S. Khaleghi, F. Jalali (Iran), and J.D. Seader (USA)
Bifurcation, Homotopy, System of nonlinear equations.
A new homotopy method, referred to here as the Fixed Point Newton (FPN) homotopy, is presented for seeking all real solutions to a system of nonlinear algebraic and/or transcendental equations. This homotopy is a linear combination of the fixed-point and the Newton homotopies. Before forming the new homotopy, the original system of equations is changed to a new system of equations by multiplying each of equations (xi - xi0), where x0 is selected as the starting point. The FPN homotopy is applied, and bifurcation points are generated. All the roots of the system of equations are obtained by switching to the other branches from these bifurcation points. If we continue this procedure, finally the coordinates of bifurcation points of the system of n equations will be found. The other bifurcation points (e.g. at x2 = x20 or x3 = x30 or...or xn = xn0) are found in the same way. After finding all bifurcation points, it is necessary to switch to other branches which are bifurcated from these points to seek all solutions to the new system of equations.
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