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5.4 Advanced Techniques 307
3. Show that the system
x ′ = x + x 3 − 2y,
y ′ = y 5 − 3x,
has no periodic solutions.
4. Show that periodic orbits, or cycles, for the dynamical system
x ′ = y,
y ′ = −ky − V ′ (x)
arepossibleonlyifk =0.
5. Consider the system
x ′ = x(P − ax + by),
y ′ = y(Q − cy + dx),
where a, c > 0. Show that there cannot be periodic orbits in the first
quadrant of the xy plane. Hint: Take β =(xy) −1 .
6. Assume f and g are continuously differentiable everywhere and suppose the
nonlinear system x ′ = f(x, t), y ′ = g(x, t) has a cycle with a single critical
point inside. One can prove the critical point cannot be a saddle point.
Why do you think this true? Convince yourself by drawing a diagram.
7. Show that the orbits of the system
x ′ = x(x 3 − 2y 3 ),
y ′ = y(2x 3 − y 3 ),
are given by x 3 + y 3 = 3Cxy, where C is an arbitrary constant, and
sketch several orbits in the phase plane. The curves are called the folia
of Descartes.
8. In Example 5.13 solve the radial equations for r and θ exactly to find the
orbits. Consider each case.
9. A system
x ′ = f(x, y)
y ′ = g(x, y),
is called a Hamiltonian system if there is a function H(x, y) for which
f = H y and g = −H x . The function H is called the Hamiltonian. Prove
the following facts about Hamiltonian systems.
a) If f x + g y = 0, then the system is Hamiltonian. (Recall that f x + g y is
the divergence of the vector field (f,g).)