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The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables $x$ and $y$. The integration time step is $h$.

$\dfrac{x_{k+1}-x_{k}}{h}=y_{k}$

$\dfrac{y_{k+1}-y_{k}}{h}=-x_{k}$

For this discrete-time system, which one of the following statements is TRUE?

  1. The system is not stable for $h > 0 \\$
  2. The system is stable for $h > \dfrac{1}{\pi} \\$
  3. The system is stable for $0 < h <\dfrac{1}{2\pi} \\$
  4. The system is stable for $\dfrac{1}{2\pi}<h<\dfrac{1}{\pi}$
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