We demonstrate novel features in the behavior of the second- and third-order nonlinearity parameters of the curvature perturbation, namely $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$, arising from nonlinear motion of the curvaton field. We investigate two classes of potentials for the curvaton—the first has tiny oscillations superimposed upon the quadratic potential. The second is characterized by a single “feature” separating two quadratic regimes with different mass scales. The feature may either be a bump or a flattening of the potential. In the case of the oscillatory potential, we find that, as the width and height of superimposed oscillations increase, both $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ deviate strongly from their expected values from a quadratic potential. $f_{\mathrm{NL}}$ changes sign from positive to negative as the oscillations in the potential become more prominent. Hence, this model can be severely constrained by convincing evidence from observations that $f_{\mathrm{NL}}$ is positive. $g_{\mathrm{NL}}$, on the other hand, acquires very large negative values. Further, this model can give rise to a large running of $f_{\mathrm{NL}}$, with respect to scale. For the single-feature potential, we find that $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$ exhibit oscillatory behavior as a function of the parameter that controls the feature. The running of $f_{\mathrm{NL}}$ with respect to scale is found to be small.

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