S.O.S. Mathematics CyberBoard

While reading the book I came across the concept of chaos. Is there any mathematical definition of chaos?

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

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.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Thank you for help.

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

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

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

I agree with shadow.

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).

_________________

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.