anomalous region very successfully though the relative displacement of s ∗ $_{im}$( ρ ) curves for different isotherms is too small. The extrema in s ∗ im define a region of anomalous entropy behaviour in the density-temperature plane that is almost identical as the region within which ( ∂S$_{2}$/∂ρ )$_{T}$ > 0. 

The overall and somewhat unexpected outcome of our instantaneous normal mode anal- ysis of a core-softened water-like fluid is that the real and imaginary frequency branches show very different sensitivities to the dynamical consequences of the interplay between two length scales in the anomalous regime of the liquid. Moreover, the entropy contribution from the imaginary frequency modes of the INM spectrum reflects the anomalous behaviour of the excess entropy and diffusivity characteristic of water-like fluids, but the real frequency branch does not. 

## Acknowledgments 

This work is is supported by the Indo-Brazil Cooperation Program in Science and Tech- nology of the CNPq, Brazil and DST, India.This work is also partially supported by CNPq, INCT-FCx. The authors would like to thank Murari Singh for help with preparing some of the figures. 

∗ Electronic address: oliveira@iceb.ufop.br 

† Electronic address: esalcedo@if.ufrgs.br 

‡ Electronic address: charus@chemistry.iitd.ernet.in 

§ Electronic address: marcia.barbosa@ufrgs.br 

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