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Quadratic Portraits

Short-time solution curves for 25 random equations of the form $\left\{\begin{array}{l} \dot{x}=a_1x^2+b_1xy+c_1y^2 \\ \dot{y}=a_2x^2+b_2xy+c_2y^2 \end{array}\right.$ with random initial points.
Mathematica Sources: QuadraticPhasePortraits.nb, QuadraticPhasePortraits.pdf.