use the definition of chaos in the introductory paper on chaos theory or the theory of dynamical systems, so I need a definition with objectives/equations, etc. Could you give me any example of chaos and say why it is chaos?

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 Post subject: Re: Chaos - definition.Posted: Thu, 17 Nov 2011 23:30:54 UTC
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KamilJ wrote:
I read it, but I still do not know how to strictly define chaos. I would like to use the definition of chaos in the introductory paper on chaos theory or the theory of dynamical systems, so I need a definition with objectives/equations, etc. Could you give me any example of chaos and say why it is chaos?

Chaos is a subject, not a mathematical construct, so it's definition is fundamentally descriptive, and not mathematical.

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 Post subject: Re: Chaos - definition.Posted: Wed, 30 Nov 2011 19:45:48 UTC
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Thank you for help.

Could you tell me one more thing - where can I find a well described Smale horseshoe?

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 Post subject: Re: Chaos - definition.Posted: Thu, 1 Dec 2011 00:27:21 UTC
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 Post subject: Re: Chaos - definition.Posted: Sat, 3 Dec 2011 18:58:29 UTC
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Chaos is a subject, not a mathematical construct, so it's definition is fundamentally descriptive, and not mathematical.

I agree with shadow.

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 Post subject: Re: Chaos - definition.Posted: Sat, 3 Dec 2011 19:19:50 UTC
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Is it the same with the bifurcation?

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 Post subject: Re: Chaos - definition.Posted: Sun, 4 Dec 2011 08:49:11 UTC
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KamilJ wrote:
Is it the same with the bifurcation?

No. Bifurcation is very precisely defined, e.g. for continuous-time dynamical systems arising from ODE, bifurcation of an autonomous system of ODE occurs at when arbitrarily close to this parameter value, there are other parameter values c that the dynamical system isn't (topological/differential/... depending on the category) conjugate to the original system.

The usual hiding behind a descriptive picture is just that you don't really want to go on a detour into defining these terms (usually defined in the second course on differential equations/dynamical systems/whatever they call it, i.e. the next course after introduction of things like Poincar\'{e}-Bendixon theorem).

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 Post subject: Re: Chaos - definition.Posted: Sun, 4 Dec 2011 18:43:00 UTC
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So what is the definition for chaotic systems?

I am after first course of differential equations, so I don't know how should I understand that. I'm reading some book about chaos theory (introduction) and I'm trying to get familiar with idea of bifurcation and bifurcation diagrams.

Maybe I should ask: how should I start with chaos theory? I've read that dynamical system is chaotic, if it is transitive and the set of periodic points is dense. I have also read about logistic map, attractors, repellers and some older student told me, that I should next know what is bifurcation. I can't help feeling, that I got lost in it.

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