Day 2 Differential Equations
See http://mathworld.wolfram.com/PhasePortrait.html for more details
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:
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.
If values are complex then the real portion
must be negative.
Steady State
A Dynamic system is said to be Unstable if the
system “blows up” as t → ∞.
What conditions will cause 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?
Stability
Unstable:
What conditions will cause unstability?
One or more positive eigenvalues.
Steady State:
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).
Difference equations vs.
Differential equations
Difference equations
- Stable if all │λ│< 1
- Steady state if one or
more λ=1 and all
other
│λ│< 1
- “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
- “blows up” if one or
more λ > 0
Phase Portrait
Phase Plane Example 1
http://www.youtube.com/watch?v=BU5yTrJ_jFA
Phase Portrait - Example 1
Phase Portrait - Example 1
Plug in some points
Phase Portrait - Example 1
Plot the results as vectors
Phase Portrait - 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.
What do the eigenvalues tell you
about a system?
What if the eigenvalues are both negative?
What if the eigenvalues are both positive?
Stability from a phase portrait
Positive distinct
eigenvalues
with eigenvector
as the asymptote
Negative distinct
eigenvalues with eigenvector as
the asymptote
Complex Eigenvalue (a + bi)
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
Complex Eigenvalue - Example 2
Complex
Eigenvalues
with Re < 0
Complex
Eigenvalues
with Re > 0
Complex Eigenvalue (a + bi)
Complex
Eigvenvalues
with Re > 0
Complex
Eigvenvalues
with Re < 0
Complex Eigenvalues (bi)
What do you think the phase portrait will
look like when the eigenvalues are purely
imaginary
Predict the eigenvalues for this
system (bi)
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
Phase Diagram - Summary
•
•
•
•
Draw in the eigenvectors.
+ve eigenvalue arrows away from the origin.
-ve eigenvalue arrows toward the origin.
eigenvalues are real and -ve the system will asymptotically
approach the origin along the eigenvectors, arrows inwards
(see slide #13 right diagram).
• eigenvalues are real and +ve, arrows outwards from origin
(see slide #13 left diagram).
• Complex eigenvalues (see slide #16):
Re portion is –ve, spiral towards origin.
Re portion is +ve spiral outwards from origin towards infinity.
Describe the phase portrait
x’t =
[ ]
-1 0
0 -3
x(t)
Describe the phase portrait
x’t =
[ ]
-1 0
0 -3
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 é
1 ù, é 0 ù
ê
ú ê
ú
ë 0 û ë 1 û
therefore the aysmptotes
will be on the x and y axis.
Homework: worksheet 8.5 1-9 all
More Info
http://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx
http://ocw.mit.edu/courses/mathematics/18-03scdifferential-equations-fall-2011/unit-iv-first-ordersystems/qualitative-behavior-phase-portraits/
http://mathlets.org/mathlets/linear-phase-portraits-matrixentry/comment-page-1/
http://www.bluebit.gr/matrix-calculator/calculate.aspx
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?
Notice that the direction of the
arrows affects stability
Single postive eigenvalue
The eigenvector is the
asymptote
Single negative eigenvalue
Eigenvector is the asymptote