MATH 175: Numerical Analysis II

Lecturer: Jomar Fajardo RabajanteIMSP, UPLB

AY 2012-2013

Systems of ODEsGeometric Analysis1. Time series in tx & ty-plane

2. Phase Trajectory in a Phase Plane (xy-plane)

t

x

t

y

),(

),(

2

1

yxfdtdy

yxfdtdx

X

y

We will consider Autonomous Systems only.

Qualitative Analysis – Systems of ODEsEquilibrium point/s: Solution/s to x’=0, y’=0Stability Diagram (in a Phase Plane):

Limit Cycles:Qualitative Analysis – Systems of ODEs

Modeling using Systems of ODEsTHE 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 JulietR<0 means Romeo hates JulietR=0 means Romeo is indifferentEtc…

1st Romeo-Juliet Love Story

Assumptions: 1. Romeo’s change in

feelings linearly depend only on Juliet’s current feelings (and vice-versa).

2. When Romeo loves Juliet, Juliet tends to love Romeo more (and vice-versa). 0,

ba

bRdtdJ

aJdtdR

Here is a coupled system of DE:

Analysis using Vector Fields

http://www.bae.ncsu.edu/people/faculty/seaboch/phase/newphase.htmlThe green end is forward in time, and red is

backwards.

Analysis using Nullclines

“Manual Analysis”: Let us analyze it using nullclines.

For fun: after sketching the behavior of the solution, let’s have some real-life interpretations.

dJ/dt or

ordR/dt

R

J Just look at the signs of the derivative per region (including the boundary)

Analysis using Nullclines

Boundary of the Regions:

Note that the intersection/s of the boundaries is/are the equilibrium point/s (since both derivatives=0).

0

0

J

aJdtdR

0

0

R

bRdtdJ

“Manual Analysis”

0,

ba

bRdtdJ

aJdtdR

R-axis

J-axis

J=0

R=0

Another example (nullclines)

yx

xy

dtdy

dtdx cos

NOTE: x & y-axes are not anymore boundaries of the regions.

There are 4 regions.

Another example (nullclines)

yx

xy

dtdy

dtdx cos

Hint in drawing arrows:Substitute extreme values (e.g. since y=cosx so -1<y<1, but x=100,000 hence

Another example (nullclines)

yx

xy

dtdy

dtdx cos

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!” 0,

ba

bJdtdJ

aRdtdR

Here is a decoupled system of DE:

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” 0,

ba

bRdtdJ

aJdtdR

Here is a coupled system of DE:

4th Romeo-Juliet Love Story

“Love is blind”. Si Juliet ay pakipot na

tapos makasarili pa!Kawawa naman si

Romeo…

0,,

cba

cJbRdtdJ

aJdtdR

5th Romeo-Juliet Love Story

“Cautious Lovers”

0,

ba

aJbRdtdJ

bJaRdtdR

6th Romeo-Juliet Love Story

“Romeo the Robot”

0,

0

ba

bJaRdtdJdtdR

WHAT IF THERE’S A THIRD PARTY???

Not a Love Story but more of Conflict

0,

ba

bRdtdJ

aJdtdR

QUESTION

0,

ba

bJdtdJ

aRdtdRMatatawag

ba itong baliw?

QUESTIONHindi! But more on ayaw nilang magkaroon ng emotion! (similar to cautious lovers)