For 1D harmonic oscillator, the sphere is 2D, and called a circle. It's rotating through time. 1D space + 1D time.
If you understood what you were trying to push now you'd recognize that adding in your time dimension to force an argument involving a circle is just a matter of the sign in the 2nd order differential equation. This also falls apart if that sign changes and you get hyperbolic solutions. and I'm sure you'll just say that's just another kind of sphere with negative curvature