Chapter 7.4, Problem 1E

Interpretation:

To show equation x¨ + μ(x2-1)x˙ + tanhx =0 has exactly one solution for μ>0 and classify its stability.

Concept Introduction:

Equation of the form x¨ +f(x)x˙+g(x)=0 is known as Lienard’s equation. This equation is equivalent to the system

x˙=y

y˙ = -g(x) - f(x)y

Lienard’s Theorem: If f(x) and g(x) satisfy the following conditions

f(x) and g(x) are continuously differentiable for all x;

g(- x)= - g(x) for all x; i.e., g(x) is an odd function;

g(x)>0 for x>0;

f(-x)= f(x) for all x; i.e., f(x) is an even function;

The odd function F(x)=∫0xf(u)du has exactly one positive zero at x=0, is negative for 0 < x < a, is positive and nondecreasing for x >a, and F(x)→∞ as x→∞

Then the system

x˙=y

y˙ = -g(x) - f(x)y has a unique, stable limit cycle surrounding the origin in the phase plane.