MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB AY 2012-2013

Systems of ODEs y x t t y X Geometric Analysis Time series in tx & ty-plane Phase Trajectory in a Phase Plane (xy-plane) y We will consider Autonomous Systems only. x t t y 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: Romeo’s change in feelings linearly depend only on Juliet’s current feelings (and vice-versa). When Romeo loves Juliet, Juliet tends to love Romeo more (and vice-versa). Here is a coupled system of DE:

Analysis using Vector Fields http://www.bae.ncsu.edu/people/faculty/seaboch/phase/newphase.html The 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. Just look at the signs of the derivative per region (including the boundary) R J or dJ/dt or dR/dt

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).

“Manual Analysis” J-axis R=0 R-axis J=0

Another example (nullclines) There are 4 regions. NOTE: x & y-axes are not anymore boundaries of the regions.

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

Another example (nullclines)

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:

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:

4th Romeo-Juliet Love Story “Love is blind”. Si Juliet ay pakipot na tapos makasarili pa! Kawawa naman si Romeo…

5th Romeo-Juliet Love Story “Cautious Lovers”

6th Romeo-Juliet Love Story “Romeo the Robot”

WHAT IF THERE’S A THIRD PARTY??? 

Not a Love Story but more of Conflict

Matatawag ba itong baliw? QUESTION Matatawag ba itong baliw?

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