If you have d^{2}x/dt^{2}+f(x)(dx/dt) + g(x) = 0

and

1.f(x), g(x) are continuosly differentiable

2. g(x) is an odd function

3.f(x) is an even function

4. g(x) > 0 for all x > 0

5.

.

6. F(x) has exactly one positive root at some valuep, whereF(x) < 0 for 0 <x<pandF(x) > 0 and monotonic forx>p.

Then the system has a limit cycle surrounding the origin. This is Lienard's Theorem.

Is it true that if these conditions do not hold there is NO such limit cycle?

Is it true that if these conditions do not hold there is no limit cycle anywhere?