Why a chaotic system always bounded?

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Discussion Overview

The discussion centers on the boundedness of chaotic systems, exploring the factors that contribute to this characteristic. Participants examine theoretical implications, specific examples, and the relationship between initial conditions and chaotic behavior.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants suggest that the unpredictability of chaotic systems implies they must be bounded, as systems that can blow up to infinity are predictable and thus not chaotic.
  • Others point to specific examples, such as the double pendulum and chaotic circuits, which are bounded by physical constraints like the lengths of components or energy sources.
  • A participant references the Logistic Map from "Nonlinear Dynamics," noting that bounded initial values lead to bounded chaotic behavior, raising the question of whether boundedness is solely due to initial conditions.
  • Another participant argues that not all chaotic systems are necessarily bounded, citing nonlinear ordinary differential equations (ODEs) that can exhibit unbounded behavior depending on initial conditions.
  • Some participants clarify that while chaotic solutions must be bounded, there may exist unbounded solutions in systems that could potentially exhibit chaotic behavior.
  • A later reply mentions that chaos requires periodic orbits to be dense in phase space, suggesting that unbounded dynamics would contradict this requirement.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between chaos and boundedness. While some agree that chaotic solutions must be bounded, others contend that systems capable of chaotic behavior may still have unbounded solutions depending on initial conditions. The discussion remains unresolved regarding the generality of boundedness in chaotic systems.

Contextual Notes

Participants acknowledge that the boundedness of chaotic systems may depend on specific initial conditions and the definitions used in their analyses. There are unresolved mathematical considerations regarding the implications of chaotic behavior and boundedness.

saravanan13
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Why a chaotic system always bounded?
What factor control the boundedness?
 
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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.
 


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?
 


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.
 


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.
 


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
 

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