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 the i equations by (x_i -x_1~0), where x~0 is selected as the starting point. The FPN homotopy is applied, and bifurcation points are generated. It is mathematically proved that one of the unknown variables of the bifurcation points equals the unknown variable of the initial point (e.g. x 1 = x_1~0). The other unknown variables of the bifurcation points are obtained by solving (n-1) nonlinear equations. The Bifurcation points of this system result from solving the (n-2) nonlinear equations (e.g. x1 = x_1~0, x2 = x_2~0). The other unknown variables equal to their unknown variables at the initial point. If this procedure is repeated, only one equation with one unknown variable is generated. The other unknown variables equal to the unknown variables of the initial point (e.g. x 1 =x_1~0,x2 =x_2~0,---,x_n = x_n~0). If the last equation is solved using the FPN homotopy, the coordinates of the bifurcation points of the system of 2 nonlinear equations are obtained. 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 =x_2~ 0 or x3 = x_3~0 or…or x_n = x_n~0) 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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