# Set of all initial conditions such that the limit as t goes to infinity for a subset of solutions exists

Consider, for $$n \in \mathbb{N}$$ and $$n \geq 1$$, the 2 dimensional system: $$\ddot{x}+x^n=0 \quad or \quad \left\{ \begin{array}{c} \dot{x} = y \\ \dot{y} = -x^n\end{array} \right.$$ This system defines a flow $$\phi (t;x,y): \mathbb{R}\times\mathbb{R}^2 \rightarrow \mathbb{R}^2$$, parametrised by time t. Solutions of the system with initial condition $$(x_0,y_0)$$ are denoted by $$(x(t;x_0,y_0),y(t;x_0,y_0))$$, orbits of the flow are denoted by $$\Gamma (x_0, y_0)$$. Assume $$n$$ is even at this stage.

I need to define a set $$S_+ \subset \mathbb{R}^2$$ of all initial conditions such that the limit of $$t\rightarrow \infty$$ for solutions $$(x(t;x_0,y_0),y(t;x_0,y_0))$$ of the system with $$(x_0,y_0) \in S_+$$ exists, and to determine $$S_+$$ explicitly. Finally, I must do the same for the set $$S_-$$, this time for the limit of $$t\rightarrow -\infty$$.

I have no idea where to start with this particular question. Looking through my textbook (Differential Dynamical Systems by J.D. Meiss), I find no mention of these sets of initial conditions $$S_\pm$$, and thus am a little stuck and confused. Any guidance is extremely welcome!

If it helps, I have determined the Hamiltonian of this system to be $$H(x,y) = \frac{y^2}{2} + \frac{x^{n+1}}{n+1}$$

The trajectories are level curves of the Hamiltonian. If $$n$$ is odd, these level curves are closed curves, so the solutions are periodic: they don't have limits unless they are constant. If $$n$$ is even, both $$x$$ and $$y$$ go to $$\pm \infty$$ with a couple of exceptions...