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RE: Nonlinear Dynamical Systems: Chaos Theory, Models and the Butterfly Effect
so then the following statement does not make any sense:
Generally, a periodic system or quasiperiodic system must have the following properties:
- At least one nonlinear factor (term) must be present in the system
- It must be at least one dimensional
I really don't understand what you mean. Are you implying that a nonlinear dynamical system cannot be periodic or quasiperiodic? because then what I have been reading in various text will be false.
What I am saying in a nutshell is that the equation of motion for nonlinear systems will have at least one term that is either a square or higher power, a product of two or more variables of the system or even a more complicated function etc.
1)Could you maybe specify why you think I am implying that a nonlinear dynamical system cannot be periodic or quasiperiodic?
2)You said
In the post you said
These statements seem to contradict. Or the use of generally in this sentence is strange.
3)the statement that all nonlinear dynamical systems show sensitive dependence on initial conditions is not true. (I thought you corrected it differently)
My post is on nonlinear dynamical systems, so I wasn't referring to linear system.
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.
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)
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