TITLE: Noninvariance for a specific nonlinear oscillator
QUESTION [5 upvotes]: Consider the nonlinear system
\begin{align*}
  \frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix}  = \begin{pmatrix} x_2(t) \\ -4x_1(t) + x_1^2(t) \end{pmatrix},
\end{align*}
which admits periodic orbits near the origin. Consider two initial states $x(0) = x_0$ and $z(0)=z_0$ where $x_0$ and $z_0$ are nonzero and satisfy $x_2(0) = - z_2(0)$. We can think of the situation as putting two particles in the vector field and watch them evolve.
I am trying to show that for any combination of initial states satisfying the above requirements, at some point in time it must happend that $x_2(t) + z_2(t) \ne 0$. This is due to the specific nonlinearity of the periodic orbits. But it seems to be very difficult to show by just looking at the equations
\begin{align*}
  x_2(t) + z_2(t) \equiv 0, \frac{d}{dt} (x_2(t) + z_2(t)) \equiv 0, \dots
\end{align*}
So I was hoping that there might be a more succinct argument involving the geometry of the periodic orbits. Any hints in the direction of a simpler argument are very welcome.

REPLY [2 votes]: As we're interested in two particles, it seems natural to introduce the coordinates $y = x + z$ and $w = x - z$. In these coordinates, we obtain the system
\begin{align}
\dot{y}_1 &= y_2,\\
\dot{y}_2 &= -4 y_1+\frac{1}{2}y_1^2+\frac{1}{2}w_1^2,\\
\dot{w}_1 &= w_2,\\
\dot{w}_2 &= -4 w_1 + w_1 y_1.
\end{align}
If I understand you correctly, the question is if, given an initial condition of the form $(y_1,y_2,w_1,w_2) = (a,0,c,d)$, there will be a time $t$ when $y_2(t) \neq 0$. The negation of this statement is equivalent with stating that $y_2(t) = 0$ for all time $t$, i.e. that the set $(a,0,c,d)$ is invariant under the above flow. However, for such an initial condition, we see that $\dot{y}_2(0) = -4a+\frac{1}{2}a^2 + \frac{1}{2}c^2 \neq 0$ for general $a,c$. Hence, the vector field at $(a,0,c,d)$ is not parallel to $(a,0,c,d)$, so the set $(a,0,c,d)$ is not invariant under the flow. Hence, there exists a time $t>0$ such that $y_2(t) \neq 0$, even when $y_2(0) = 0$.