Math 119 B, Midterm


ODEs and Dynamical Systems

1.

Construct a chaotic solution of the damped and driven Duffing's equation

$$\ddot x +\delta \dot x -4 x + x^3 = \epsilon \cos(\omega t),$$

(a)

Describe how you construct the sequence corresponding to this solution and what its properties are.

(b)

Describe what the corresponding Poincaré map is doing.

2.

Show that the solutions of the system

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

have global existence.

3.

Show that the origin is an asymptotically stable solution of the equation

$$\ddot x +\delta \dot x + x + x^3 = 0,$$

4.

What conclusions can you draw if the Melnikov function M(t_o)of the damped and driven nonlinear pendulum

$$\ddot x +\delta \dot x + \sin(x) = \epsilon \cos(\omega t),$$

has simple zeros and how does the resulting behaviour manifest itself in the flow?