A new method is offered here for global analysis of nonlinear dynamical systems. It is based upon the idea of constructing the associated cell-to-cell mappings for dynamical systems governed by point mappings or governed by ordinary differential equations. The method uses an algorithm which allows us to determine in a very effective manner the equilibrium states, periodic motions and their domains of attraction when they are asymptotically stable. The theoretic base and the detail of the method are discussed in the paper and the great potential of the method is demonstrated by several examples of application.

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Based on dynamical systems theory, a computational method is proposed to locate all the roots of a nonlinear vector function. The computational approach utilizes the cell-mapping method. This method relies on discretization of the state space and is a convenient and powerful numerical tool for analyzing the global behavior of nonlinear systems. Our study shows that it is efficient and effective for determining roots because it minimizes and simplifies computations of system trajectories. Since the roots are asymptotically stable equilibrium points of the autonomous dynamical system, it also provides the domains of attraction associated with each root. Other numerical techniques based on iterative and nomotopic methods can make use of these domains to choose appropriate initial guesses. Singular manifolds play an important role in limiting the extent of these domains of attraction. Both a theoretical basis and a computational algorithm for locating the singular manifolds are also provided. They make use of similar state-space discretization frameworks. Examples are given to illustrate the computational approaches. It is demonstrated that for one of the examples (a mechanical system), the method yields many more solutions than those previously reported. The above equation contains variables x¡, i = 1,2 N, which are nonlin-early coupled. The problem of determination of the roots or zeros of a system of nonlinear algebraic equations is often encountered in science and engineering, especially when dealing with nonlinear phenomena. The same problem is posed when looking for periodic solutions of nonlinear differential equations and fixed points of maps. One notes that, in general, Equation (1) may possess more than one root, and two questions often arise when dealing with it. The first question is concerned with how to obtain just any one solution of (1). The second question refers to locating all the solutions. This paper primarily focuses on the second question, which has significant bearing on the first question. A theory for locating all the zeros of (1) has been proposed by Zufiria and Guttalu [30], and is based on a dynamical system approach. Computational aspects of finding all zeros will be treated in this paper by making use of the theory developed there. There exists an extensive literature dealing with both theoretical and computational aspects of obtaining a solution to (1). We give only a brief account of it here. Iterative techniques, usually based on the contractionmapping theorem (see Marsden [23]), are a notable class of such methods. The literature is vast on the analysis of local convergence properties of various iterative procedures; for details refer to Ortega [25], Ostrowski [28], and Rheinboldt [29]. In general, all the iterative techniques require the initial guess to be inside a regular neighborhood (radius of convergence) of the root. Results given by Axelsson [1], Brent [3], Garcia and Zangwill [7], Griewank and Osborne [9], Hirsch and Smale [13], Ortega [26], and Rheinboldt [29] guarantee co...

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A new type of cell mapping, referred to as an adjoining cell mapping, is developed in this paper for autonomous dynamical systems employing the cellular state space. It is based on an adaptive time integration employed to compute an associated cell mapping for the system. This technique overcomes the problem of determining an appropriate duration of integration time for the simple cell mapping method. Employing the adjoining mapping principle, the first type of algorithm developed here is an adaptive mapping unraveling algorithm to determine equilibria and limit cycles of the dynamical system in a way similar to that of the simple cell mapping. In addition, it is capable of providing useful information regarding the behavior of dynamical systems possessing pathological dynamics and of systems with rapidly changing vector field. The adjoining property inherent in the adjoining cell mapping method, in general, permits development of new recursive algorithms for unraveling dynamics. The required computer memory for a practical implementation of such algorithms is considerably less than that required by the simple cell mapping algorithm since they allow for a recursive partitioning of state space for trajectory analysis. The second type of algorithm developed in this paper is a recursive unraveling algorithm based on adaptive integration and recursive partitioning of state space into blocks of cells with a view toward its practical implementation. It can find equilibria of the system in the same manner as the simple cell mapping method but is more efficient in locating periodic solutions.