Why a chaotic system always bounded?

[saravanan13]

Why a chaotic system always bounded?
What factor control the boundedness?

 

[eczeno]

well, and this is not rigorous, but the boundedness of a chaotic system seems to follow from the fact that it is 'unpredictable'. that is, if we know a system blows up to infinity at some point, well, we can predict what that system is going to do, and hence it is not chaotic.

the factors involved would depend on the particular problem. for instance, the double pendulum is physically bounded by the lengths of the two parts, chaotic circuits are bounded by the energy source and so on.

again, not rigorous or anything, but i hope this gives some insight.

 

[saravanan13]

Well I accept your answer. But in the book "Nonlinear Dynamics" written by M. Lakshmanan and S. Rajasekar, they investigated the Logistic Map. In that they fix the initial value for 'x' lies between 0 and 1, finally they conclude the topic with Lyapunov exponent. At last in the definition it is said chaos in a bounded phenomenon. It is because of this initial value it is bounded?

 

[Skrew]

I don't know much about this topic area but I'm pretty sure a system which could be chaotic is not alway bounded.

IE a nonlinear ODE could be unbounded, depending on the initial conditions, etc.

 

[eczeno]

yes, the boundedness of the solution to a particular system is dependent on the ic's, and so a system may or not be bounded for particular ic's. i guess what i should have said is that if the solution is to display chaotic behavior, then the system is bounded for those particular ic's. if the solution blows up, i.e. the system is not bounded, then there can be no chaotic behavior.

so, a system which could be chaotic may have some unbounded solutions, but all chaotic solutions will be bounded.

 

[lavinia]

According to the Wikipedia article chaos requires that its periodic orbits are dense in the phase space. If the dynamics were unbounded it seems ely intuitivthat this could not happen