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We explicitly derive the proper time $(\tau)$ principal Lyapunov exponent($\lambda_{p}$) and coordinate time ($t$) principal Lyapunov exponent($\lambda_{c}$) for Reissner Nordstr{\o}m (RN) black hole (BH) . We alsocompute their ratio. For RN space-time, it is shown that the ratio is$\frac{\lambda_{p}}{\lambda_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ fortime-like circular geodesics and for Schwarzschild BH it is$\frac{\lambda_{p}}{\lambda_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. Wefurther show that their ratio $\frac{\lambda_{p}}{\lambda_{c}}$ may vary fromorbit to orbit. For instance, Schwarzschild BH at innermost stable circularorbit(ISCO), the ratio is$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginallybound circular orbit (MBCO) the ratio is calculated to be$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RNBH the ratio at ISCO is$\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$.We also further analyse the geodesic stability via this exponent. By evaluatingthe Lyapunov exponent, it is shown that in the eikonal limit , the real andimaginary parts of the quasi-normal modes of RN BH is given by the frequencyand instability time scale of the unstable null circular geodesics.

Stability analysis and quasinormal modes of Reissner Nordstr{\o}m Space-time via Lyapunov exponent