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RE: Nonlinear Dynamical Systems: Chaos Theory, Models and the Butterfly Effect
So what does linear factor mean in the sentence** one linear factor must be present in the system**
As a note on chaos. There is no universal definition of chaos. Furthermore, there are many non-equivalent definitions of chaos. The definition for chaotic dynamics using Lyapunov exponents is just one of many.
This statement all dynamical systems show sensitive dependence on initial conditions is not true
Thank you @mathowl for the correction. As you might have noted, I was talking about nonlinear dynamical system even from the sub-heading. I must have unconsciously omitted the word "nonlinear" when stating that.
I just realized I kept saying linear as against nonlinear that I am discussing. Quite disheartening. Imagine the word document for this post I wrote some months back has nonlinear all through. I have made the corrections. You can peruse now and let me know what you think.
Yes. Lyapunov exponent is one of many. I recently had a discussion with some who worked on chaos and has never used the Lyapunov exponent.
Thanks once again!
Good job in correcting it.
I am still not sure what you mean with nonlinear factor. A linear and a nonlinear system can both exhibit periodic behaviour. In the linear case just consider the discrete dynamical system with phase space \mathbb{R} / \mathbb{Z} and evolution operator x \mapsto a x where a is a real non-zero constant. The evolution operator is linear and it exhibits periodic behaviour.
You are right; a periodic system can be linear or nonlinear. With nonlinear term, I am referring to the function that specifies the change in the system. That is, the rule governing the system must contain nonlinear term.
So sorry for my belated response, I have to teach kids in a rural community every day where we hardly get internet connection. I only see comments and read post when I get home around this time.
so then the following statement does not make any sense:
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: