hopf bifurcation for an ode

Question

I understand how to analyse a system of equations like

$x'(t) = f(x,y)$

$y'(t) = g(x,y)$

set $x'$ and $y'$ to zero and find the fixed points etc, and find the stability.

What Im am not sure of is analyzing an equation of the form

$x'''(t) + x''(t) + x'(t) + a sin(x) = 0$

I want to find the stability and hopf bifurcation parameter $a_h$ for the above system.

Answer 1

You make this differential equation into a system (in this case three-dimensional) by defining new dependent variables to correspond to $x'$ and $x''$. $$ \eqalign{x' &= v\cr v' &= w\cr w' &= - w - v - a \sin(x)\cr}$$ The linearization at the equilibrium point $(0,0,0)$ corresponds to the matrix $$ A = \pmatrix{0 & 1 & 0\cr 0 & 0 & 1\cr -a & -1 & -1\cr }$$ The characteristic polynomial is $P(\lambda) = \lambda^3 + \lambda^2 + \lambda + a$.