Stability on oscillatory system

• 1D spring system: \(F(p^k,v^k,t^k)=-K\,p^{k}\)
• \(p^{k+1}= 2\, p^k - p^{k-1} - h^2\,K/m\,p^k\)
• \(\Rightarrow p^{k+1}= (2-h^2\,K/m)\, p^k - p^{k-1}\)

• Stable and permanent oscillation when \(h<\frac{2}{\sqrt{K/m}}=\frac{2}{\omega}\)

• Q. How can we demonstrate it ?

• Note: \(h<2/\omega\) only valid for one 1D spring. Coupled 3D springs may require smaller value of h.