RE: Nonlinear Dynamical Systems: Chaos Theory, Models and the Butterfly Effect

[obamike]

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.

Comments

[mathowl]

so then the following statement does not make any sense:

Generally, a periodic system or quasiperiodic system must have the following properties:

1. At least one nonlinear factor (term) must be present in the system
2. It must be at least one dimensional

[obamike]

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.

I hope you know I am talking about periodic and quasiperiodic systems in terms of nonlinear system as my topic imply and not linear system?

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.

[mathowl]

1)Could you maybe specify why you think I am implying that a nonlinear dynamical system cannot be periodic or quasiperiodic?

2)You said

a periodic system can be linear or nonlinear.

In the post you said

Generally, a periodic system or quasiperiodic system must have the following properties:

1. At least one nonlinear factor (term) must be present in the system
2. It must be at least one dimensional

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)

[obamike]

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...

[mathowl]

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:

A periodic system or quasiperiodic system must have the following properties:

1. At least one nonlinear factor (term) must be present in the system
2. It must be at least one dimensional

[obamike]

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:

all animals depend on plants and algae for food?

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.

[mathowl]

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.