I want to study the stability of the equilibrium points of the following ODE in $n=2$:
$x'(t) = y(t)$
$y'(t) = −kx(t) + cx^2(t)$
Where $k,c\geq0$
From the system:
$y = 0$
$−kx + cx^2 = 0$
we find that equilibrium points are $P_1 = (0, 0)$ e $P_2 = (k/c,0)$. Evaluating the jacobian of the equations at $P_1$ we get that it has two eigenvalues with real part equal to $0$. My professor claims that $P_1$ is stable but not asymptotically, but the Hartmann-Grossman theorem says the following:
Consider the nonlinear autonomous ODE in dimension $n$ and assume that $f ∈C^1(X;R)$. Let $\bar x ∈ {\rm Int}X$ be an equilibrium point such that $\max\{Reλ, λ ∈ σ(D f(x)\} \neq 0$. Then :
• If $\max\{Reλ, λ ∈ σ(D f(x))\}> 0$ then $\bar x$ is unstable
• If $\max\{Reλ, λ ∈ σ(D f(x))\}< 0$ then $\bar x$ is stable and AS.
So it looks to me that we can not say anything a-priori looking only at the eigenvalues since the real part is zero.
Can you help me?
Your system is conservative, all trajectories lie on level curves of the first integral/energy function $$E(x,y)=\frac12y^2+\frac{k}2x^2-\frac{c}3x^2,$$ which means that you do not get source or sink equilibrium points, but only centers and saddle points. With $f'(x,y)=\pmatrix{0&1\\-k+2cx&0}$ you get a center at $(x,y)=(0,0)$ and a saddle point at $(x,y)=(\frac kc,0)$.