Math 331, Fall 2002: Problems 17-20

17.

(expires 10/28)     Consider the differential equation $\dot{\mathbf {z}}(t) = \mathbf {F}(\mathbf {z}(t))$, where the vector $\mathbf {z}(t) =(x(t),y(t))$ and the field $\mathbf {F}(x,y) = (-y,x-y)$. Plot a few solutions. What happens to them when $t \to +\infty$? Give a ``Maple-proof'' that this is a general fact for every solution. [A ``Maple-proof'' is an argument that is rigorous once we accept Maple results as incontrovertibly true.]

18.

(expires 10/28)    (No Maple.) For the equation $\dot{\mathbf {z}} = \mathbf {F}(\mathbf {z})$, $\mathbf {z} =(x,y)$, with the vector field

$$\mathbf {F}(x,y) = \left\langle -x(x^4+y^4)-y \, , \, x-y(x^4+y^4) \right\rangle,$$

prove that the origin is an attractor in the future, i.e., every solution verifies

$$\lim_{t \to +\infty} \mathbf {z}(t) = 0.$$

[You can ask around how to do this, but then you have to show clearly that you have understood it.]

19.

(expires 10/28)     For the system of differential equations of prob. #23,

$$\left\{ \begin{array}{rcl} \dot{x} &\!=\!& x^2 + y, \\ \dot{y} &\!=\!& x (y^2 - 1), \end{array} \right.$$

find the eigenvalues and eigenvectors of the Jacobian at the fixed points. [This is a give-away if you have done #16.]

20.

(expires 10/28)     Consider the equations of the glider with no drag term ($R=0$). Use dsolve, type=numeric to solve them numerically with initial conditions $\theta(0)=0$, $v(0)=0.8$. Then solve exactly the linearized system around the fixed point $(\theta_0,v_0) = (0,1)$, with the same initial conditions. Graph the two functions for $0 \le t \le 5$, and give a good estimate of their maximum difference. What happens if we take a larger $t$-range?