Math 119 A, Final


ODEs and Dynamical Systems

1.

Consider the system

$$\begin{eqnarray*}\dot x &=& x\\ \dot y &=& y, \end{eqnarray*}$$

(a)

Find an integral (energy) of the motion.

2.

Determine the stationary solutions of the system

$$\begin{eqnarray*}\dot x &=& x^2-y^3\\ \dot y &=& 2x(x^2-y), \end{eqnarray*}$$

(a)

Determine the stability of the stationary solutions.

(b)

Do periodic solutions exist?

3.

Do the qualitative analysis of the Duffing's equation,

$$\ddot x -4 x + x^3 = 0,$$

(a)

Find the stationary solutions.

(b)

Determine the stability of the stationary solutions.

(c)

Draw the phase portrait of the Duffings equation.

(d)

Use the phase portrait and (a) and (b) to identify four different types of solutions of the Duffing's equation and describe their qualitative behaviour.