Kernel: SageMath (stable)
Video link https://youtu.be/CApKimXcvJk
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 |
Example
In [3]:
In [4]:
Estimate slope of from
In [38]: