We investigate the basin of attraction properties and its boundaries for chimera states in a circulant network of Henon maps. Chimera states, for which coherent and incoherent domains coexist, emerge as a consequence of the coexistence of basin of attractions for each state. It is known that the coexisting basins of attraction lead to a hysteretic behaviour in the diagrams for the density of incoherent and coherent states as a function of a varying parameter. Consequently, the distribution of chimera states can remain invariant by a parameter change, as well as it can suffer subtle changes when one of the basin ceases to exist. A similar phenomenon is observed when perturbations are applied in the initial conditions. By means of the uncertainty exponent, we characterise the basin boundaries between the coherent and chimera states, and between the incoherent and chimera states, and uncover fractal and riddled boundaries, respectively. This way, we show that the density of chimera states can be not only moderately sensitive but also highly sensitive to initial conditions. This chimeras dilemma is a consequence of the fractal and riddled nature of the basins boundaries.
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