(142f) Locating Bifurcation Points for Seeking All Real Roots to System of Nonlinear Equations with a New Homotopy
Abstract A new homotopy method, Fixed-Point Newton (FPN) homotopy, is presented for seeking all real solutions to nonlinear equations. This homotopy is a combination of the fixed-point and the Newton homotopies. The original system of equations is first changed to the new system of equations by multiplying by an (x-x0) term. When x0 is selected as the starting point and the FPN homotopy is applied, bifurcation points are generated. It is mathematically proven that one of the unknown variables of the bifurcation points equals the unknown variable of the initial point. The other unknown variables of bifurcation points for the new system of equations are obtained by solving the (n-1) nonlinear equations. The Bifurcation points of this system are obtained from solving the (n-2) nonlinear equations. The other unknown variables are 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 are equal to their unknown variables at the initial point. If the last equation is solved using the FPN homotopy, the coordinates of the bifurcation points of the system of two nonlinear equations are obtained, so we can get all roots of this system of equations by switching to the other branches from these bifurcation points. If we continue this procedure, finally the coordinates of the bifurcation points of a system of n equations will be found. After finding all bifurcation points, it is necessary to switch to other branches that are bifurcated from these points to seek all solutions to the new system of equations. This method has been applied to five single readily obtained. Besides solving these problems, this method has been used to find all solutions of two stability problems.
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