Linear System (Revisit)
We have seen system in the following form:
However, this it sometimes hard to define!!!!
Sometimes we only know the "Propensity" (More tuna, more sharks, too many tuna, not enough food, less tuna)
For these cases, we can use the continuous model!
Continuous Time Model v.s. Discrete Time Model
Discrete Time:
Knows the direct value of
Eigenvalues unstable, stable
Longterm behavior converges to dominate eigenvalue's eigenvector
Continuous Time
Only knows the change
Eigenvalues
Equilibrium
Most of the time, a continuous time system is written in the following form:
This is not a matrix!!! How can we analyze the stability?
Jacobian
J =
Partial Differential (Only differentiate one of the variable, treat others like constant)
For
Recall: Stable/ Unstable Equilibrium Points
For , which of the following points is stable?
(a)
(b)
(c)
For , it has slope(gradient) in 2 directions, what should be ????
Real Parts of Eigenvalues of the Jacobian <0 Stable
Steps of Analyzing Stability
Find the equilibrium points
Get the Jacobian (symbolic)
Calculate the Jacobian at equilibrium points (value)
Calculate the Jacobian's eigenvalues!
Unstable node | Line of equilibria | stable node | |
Unstable spiral | Center | stable spiral |
Why?
What is continuous time system?
The solution is
Continuous time system is written in the following way:
The solution is still:
(yes, the power can be a matrix)
What is the criterion that x(t) to be stable?
for , if , as times goes on, this will decay to 0 (e.g., )
For matrix cases, the term is about its "Eivenvalues' real parts <0
But this is a linear case!! What if the system is nonlinear?
Can we still find the eigenvalues of it?
Yes, but no.........
All we want is to write:
From:
In other words, we want to express linear combination of x and y
But we can try to linearize it:
If we have two points,
We know and
So what is ????
Although we know it is , this is not linear.
We can approximate as:
If is equilibrium point!
At equilibrium point,
we get:
Let's define a new coordinate:
, is a constant
, is a constant
We can now rewrite the above equation as:
or we can use a linear function to represent it:
Unstable node | Line of equilibria | stable node | |
Unstable spiral | Center | stable spiral |
Programming
How to get Jacobian??
Tangent Plane
Suppose we have a surface,
Any plane in 3D can be defined by the following function:
How to find the tangent plane that passes through the red point?
1. Define XZ-plane and YZ-plane at that point
2. On both planes, the projection of this tangent plane is a line
3. The slope of this line is the same as the slope of the projection of the surface.
Projection of on xz-plane at (2,1,13)
=>
Projection of on yz-plane at (2,1,13)
=>
Slope on xz plane =
Slope on yz plane =