Chaotic system w/ initial condition

In summary, changing the initial conditions of a chaotic system can indeed lead to it becoming non-chaotic. Chaotic systems are sensitive to initial conditions, so changing them can have an effect on the chaotic behavior. This is demonstrated by the example of a damped driven pendulum and Duffing Oscillator, where changing the initial conditions resulted in a non-chaotic system. However, it is important to note that for a dynamic system with fixed parameters, there may be multiple attractors, including both chaotic and non-chaotic ones, and the initial conditions will determine which type of trajectory is followed.
  • #1
Brown Arrow
101
0
so i have been studying chaotic system in class, and i just want to know if we change the initial conditions of a chaotic system can it become non-chaotic?

I think yes because, chaotic system is sensitive to initial condition hence it would have an effect on the chaotic behavior.

I'm I right? i have a feeling I'm wrong.


:/ I'm contradicting my self :confused:
 
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  • #2
Brown Arrow said:
so i have been studying chaotic system in class, and i just want to know if we change the initial conditions of a chaotic system can it become non-chaotic?

I think yes because, chaotic system is sensitive to initial condition hence it would have an effect on the chaotic behavior.
Hi Arrow. I'm no expert on chaotic systems but isn't a pencil balanced on its point a chaotic system?
Ref:
http://www.rigb.org/christmaslectures06/pdfs/why_does_it_always_rain.pdf
The Frontiers of Science
If so, then what happens if the initial conditions of the perfectly vertical pencil were to be changed?
 
  • #3
thanks for the reply Q Goest.

umm yes that is true it will no longer be chaotic.

guess i did not specify the system :/

I had a damped driven pendulum and Duffing Oscillator,

I ran some plotting (in python) for it and changed the initial condition. what once was a chaotic system became non-chaotic after the change in initial condition.

so is it safe to assume that chaotic system can become non-chaotic depending on the initial condition? from the reference you gave Q Goest i think the answer is yes.
 
  • #4
In short, for a (dissipative) dynamic system with a given fixed set of parameters there can in general be one or more attractors (with at least one of these being a chaotic attractor a.k.a. strange attractor if the system is to be chaotic) each with an associated basin of attraction. If the system in addition to the chaotic attractor(s) has a non-chaotic attractor (say, a fix point) then there obviously must be some initial conditions, namely those in the basic of attraction for this non-chaotic attractor, that will lead to a non-chaotic trajectory.

For the Duffing Oscillator (Duffing's Equation) I believe there are parameters for which the system has both chaotic and non-chaotic trajectories, and in those cases you will get chaotic or non-chaotic trajectory depending on the initial conditions. For instance, it looks like there should be both a chaotic and non-chaotic attractor for k = 0.2 and B = 1.2 (liftet from Ueda's parameter map for Duffing's Equation as it is shown in [1]).

[1] Nonlinear Dynamics and Chaos, Thompson and Steward, Wiley, 2002.
 
  • #5


I can say that your thinking is on the right track. Changing the initial conditions of a chaotic system can indeed affect its behavior and potentially lead to a non-chaotic outcome. However, it is important to note that chaos is not simply determined by initial conditions alone. There are other factors at play, such as the system's parameters and external influences, that can also impact its behavior. So while changing initial conditions may have an effect, it is not a guarantee that the system will become non-chaotic. Further research and analysis would be needed to fully understand and predict the behavior of a chaotic system.
 

1. What is a chaotic system with initial condition?

A chaotic system with initial condition refers to a non-linear system that exhibits unpredictable and chaotic behavior over time, which is highly sensitive to its initial conditions. This means that even a small change in the initial conditions can result in significantly different outcomes or trajectories.

2. How is chaos theory related to chaotic systems with initial condition?

Chaos theory is a branch of mathematics that studies the behavior of chaotic systems. It helps to understand and predict the dynamics of these systems, which are highly sensitive to initial conditions and exhibit unpredictable behavior.

3. What are some real-life examples of chaotic systems with initial condition?

Some real-life examples of chaotic systems with initial condition include weather patterns, population dynamics, stock market fluctuations, and the movement of celestial bodies. These systems are highly complex and are affected by numerous variables, making their behavior unpredictable.

4. Can chaotic systems with initial condition be controlled or predicted?

It is challenging to control or predict the behavior of chaotic systems with initial condition due to their sensitivity to initial conditions. However, chaos theory provides techniques such as Lyapunov exponents and bifurcation diagrams, which can help in understanding the system's dynamics and making predictions to some extent.

5. How are chaotic systems with initial condition relevant in science and technology?

Chaotic systems with initial condition have numerous applications in science and technology. They are used in fields such as meteorology, physics, biology, economics, and engineering to model and analyze complex systems. In technology, chaos theory has been applied in the development of secure communication systems and data encryption methods.

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