1. Phase Diagrams
See for more details

2. Stability
A Dynamic System is said to be stable if the system approaches zero as t→∞.
What conditions are necessary for a system to be stable?
Hint: what kinds of values do we need for λ?
Recall:

3. Stability
A Dynamic System is said to be stable if the system approaches zero as t→∞.
What conditions are necessary for a system to be stable?
All eigenvalues must be negative.
For differential equations
If values are complex then the real portion must be negative

4. Steady State
A Dynamic system is said to be Unstable if the system “blows up” as t→∞.
What conditions are needed for unstability?
A Dynamic System is said to represent a steady state if the system neither approach zero nor infinity as t →∞.
What conditions are necessary for a steady state?

5. Stability What conditions are needed for unstability?
One or more positive eigenvalues
A Dynamic System is said to represent a steady state if the system approaches a constant value (other than zero) at t →∞.
eigenvalues all negative (real portion is negative in the case of complex numbers)

6. Difference equations vs. Differential equations
Stable if all │λ│<1
Steady state if one or more
λ=1 and all other
│λ│<1
Unstable or
“blows up” if one or more │λ│>1
Differential equations
Stable if all λ <0
Steady state if one or more λ = 0 and all other λ <0
Unstable or
“blows up” if one or more λ > 0

7. Example 1

8. Example 1
Plug in some points

9. Example 1
Plot the results
as vectors

10. Example 1
Follow some of the paths from different starting points (different initial conditions)
Notice that the path varies greatly depending upon the initial conditions
Usually just the paths
are drawn without the
vector field
One negative and one positive eigenvalue results in a saddle point

11. What do the eigenvalues tell you about a system?
What if the eigenvalues are both negative?
What if the eigenvalues are both positive?

12. Stability from a phase portrait
Positive distinct
eigenvalues
with eigenvector
as the asymptote
Negative distinct
eigenvalues with eigenvector as the asymptote

13. From your knowledge of complex numbers
From your knowledge of complex numbers. What do you think the phase portrait will look like if the eigenvalues are complex?
Hint consider
Complex Eigenvalues with Re > 0
Complex Eigenvalues with Re < 0

14. Example 2 Complex Eigenvalues with Re < 0 Complex Eigenvalues

15. Complex
Eigvenvalues
with Re > 0
Complex
Eigvenvalues
with Re < 0

16. What do you think the phase portrait will look like when the eigenvalues are purely imaginary

17. Predict the eigenvalues for this system
Eigenvalues are purely complex
This is because you end up with
eix in your answer
e ix = cosx + isinx
acosx + bsinx forms and ellipse
Center is stable

18. Describe the phase portrait
[ ]
x’t = x(t)

19. Describe the phase portrait
[ ]
x’t = x(t)
By inspection the eigen values are -1 and -3
Therefore the system is asymptotically stable the phase portrait will consist of curves that run towards the origin of the form at the right
The eigenvectors are <1,0> and <0,1> therefore the aysmptotes will be on the x and y axis

20. Homework: worksheet 8.5 1-9 all

21. What do you think the phase portrait will look like when there is one positive (a repeated eigenvalue) eigenvalue?
What do you think the phase portrait will look like when there is one negative eigenvalue?

22. Notice that the direction of the arrows affects stability
Single positive eigenvalue
The eigenvector is the asymptote
Single negative eigenvalue
Eigenvector is the asymptote

23. Learn more at: