Cell Mapping Method Research Articles

Application of Cell Mapping methods to a discontinuous dynamic system

The Cell Mapping method is a robust tool for investigating nonlinear dynamic systems. It is capable of finding the attractors and corresponding basins of attraction of a system under investigation. To investigate the applicability of the Cell Mapping method to discontinuous systems, a forced zero-stiffness impact oscillator is chosen as an application. The numerical integration algorithm, the basic element in the Cell Mapping method, is adjusted to overcome the discontinuity. Four types of Cell Mapping techniques are applied: Simple Cell Mapping, Generalized Cell Mapping, Interpolated Cell Mapping, and Mixed Cell Mapping. The last type is a new modification to existing types. Each type of Cell Mapping is briefly explained. The results are compared to the exact solutions. The Interpolated Cell Mapping and Mixed Cell Mapping methods are found to produce the most accurate results for this case.

A fuzzy cell mapping method for a sub-optimal control implementation

This paper presents a fuzzy cell-to-cell mapping method for implementing a sub-optimal control law. The method preserves the simplicity and flexibility properties of the original cell mapping technique but it relaxes the requirement on the number of cells. Since the new method generates smooth control efforts, it does not create the chattering phenomena. Furthermore, the new method eliminates the problem of steady-state error accumulation caused by the inexact state description. Experimental studies on the control of a DC servo motor show that finer grid sizes achieve trajectories that are closer to the analytical solution. Different loadings were also tried on the motor to demonstrate the robustness of the method.

A cell mapping method for general optimum trajectory planning of multiple robotic arms

This paper proposes a method that uses cell state space and cell mapping based techniques for planning general optimum trajectories along given geometric paths for coordinated multiple robotic arm systems. The major advantages of this method include its simplicity and applicability to a wide range of problem formulations. In particular, three performance indices for optimum trajectory specification are investigated, i.e., minimum-energy, minimum-jerk, and minimum-time formulations. A simple search strategy is constructed using cell-to-cell mapping to find optimum trajectories. A special feature of this search algorithm is its ability to generate all optimum trajectories for all possible initial conditions through a single search. The computational complexity is analyzed for the search algorithm and its hierarchical implementation. Parallel execution of the hierarchical search method is discussed and the results indicate that it can improve the cell-mapping search efficiency significantly.

The adjoining cell mapping and its recursive unraveling, part II: Application to selected problems

Several applications of the adjoining cell mapping technique are provided here by employing the adaptive mapping unraveling algorithm to analyze smooth and pathological autonomous dynamical systems. The performance of an implementation of recursive unraveling algorithm is also illustrated regarding its low memory requirements for computational purposes when compared with the simple cell mapping method. The applications considered here illustrate the effectiveness of the adjoining cell mapping technique in its ability to determine limit cycles and to unravel nonstandard dynamics. The advantages of this new technique of global analysis over the simple cell mapping method are discussed.

Domain of stability of a synchronous machine with excitation control—a cell mapping approach

This paper presents the application of the cell mapping method to determine the domain of stability of a synchronous machine with excitation control. Excitation control and the synchronous machine dynamics are formulated mathematically and solved by step-by-step integration. At each step, the solution is converted to the form of cells. The cell mapping algorithm is used to find all the asymptotically stable equilibrium points and periodically stable equilibrium states and their domains of stability. Then a compaction scheme is used to plot all equilibrium states and their domains of stability. A description of the cell mapping method, construction of the cell map, compaction scheme, cell mapping algorithm, test system, dynamic equations and the results are presented in this paper. The results show that the domain of stability is expanded due to the introduction of excitation control. Several advantages of the cell mapping approach to find the domain of stability of a dynamic system, such as the simplicity and versatility of the cell mapping algorithm, adaptability of the computer program to any higher order non-linear dynamic system, capability to handle non-linearities in equations, and complete elimination of all the manual calculations, are shown.

Dynamic Stability and Chaos of System with Piecewise Linear Stiffness

The dynamic stability behavior of a single‐degree‐of‐freedom system with piecewise linear stiffness is considered. The analytical expression representing the divergence of perturbed trajectories is derived. The mechanism triggering dynamic instability of trajectories and the cause of chaotic behavior are then studied. Liapunov exponents are used as a quantitative measure of system stability. Numerical results including bifurcation diagrams and largest Liapunov exponents of a system with symmetric bilinear stiffness are presented. Several different types of bifurcation and nonlinear phenomena are identified, including pitchfork, fold, flip, boundary crisis, and intermittency of type 3. To illustrate the initial‐condition dependent nature of the problem, basins of attraction of multiple steady‐state responses are determined on the phase plane using the simple cell mapping method.

The adjoining cell mapping and its recursive unraveling, part I: Description of adaptive and recursive algorithms

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.

GLOBAL ANALYSIS BY CELL MAPPING

This paper deals with cell mapping methodology for global analysis of nonlinear dynamical systems. It serves a mixed set of purposes. It is basically a tutorial paper on cell mapping. But, it also reports on certain new developments in cell mapping and includes a summary of recent publications on the topic. Presented in Sec. 1–3 and 5 are the basic concepts and theory of cell mapping. Two types of cell-to-cell mapping are discussed, namely: simple cell mapping and generalized cell mapping. Once a dynamical system has been cast in the form of a cell mapping, one needs to extract the system behavior from the mapping. In Secs. 4 and 6, computation algorithms for simple cell mapping and generalized cell mapping are discussed in detail. Moreover, a workshop-type example is included to guide the reader if he wishes to gain a working knowledge of the methodology. The new developments presented in the paper include an algorithm for processing simple cell mappings and a theory of subdomain-to-subdomain global transient analysis of generalized cell mapping. These are reported in Secs. 4–6. Listed in the last section are publications on cell mapping, including applications in many rather diversified areas of dynamics. Hopefully, the breadth of the examples of application will indicate the potential of the cell mapping method.

Domains of Attraction of System of Nonlinearly Coupled Ship Motions by Simple Cell Mapping

A nonlinear dissipative dynamical system can often have multiple attractors. In this case, it is important to study the global behavior of the system by determining the global domain of attraction of each attractor. In this paper we study the global behavior of a two-degree-of freedom system. The specific system examined is a system of nonlinearly coupled ship motions in regular seas. The system is described by two second-order nonlinear nonautonomous ordinary differential equations. When the frequency of the pitch mode is twice the frequency of the roll mode, and is near the encountered wave frequency, the system can have two asymptotically stable steady-state periodic solutions. The one solution has the same period as the encountered wave period and has the pitch motion only. The other solution has twice the period of the encountered wave period and has pitch and roll motions. The harmonic and second-order subharmonic solutions show up as period-1 and period-2 solutions, respectively, in a Poincare´ map. We show how the method of simple cell mapping can be used to determine the two four-dimensional domains of attraction of the two solutions in a very effective way. The results are compared with the ones obtained by direct numerical integration.

Global sensitivity analysis of a nonlinear system using animated basins of attraction

A computer program is described that allows the effect of parameter changes on the global behavior of a nonlinear system to be studied. Steady‐state motions and their basins of attraction can be rapidly obtained and visualized for given parameter values. The program is applied to the study of a differential equation modeling a nonlinear oscillator forced parametrically through its bifurcation parameter. The algorithm used in our program is based on the Interpolated Cell Mapping (ICM) method developed by Tongue [B. H. Tongue, Physica D 28(3), 401–408 (1987)]. It is shown how this method, when implemented on a vector/parallel architecture machine, can be combined with animation to perform global sensitivity analyses of nonlinear systems. A high‐resolution (1152×900 pixel) animation is created which shows the evolution of the basins of attraction for a specific pair of periodic solutions as the forcing frequency and forcing amplitude are varied over a closed path in the parameter space. The movie consists of 720 pictures, each of which requires 518 400 separate simulations. In order to accomplish this task, the basic algorithm is modified, using IBM Parallel Fortran, to exploit the architecture of the IBM 3090‐600S. Of particular interest is the way the animation reveals phenomena that are difficult or impossible to see in the still images. Detailed benchmarks are presented which show the performance of different configurations of the code in a typical university supercomputing environment.

Effects of small random uncertainties on non-linear systems studied by the generalized cell mapping method

The effects of small random uncertainties on certain non-linear systems under parametric and external excitation are studied by carrying out global analyses using the generalized cell mapping (GCM) method. The solutions to some fundamental problems in non-linear systems such as bifurcations, steady state solutions, limiting probability distribution of steady state solutions, domains of attraction and basin boundaries are obtained. It is found that system uncertainties increase the unpredictability of the system, and blur the geometrical structure of solutions of the deterministic system by making more multipledomicile cells near the boundaries of domains of attraction, and producing larger persistent groups representing stable steady solutions of the system. Furthermore, the noise fluctuations of the system induce early transition to chaos for a system undergoing a sequence of period doubling bifurcations to chaos, and destabilize first the solution with “weaker protection” for a system having multiple stable steady state solutions. This result is of importance to the reliability design of mechanical and structural systems that have multiple stable steady state solutions and are expected to endure uncertainties.

