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In this paper, we determine and specify the type of curves formed by thevanishing discriminant of some specified family of complex-valued harmonicpolynomials with two parameters. We also classify the region formed by curvesas bounded and unbounded connected components which in turn used to count thezeros of the complex-valued harmonic polynomials. Our study is a computer-aidedcase and the result shows that the curves formed are a teacup curve, aparabolic curve and a swallowtail catastrophe curve. These curves come uptogether to form a butterfly catastrophe.

Curves formed by Vanishing Discriminant and Roots of Complex-valued Harmonic Polynomials (Computer-Aided Case Study)