MATH 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
AY 2012-2013
Systems of ODEs
We will consider
Autonomous
Systems only.
 dx
 dt  f1 ( x, y )

 dy  f ( x, y )
2
 dt
Geometric Analysis
1. Time series in tx & ty-plane
y
x
t
t
2. Phase Trajectory in a Phase
y
Plane (xy-plane)
X
Qualitative Analysis – Systems of ODEs
Equilibrium point/s: Solution/s to x’=0, y’=0
Stability Diagram (in a Phase Plane):
Qualitative Analysis – Systems of ODEs
Limit Cycles:
Modeling using Systems of ODEs
THE ROMEO AND JULIET LOVE STORY:
Let R(t) be the degree of love of Romeo to Juliet at
time t.
Let J(t) be the degree of love of Juliet to Romeo at
time t.
R>0 means Romeo loves Juliet
R<0 means Romeo hates Juliet
R=0 means Romeo is indifferent
Etc…
1st Romeo-Juliet Love Story
Assumptions:
1. Romeo’s change in
feelings linearly depend
only on Juliet’s current
feelings (and viceversa).
2. When Romeo loves
Juliet, Juliet tends to
love Romeo more (and
vice-versa).
Here is a coupled
system of DE:
 dR

aJ
 dt

dJ
  bR
 dt
a, b  0
Analysis using Vector Fields
http://www.bae.ncsu.edu/people/faculty/seaboch/ph
ase/newphase.html
The green end is forward in time, and red is
backwards.
Analysis using Nullclines
“Manual Analysis”: Let us analyze it using nullclines.
Just look at the signs of the
derivative per region
(including the boundary)
J
R
dJ/dt
dR/dt
or
or
For fun: after sketching the behavior of the solution,
let’s have some real-life interpretations.
Analysis using Nullclines
Boundary of the Regions:
dR
 aJ  0
dt
J 0
dJ
 bR  0
dt
R0
Note that the intersection/s of the boundaries is/are
the equilibrium point/s (since both derivatives=0).
“Manual Analysis”
 dR
 dt  aJ

 dJ  bR
 dt
a, b  0
J-axis
R=0
R-axis
J=0
Another example (nullclines)
dx

 dt  y  cos x
 dy

x

y

dt

There are 4
regions.
NOTE: x & y-axes are
not anymore boundaries
of the regions.
Another example (nullclines)
dx

 dt  y  cos x
 dy

x

y

dt

Hint in drawing
arrows:
Substitute extreme
values (e.g. since
y=cosx so -1<y<1,
but x=100,000 hence
Another example (nullclines)
dx

 dt  y  cos x
 dy

x

y

dt

2nd Romeo-Juliet Love Story
Assumption:
Romeo’s change in feelings
linearly depend only on
his current feelings
(same with Juliet).
They respond more to
their own emotions
than to each other’s
emotions.
“Self-centered lovers!”
Here is a decoupled
system of DE:
 dR

aR
 dt

dJ
  bJ
 dt
a, b  0
3rd Romeo-Juliet Love Story
Assumption:
Romeo’s change in feelings
linearly depend on
Juliet’s current feelings.
But Juliet tends to dislike
Romeo when Romeo is
loving her more. But she
tends to charm Romeo
when Romeo’s feeling is
downbeat.
“Responsive Romeo, Fickle
Juliet”
Here is a coupled
system of DE:
 dR

aJ
 dt

dJ
  bR
 dt
a, b  0
4th Romeo-Juliet Love Story
“Love is blind”.
Si Juliet ay pakipot na
tapos makasarili pa!
Kawawa naman si
Romeo…
 dR

aJ
 dt

dJ
  bR  cJ
 dt
a , b, c  0
5th Romeo-Juliet Love Story
“Cautious Lovers”
 dR


aR

bJ
 dt

dJ
  bR  aJ
 dt
a, b  0
6th Romeo-Juliet Love Story
“Romeo the Robot”
 dR

0
 dt

dJ
  aR  bJ
 dt
a, b  0
WHAT IF THERE’S
A THIRD
PARTY??? 
Not a Love Story but more of Conflict
 dR


aJ
 dt

dJ
  bR
 dt
a, b  0
QUESTION
Matatawag
ba itong
baliw?
 dR


aR
 dt

dJ
  bJ
 dt
a, b  0
QUESTION
Hindi! But more on ayaw nilang magkaroon ng emotion!
(similar to cautious lovers)