You are viewing a single comment's thread from:
RE: Understanding Dynamical Systems: Introduction to Chaos Theory and Its Real Life Applications
Could you indicate how a dynamical system is induced by the equation m a =F where m is mass, a is the acceleration and F is a constant force?
I don't understand this:
these systems produce an output (future state) by updating the rules at a much smaller updating time which makes it appear continuous.
what do you mean with appear continuous?
F=ma represent the net force acting on a system. I mean that the acceleration of the swinging pendulum is determined by its angle, angular velocity and it will evolve following the rule of the equation above.
these systems produce an output (future state) by updating the rules at a much smaller updating time which makes it appear continuous.
Much smaller updating time that we view as a continuous time. I am implying a limit of discrete system in which the updating time difference is so small, that the system evolves continuously in time.
To call it a rule is a bit vague. In the case of the pendulum F=ma can be written as two first order equations. This induces a flow on the phase space associated to the angle and angular velocity variable.
It is easier to start from the continuous case and then continue with the discrete case since a dynamical system with a continuous time set naturally induces a dynamical system with a discrete time set.
Vague! I really don't think it is. Is F=ma not a rule?
F describes the net force on a system. Whatever form it is written, it still centers around that equation.
On this I agree. every system evolves discretely in time. But depending the updating internal at the point of observation, it could be discrete or continuous as the case may be.
In case of a deterministic (continuous) dynamical system the future states are determined by the initial state. F=ma does not directly specify a deterministic dynamical system since it is a second order ODE. Once you have written it as two first order ODEs and then consider the corresponding flow you obtain the (deterministic) dynamical system.
Do you get what I mean by vague? I am talking about vague in the sense how your rule induces a dynamical system.
I don't think it is.
The differential equation model of the swinging pendulum corresponds to a two-dimensional map. We assume that the pendulum is free, such that it can swing through 360° (frictionless pendulum). However, the motion of the pendulum is constrained to a circle whose radius corresponds to the length l, of the pendulum rod, while the angle of the pendulum is θ which is measured in radians.
Newton's law of motion F=ma can then be used to find the pendulum equation.
The differential equation governing the pendulum becomes:
Therefore, the pendulum requires a two-dimensional state space.
For a solution to this differential equation (second order), two initial conditions must be defined i.e θ (0) and θ ̇(0) (first derivative). If just one is specified, then we cannot predict the future state of the pendulum.
However, if θ and θ ̇(first derivative) are specified at time t=0. We can uniquely determine θ (t) at the next instant.
You can check out:
I said : F=ma does not directly specify a deterministic dynamical system since it is a second order ODE.
Observe that
is a second order ODE. Observe that Ө' is not a dependent variable of this ODE. So you need to define it as a dependent variable then you can write the system as two first order ODEs and apply the uniqueness theorem of first order ODEs which induces flow.
I am trying to point out that if you have the equation
Then this does not induce a unique dynamical system since for example i could consider the system corresponding to the dependent variables defined by x=Ө , y=Ө' but also v=Ө , w=2Ө'
Is this clear?
Your explanation is clear, but I just want you to get my point. This example is simplified with assumptions to make it fit the context in which I used it above. Those two initial conditions must be specified. There are still other systems which corresponds to a two dimensional state space and two initial conditions need to be specified for determinism and the law of motion is still F=ma
In fact, in the case of nonlinear methods,if the methods are used only when determinism is strong then there would be a limitation, but it is used even when determinism is weak.
interesting intellectual argument going on here, I will just observe.
Yeah I understand it is a simplifiction.
What do you mean with
More specifically, what is strong determinism and what is weak determinism?