Describe how their relationship evolves with time.
3. Consider the following system describing the motion of a particle in a potential field
ẋ = y
y˙ = x2 − 1
a. Find all fixed points and classify them according to
their linear stability. Compute eigenvectors for any
b. Find V(x) such that the quantity
E(x, y) = 2 y V (x) is conserved along
1 2 +
c. Sketch the phase portrait.
d. Are there any periodic orbits? If yes, for what
values of E?
e. Is there a homoclinic orbit? If yes, for what value of
4. Show that the system
ẋ = y − x3
y˙ = − x − y3
Has no periodic orbits. Justify your argument. (Hint: Use a Liapunov Function
L(x, y) = ax2 + by2 with suitable a, b
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