Why a chaotic system always bounded?

In summary, the boundedness of a chaotic system is closely related to its unpredictability, as knowing the system's behavior would make it non-chaotic. Factors such as physical constraints and initial conditions also play a role in determining the boundedness of a chaotic system. While some solutions may be unbounded, all chaotic solutions are bounded. Additionally, chaotic behavior requires dense periodic orbits in the phase space, which would not be possible in an unbounded system.
  • #1
saravanan13
56
0
Why a chaotic system always bounded?
What factor control the boundedness?
 
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  • #2


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.
 
  • #3


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?
 
  • #4


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.
 
  • #5


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.
 
  • #6


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
 

1. Why is a chaotic system always bounded?

A chaotic system is always bounded because it follows the laws of physics and mathematics. These laws dictate that energy and matter cannot be created or destroyed, but can only be transformed. In a chaotic system, the interactions and transformations of energy and matter are constantly occurring, leading to a bounded behavior.

2. How does chaos theory explain the boundedness of a system?

Chaos theory explains the boundedness of a system through the concept of sensitive dependence on initial conditions. This means that even a small change in the starting conditions of a chaotic system can result in drastically different outcomes. As a result, the system is limited in its behavior and cannot exceed certain boundaries.

3. Can a chaotic system become unbounded?

No, a chaotic system cannot become unbounded. This is because, as mentioned before, the laws of physics and mathematics dictate that energy and matter cannot be created or destroyed. Therefore, the system will always be limited in its behavior and cannot exceed certain boundaries.

4. How do scientists study the boundedness of chaotic systems?

Scientists study the boundedness of chaotic systems by using mathematical models and simulations. These models help them understand the behavior of the system and predict its future states. They also use data analysis techniques to observe patterns and identify boundaries in the system's behavior.

5. Are there any real-life examples of chaotic systems that are bounded?

Yes, there are many real-life examples of chaotic systems that are bounded. Some examples include weather patterns, stock market fluctuations, and population dynamics. These systems exhibit chaotic behavior, but they are still limited by certain boundaries and follow the laws of physics and mathematics.

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