Bifurcation parameter

Question

in these following images there is value of gamma=1.00786 on which fixed point change its behaviour. i am not able to calculate this value. Please help how i can caluclate this value . Thanks in advance.

Answer 1

Consider a mono-dimensional system like this:

$$\dot{x} = f(x, \theta),$$

where $x$ is the state variable and $\theta$ is a parameter.

1. Find the steady states $x^* : f(x^*, \theta) =0$. Find also their field of existence with respect to the parameter $\theta$ (i.e. if $x^* = \sqrt{\theta}$, then it is real for $\theta \geq 0$).
2. Evaluate the derivative of $f$ in each $x^*$.
3. Study the sign of these derivatives with respect to $\theta$. Recall that a negative derivative means that the steady state $x^*$ is stable, while positive derivative means instability. For nonlinear systems, $f' = 0$ is not conclusive for the stability.

In general, a bifurcation occurs at $\theta = \bar{\theta}$ if:

1. A couple of steady states born or die, and/or
2. At least one steady state changes its stability (i.e. it passes from stable to unstable, or viceversa)