Classifying critical points (using det(A) and tr(A) (, ), using…
Classifying critical points
using det(A) and tr(A)
means that there are no straight line solutions
and therefore no asymptotes. so the trajectories will go round and round the CP
Has real part
Sign of real part indicates stability
the name spells the game
concentric ellipses around the cp
Sign indicates stability
-> UNSTABLE bc from any trajectory, the system goes "away" from the CP (bc exp growth)
-> STABLE since regardless of initial position (which trajectory) it always eventually goes "into" the CP
Types of phase portrait
evals are equal
relationship between y1 and y2
for any c is constant.
so there are only straight line solutions.
has one eigenvector
=>one straight line sol
evals are different
hey am approaching the cp
oh oops im going off into infinity again
6 types of phase portraits
Procedure of drawing phase portraits
(1) Find eigenvalues, eigenvectors, general soln and .'. characterise the critical points
(2) if eigenvalues are complex, then...
(2) If eigenvalues are real, then there exist straightline solns. Sketch these straightline solns by considering when c1=0 and then when c2=0.
(3) Let f(
) = the column vector that comes from applying A to
(4) Define gradient field expression: z'/y' = blah
(5) Calc nullclines i.e., when gradient field = 0 or undefined i.e., when z'=0 (horizontal tangent lines) and y'=0 (vertical tangent lines)
(6) To add directions to the horizontal and vertical lines of then nullclines, see what end z approaches as y approaches pos infinity.
(7) If need more lines to draw trajectories, then consider y=z. Draw the line and to find the gradient of the tangents coming off of it, sub y=z into gradient field eqn.
To find critical points (y
), solving for (y',z') = (0,0)