Find transfer function from Bode plot
Background

The solution given in the book is wrong because the numerator in the given solution would make the notch frequency at \$\sqrt{1001}\$ and clearly it is at about 1 radian per second.

Answer


  how can I find the damping ratios ?


Pictorial analysis of the bode plot: -



I've drawn the red lines on to show what I consider to be flow of the frequency response should the peaks and nulls be subdued. This allows me to say that the resonant peak at 10 rad/s is about 12 dB and ditto at 100 rad/s.

Knowing that for a fairly undamped filter, Q (quality factor) is the peaking value as per this graph on this answer: -



You could use the more precise formula detailed lower down in that picture but I suspect assuming the peaking amplitude = Q is good enough.

So, we can say that Q is approximately 12 dB converted to a real number i.e. about 4. Because Q = 1/2\$\zeta\$, \$\zeta\$ = ~0.125. 

We can also fairly well say that with the three resonances at factors of ten difference there is little interaction to muddy the waters too much.