For a second order system with denominator \(s^{2}+2 \delta w_{n} s+w_{n}^{2} ; w_{n}^{2}>0\) the roots are complex conjugates when

Concept:

The standard second-order system has the characteristic equation:

\( s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \)

The roots of this equation determine the nature of the system (overdamped, underdamped, critically damped).

Condition for Complex Conjugate Roots:

Roots are complex conjugates if the discriminant is negative:

\( (2\zeta\omega_n)^2 - 4\omega_n^2 < 0 \Rightarrow \zeta^2 < 1 \)

So, complex roots exist when:

\( 0 \leq \zeta < 1 \)

Such systems are called underdamped systems.

Therefore,  the correct answer is   \( 0 \leq \zeta < 1 \)