Showing papers on "Homotopy analysis method published in 2000"

A coupling method of a homotopy technique and a perturbation technique for non-linear problems

Abstract: In this paper, a coupling method of a homotopy technique and a perturbation technique is proposed to solve non-linear problems. In contrast to the traditional perturbation methods, the proposed method does not require a small parameter in the equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter p∈[0, 1] is constructed, and the imbedding parameter is considered as a “small parameter”. So the proposed method can take full advantage of the traditional perturbation methods. Some examples are given. The results reveal that the new method is very effective and simple.

Algorithm 801: POLSYS_PLP: a partitioned linear product homotopy code for solving polynomial systems of equations

Abstract: Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. POLSYS_PLP consists of Fortran 90 modules for finding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 90 derived data types to support a partitioned linear product (PLP) polynomial system structure. PLP structure is a generalization of m-homogeneous structure, whereby each component of the system can have a different m-homogeneous structure. The code requires a PLP structure as input, and although finding the optimal PLP structure is a difficult combinatorial problem, generally physical or engineering intuition about a problem yields a very good structure. POLSYS_PLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different PLP structures and their corresponding Bezout

Homotopy Classes for Stable Periodic and Chaotic¶Patterns in Fourth-Order Hamiltonian Systems

Abstract: We investigate periodic and chaotic solutions of Hamiltonian systems in ℝ4 which arise in the study of stationary solutions of a class of bistable evolution equations. Under very mild hypotheses, variational techniques are used to show that, in the presence of two saddle-focus equilibria, minimizing solutions respect the topology of the configuration plane punctured at these points. By considering curves in appropriate covering spaces of this doubly punctured plane, we prove that minimizers of every homotopy type exist and characterize their topological properties.

Constructive homotopy methods for finding all or multiple DC operating points of nonlinear circuits and systems

Abstract: A comprehensive analysis of homotopy methods on the computation of all DC operating points of nonlinear circuits and systems is conducted. Several sufficient conditions for the connectivity of all the solutions along a single homotopy path are derived. These conditions offer criteria to determine a starting point from which one can find all the solutions along one homotopy path. For the class of nonlinear circuits and systems in which all the solutions lie on several homotopy paths, a new systematic method to explicitly construct a starting point for each homotopy path is developed, From a practical viewpoint, the constructive method developed does not require the difficult task of finding a good initial guess and is applicable to general nonlinear circuits and systems. From a methodological viewpoint, the constructive method developed is applicable to general homotopy methods with different homotopy functions.

A new method to calculate multiple power flow solutions

Abstract: This paper presents a new method to calculate multiple power flow solutions. In using the concept of the homotopy continuation method and toroidal mapping, a general method to calculate multiple solutions of nonlinear equations is introduced. Using this method in the simple three nodes example, the decreasing of the number of solutions can be seen clearly as the load increases.

Convergent regions of Newton homotopy methods for nonlinear systems: theory and computational applications

Abstract: This paper introduces the concept of the convergent region of the Newton homotopy method. It is shown that convergent regions of the Newton homotopy method are equal to the stability regions for the Newton flow x/spl dot/=-adj(DF(x))F(x). A quite complete algebraic characterization of a convergent region and its boundary for a large class of nonlinear systems is derived and this characterization, which is explicit and computationally feasible, leads to the development of a numerical method to determine the convergent region and to the construction of simple criteria to avoid revisitations of the same solutions from different initial guesses. Two examples are given to illustrate the theoretical prediction.

An optimal reordering schema of homotopy equations for the analysis of nonlinear resistive circuits

Abstract: Homotopy-based simulation has been recently introduced in order to cope with the problem of finding the DC operating point of nonlinear resistive networks, when their equilibrium equations possess more than one solution. Although homotopy methods can overcome most of the convergence problems associated with the Newton-like methods, they still show several drawbacks, such as a slower speed of convergence. This paper introduces a method focussed on speeding up homotopy simulations. The method is based on an efficient reordering scheme of the nonlinear algebraic equations emanating from the circuit.

Electrostatic potential of strongly nonlinear composites: homotopy continuation approach

Abstract: The homotopy continuation method is used to solve the electrostatic boundary-value problems of strongly nonlinear composite media, which obey a current-field relation of J = σE + χ|E|2E. With the mode expansion, the approximate analytical solutions of electric potential in host and inclusion regions are obtained by solving a set of nonlinear ordinary different equations, which are derived from the original equations with homotopy method. As an example in dimension two, we apply the method to deal with a nonlinear cylindrical inclusion embedded in a host. Comparing the approximate analytical solution of the potential obtained by homotopy method with that of numerical method, we can obverse that the homotopy method is valid for solving boundary-value problems of weakly and strongly nonlinear media.

Effect of ordering on homotopy equations for calculating multiple DC operating points

Abstract: This paper presents a homotopy method focused on finding multiple solutions of the nonlinear algebraic equations (NAEs) emanating from nonlinear resistive circuits. This homotopy method exploits the differences of nonlinearity degrees among the equations in the system of NAEs, with the aim of speeding-up the process of determining the homotopic paths.