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All Real Roots of a Nonlinear System of Equations

Consider the system of nonlinear algebraic equations: and , where you can vary and .
This Demonstration finds all roots of this system in the domain using Seader's approach [1] and the arc length continuation technique.
The problem considered is described as follows: (the function was first proposed in [1]), , and (i.e., the auxiliary equation). Using the built-in Mathematica function WhenEvent all roots of the system are readily obtained when . A list of all roots is provided.

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Reference
[1] S. K. Rahimian, F. Jalali, J. D. Seader, and R. E. White, "A New Homotopy for Seeking all Real Roots of a Nonlinear Equation," Computers and Chemical Engineering, 35(3), 2011 pp. 403–411. doi:10.1016/j.compchemeng.2010.04.007.
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