Solve Bifurcation Diagrams: Find Critical Values of r

[glebovg]

Homework Statement

For each of the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of r.

ẋ = rx + cosh(x)

ẋ = x(r - sinh(x))

ẋ = rx - xe^-x2

The attempt at a solution

Fixed points satisfy

f'(x) = r + sinh(x) = 0 ⇒ x_* = arcsinh(-r) = -arcsinh(r).

f'(x) = r - sinh(x) - xcosh(x) = 0.

f'(x) = r + 2xe^-x2 - e^-x2 = 0.

 

[glebovg]

Fixed points satisfy

ẋ = rx + cosh(x) = 0

ẋ = x(r - sinh(x)) = 0 ⇒ x_* = 0, x_* = arcsinh(r)

ẋ = rx - xe^-x2 = 0 ⇒ x_* = 0, x_* = ±√[-ln(r)].