Quiz 2 Flashcards
What is a system?
A differential equation
What is the state of the system?
u
What is a dynamic?
f(u)
If the initial value of u(t)=u(0), what is the long-term behavior of the system?
What happens to lim u(t) as the t approaches infinity?
If the initial value of u(t)= u(0) what is the short term behavior of the system?
Dependent on the application
What are the axes for a dynamics plot?
x=u, y= f(u)
What is an equilibrium point?
a constant solution for the function f(u)=0.
If u* is an equilibrium point, what is f(u*) equal to?
f(u*)=0
What does it mean for an eq pt to be isolated?
There is a small neighborhood containing u* s.t. no other U* exist within this neighborhood.
What do eq pts show?
The direction of the movement of as time increases.
What is a phase line?
Plots the growth rate vs the population. p on left p’ on right
What is a phase line used for?
To determine if the points in the neighborhood containing u* move toward or away from the eq pt
What is an attractor?
A stable eq pt. The points in the neighborhood (on both sides of u) are attracted to u
What is a repeller?
An unstable eq pt. The points in the neighborhood (on both sides of u) are repelled or pushed away from u
What is a semi-stable eq pt?
The points in the neighborhood containing u* are attracted to u* on one side and repelled on the other
When is an eq pt globally stable?
When at no matter what value, all points go back to the eq. pt.
Why do we want to work with a dimensionless model?
They’re easier to work with because they eliminate variables so that we can model the function.
What happens when f’(u*)<0? (Stability Theorem)
u* is a stable eq. pt if f is autonomous and u* is isolated
What happens when f’(u*)>0? (Stability Theorem)
u* is an unstable eq. pt if f is autonomous and u* is isolated
What happens when f’(u*)=0? (Stability Theorem)
No information. Compute the next derivative.
What is a bifurcation?
a sudden change in qualitative behavior of a system with a small change in behavior
What is a Creation-Annihilation Bifurcation?
one or more eq pts. is either created/destroyed with a small change in parameter.
What is a bifurcation diagram?
It plots u* versus h. Then, we draw a phase line through it and plug the values into f’(u*) to determine if the eq. pts. are stable, unstable, or semi-stable.
What do you need to look for in a bifurcation diagram?
Changes in sign/stability.