The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation

A short-time Gaussian approximation scheme is proposed in the paper. This scheme provides a very efficient and accurate way of computing the one-step transition probability matrix of the previously developed generalized cell mapping (GCM) method in nonlinear random vibration. The GCM method based upon this scheme is applied to some very challenging nonlinear systems under external and parametric Gaussian white noise excitations in order to show its power and efficiency. Certain transient and steady-state solutions such as the first-passage time probability, steady-state mean square response, and the steady-state probability density function have been obtained. Some of the solutions are compared with either the simulation results or the available exact solutions, and are found to be very accurate. The computed steady-state mean square response values are found to be of error less than 1 percent when compared with the available exact solutions. The efficiency of the GCM method based upon the short-time Gaussian approximation is also examined. The short-time Gaussian approximation renders the overhead of computing the one-step transition probability matrix to be very small. It is found that in a comprehensive study of nonlinear stochastic systems, in which various transient and steady-state solutions are obtained in one computer program execution, the GCM method can have very large computational advantages over Monte Carlo simulation.

Global analysis of nonlinear dynamical systems with fuzzy uncertainties by the cell mapping method

In this paper we introduce a fuzzy set cell mapping method for nonlinear dynamical systems with fuzzy uncertainties. This method comprises a generalization, based upon Zadeh's extension principle, of the previously developed cell mapping methods for crisp deterministic and stochastic dynamical systems. The systems studied here are assumed to have a parameter modeled as a fuzzy number with a specified membership function. Some simple examples of fuzzy nonlinear dynamical systems are studied. The results presented herein are, in general, difficult to obtain analytically. A comparison is carried out between the cell mapping results and those from direct simulation. The study shows that the fuzzy set cell mapping method is an appropriate and efficient method for the analysis of fuzzy dynamical systems because of its intrinsic capability of dealing with the global nature of the response of the system.

A Computational Method for Finding All the Roots of a Vector Function

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 homotopic 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.

Random vibration of hinged elastic shallow arch

The response to Gaussian white noise excitation of a hinged elastic shallow arch is studied. The steady state solutions are obtained by means of the Fokker-Planck equation. A class of transient solutions, viz. the first-passage time probability for snap-through of a sinusoidal shallow arch, is studied numerically by the generalized cell mapping method and direct Monte Carlo simulation. In obtaining the global information of first-passage time probability of the system, there are large computational advantages of the generalized cell mapping method over direct Monte Carlo simulation.

Application of a cell-mapping method to optimal control problems

Abstract The cell-mapping method previously used to study the global behaviour of strongly non-linear systems is extended to optimal control problems. Approximate optimal control results are extracted from the family of controlled mappings by a systematic search. Provisions are made to account for rigid barriers to flow in the continuum state space. A method of implementation is also proposed, which involves linear interpolation of the discrete optimal controls.

Application of a cell-mapping method to optimal control problems

Abstract The cell-mapping method previously used to study the global behaviour of strongly non-linear systems is extended to optimal control problems. Approximate optimal control results are extracted from the family of controlled mappings by a systematic search. Provisions are made to account for rigid barriers to flow in the continuum state space. A method of implementation is also proposed, which involves linear interpolation of the discrete optimal controls.

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.

Crises in a driven Josephson junction studied by cell mapping.

We use the method of cell-to-cell mapping to locate attractors, basins, and saddle nodes in the phase plane of a driven Josephson junction. The cell-mapping method is discussed in some detail, emphasizing its ability to provide a global view of the phase plane. Our computations confirm the existence of a previously reported interior crisis. In addition, we observe a boundary crisis for a small shift in one parameter. The cell-mapping method allows us to show both crises explicitly in the phase plane, at low computational cost.

A Statistical Study of Generalized Cell Mapping

In this paper a statistical error analysis of the generalized cell mapping method for both deterministic and stochastic dynamical systems is examined, based upon the statistical analogy of the generalized cell mapping method to the density estimation. The convergence of the mean square error of the one step transition probability matrix of generalized cell mapping for deterministic and stochastic systems is studied. For stochastic systems, a well-known trade-off feature of the density estimation exists in the mean square error of the one step transition probability matrix, which leads to an optimal design of generalized cell mapping for stochastic systems. The conclusions of the study are illustrated with some examples.