Consider the scalar dynamical system given by
$$ \dot{x} = -x^3 + \cos(-x) \,. $$
I would like to analyse stability of this system using Lyapunov's direct method. For this I first have to transform the system to the origin such that $\dot{x} = 0$ for the equilibrium $x^{*}.$
However,
$$ -x^3 + \cos(-x) = 0 $$
doesn't seem to have an anlytical solution? Wolfram Alpha as well just gives a numerical approximation of around $x^{*} \approx 0.865.$
How to proceed from here? Is a transformation to the origin not possible? Or should one somehow use the approximate value?
If the aim is to study the stability of the equilibrium point $x^*$ one can proceed with a qualitative study of the differential equation $x'=f(x)$ with $f(x)=-x^3+\cos(x)$.
We already know that there is only one equilibrium point, that is $x^*>0$. Moreover, since $f\in C^1$, the equation verifies local existence and uniqueness of a solution for every choiche of initial values $x(t_0)=x_0$. Also we have: $f(x)>0$ if and only if $x<x^*$, and $f(x)<0$ if and only if $x>x^*$. Hence for every choice of initial conditions $(t_0,x_0)$ with $x_0<x^*$, the solution will be increasing tending to $x^*$ as $t$ increases; while for every choice of initial conditions $(t_0,x_0)$ with $x_0>x^*$, the solution will be decreasing tending to $x^*$ as $t$ increases. We conclude that $x^*$ is an asymptotically stable equilibrium point.