Discrete-Time Dynamical Systems Flashcards
Population growth and heart and lung dynamics are examples of what type of change?
Discrete change
A discrete-time dynamical system models what type of change?
Discrete change
What relationship does a discrete-time dynamical system describe?
The relationship between a quantity at the beginning and end of a time interval.
In a discrete-time dynamical system, what does m(t) represent?
A measurement at the beginning of the time interval
In a discrete-time dynamical system, what does (t + 1) represent?
The time one step into the future
In a discrete-time dynamical system, what does m(t+1) represent?
A measurement at the end of an experiment
What types of numbers make up the domain of a discrete-time dynamical system?
Whole numbers only
How do you graph a discrete-time dynamical system?
By plotting points on the graph. This is because the domain only contains those points.
How do you find the point of equilibrium of a DTDS?
Sub in L (limit) for m(t) and m(t + 1). Solve for L.
What is the updating function of a DTDS?
The formula of the dynamic rule.
How do you find the updating function?
Replace m(t) with x and m(t + 1) with f(x)
What is cobwebbing?
A way to find values for m(t + 1) using graphing.
In cobwebbing, which two functions are graphed?
The updating function and y = x.
What do the x-coordinates where the updating function intersects with y = x signify?
Where the DTDS reaches equilibrium.
Given the updating function x(t + 1) = f(xt), what is the equilibrium of the DTDS?
The value L for which L = f(L)
If you start near x = L and the points approach x = L, is the equilibrium stable or unstable?
Stable
If |f’(L)| < 1, is the equilibrium stable or unstable?
Stable
If |f’(L)| > 1, is the equilibrium stable or unstable?
Unstable
If |f’(L)| = 1, is the equilibrium stable or unstable?
Could be either. Need to use cobwebbing
In medication concentration problems, is the equilibrium stable or unstable?
It is always stable
In population growth problems, is the equilibrium always stable or unstable?
Could be stable or unstable
