Regarding Bode plots; $H(s)$ and $H(j\omega)$
Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{s_{\infty}t}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma&lt;0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.