Problem 3: Consider the logistic equation with periodic...

Problem 3: Consider the logistic equation with periodic coefficients $x' = r(t)x\left(1 - \frac{x}{K(t)}\right)$. Denote the common period of $r(t)$ and $K(t)$ by $T$. Show that the Poincaré map of this system is of the form $P(x_0) = \frac{Ax_0}{1 + Bx_0}$. Give the expressions for $A$ and $B$ in terms of $r$ and $K$. Find the fixed points of this Poincaré map. What can you say about stability?

Transcript

00:01 We are given three differential equations for x, y, prime, and z prime, and we have to find their fixed points.

00:08 Now the fixed points are given by the points which satisfy these conditions.

00:13 X prime equal to 0, y prime equal to 0, and z prime equal to 0.

00:18 So solving these three equations, we get two fixed points.

00:22 The first one has the coordinates, 0 ,0, 10, and the second one has the coordinates, the coordinates, 4 ,0, 10.

00:30 Coordinates 1 by 9 1 by 10 and 10 so we call the first front p1 and the first and the second one p2 now first let us consider fixed point p1 and analyze its stability now linearizing the equation about the fixed point we get delta x prime delta y prime delta z prime linearizing these equations about p1 gives us one minus one zero zero zero zero minus one delta x delta y delta z so basically we have taken this equation and expanded them as say zero plus zero plus delta x minus 10 delta x delta y this is for example for the delta x prime term and then kept up to linear order which just gives us delta x and similarly we did this for delta y prime and delta z prime now let us call this matrix m this is the stability matrix now if we find the eigenvalues of m these eigenvalues are given by 1 minus 1 and minus 1 now since there is a positive eigenvalue present here this fixed point is an unstable fixed point now let us consider p 2 now for p 2 we have to linearize this equation if for x y x prime y prime about this point 1 9 1 10th and and doing that, again, writing in this form, the stability matrix m this time is given by 0, minus 10 by 9, 0, 81 by 100, minus 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...

Question

Please give Ace some feedback