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RE: Nonlinear Dynamical Systems: Chaos Theory, Models and the Butterfly Effect
My post is on nonlinear dynamical systems, so I wasn't referring to linear system.
the statement that all nonlinear dynamical systems show sensitive dependence on initial conditions is also not true.
Depending on the initial condition, a nonlinear system can be periodic or chaotic which is a dependence on initial condition. I mean dealing with real systems.
I believe generally could mean usually or widely except my knowledge of the English language has failed me. You have to forgive me. You know I am from Africa and English is not our first language.
It's like saying generally dogs eat bones...
So if that is the case:
1)Do you agree that the dynamical system induced by the ODE x'=-x^3 has no initial conditions which correspond to sensitive dependence. (EDIT: changed the vector field from -x^2 to -x^3)
2)Do you agree that the following statement is incorrect:
1)i think I have tried to clarify what I mean by that statement: depending on the initial condition, a single nonlinear system can be periodic or chaotic
So as an analogy, is it wrong to say:
If it is wrong then I agree
2)In terms of linear systems I agree. But in terms of nonlinear systems on which my post is based, well...I will like you to suggest a textbook I can read.
2)you do realise that all nonlinear systems are of dimension one and have a nonlinear term right? So the properties 1 and 2 are useless in the setting of a nonlinear system.
Please state a reference material or textbook I can consult for the above statement. Or is it to the best of your knowledge?
So what of Poincare map? Henon map?
I hope you know that in terms of dimension I am referring to map(mathematical transformation) or you could explain what the dimension you are talking about is?
Please I will like to see a source or reference book in your next comment,. After all, an information is as good as the source.
I am still talking about this part of your post->
So you said you were only considering non-linear systems for this statement.
My claim is that this statement gives no information for non-linear systems since all non-linear systems automatically satisfy both 1 and 2.
I have no idea for a reference but the proof is almost immediate. Since you only need to show that non-linear dynamical systems cannot have dimension 0.
(Note that in the above I am considering a phase space which is an open subset of \mathbb{R}^n. I am also assuming that you are considering a connected phase space)
Made with http://quicklatex.com/ :)
Teacher(me): students the characteristics of living things include: Movement, Respiration, Irritability, growth, etc..
I then ask a question, what are the characteristics of a dog?
Student: movement, respiration, irritability, growth...etc
Teacher (me): It doesn't make sense since all living things has this properties.
This gives no information for a dog
Alright, I didn't just stop after mentioning this properties for periodic and quasiperiodic systems, I also say the same for chaotic and hyperchaotic systems. I have not left any nonlinear dynamical system out in my description so what is my crime?
You can tell that is a general property, what if I have left that out and some other reader who is not well vast or doesn't even know about nonlinear system as much as you reads and that information is not there?
Whatever I write are things I read in the course of my study and I am still studying.
With due respect sir, I hope you don't have something against me? because I can't help but noticed that you are dragging me back and forth.
If you have something you wish to tell me please feel free...I love honest feedback.
I just want your post to be correct and have meaning.
Using your example it is not usefull to say that a dog is a thing. This gives almost no information. The same is true for the point 1 and 2 in the earlier statement.
So my post is not correct and doesn't have meaning?
please note, it is useful to say that a dog is a living thing! What is useful is relative!