C.S. Hsu

Equilibrium configurations of a shallow arch of arbitrary shape and their dynamic stability character

Abstract The behavior of a shallow arch is inherently non-linear. Offered here is an exact and complete analysis which allows us to determine all the possible equilibrium configurations and their dynamic stability character. The arch may be of any arbitrary shape. Both simply-supported and clamped end conditions are treated. The results have immediate applications to the snap-through stability of arches when subjected to impulsive loads or time-varying loads of finite duration.

Chaotic rounding error in digital control systems

Chaotic behavior due to the round-off effect in digital control systems is called the chaotic rounding error. First, we model digital control systems with finite-wordlength digital compensators by mixed mappings. A mixed mapping system is described jointly by a point mapping and a cell mapping, that is, its state consists of real numbers and integers. After presenting some definitions and fundamental results of mixed mapping systems, we prove certain sufficient conditions for the existence of the chaotic rounding error. Finally, as an illustration, we discuss the chaotic rounding error of a system with a second-order plant and first-order digital compensator.

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.

Domains of attraction for multiple limit cycles of coupled van der pol equations by simple cell mapping

Abstract One of the most difficult tasks in non-linear analysis is to determine globally various domains of attraction in the state space when there exist more than one asymptotically stable equilibrium states and/or periodic motions. The task is even more demanding if the order of the system is higher than two. In this paper we consider two coupled van der Pol oscillators which admit two asymptotically stable limit cycles. For systems of this kind we show how the method of cell-to-cell mapping can be used to determine the two four-dimensional domains of attraction of the two stable limit cycles in a very effective way. The final results are shown in this paper in the form of a series of graphs which are various two-dimensional sections of the four-dimensional state space.

Domain of stability of synchronous generators by a cell mapping approach

In this paper, the theory of simple cell mapping is applied to the determination of domains of stability of synchronous generators. The generators are represented by two-dimensional differential dynamical systems including the effects of variable damping and saliency. First, the differential equation governing the behaviour of a generator is reduced to a cell-to-cell mapping. A scheme of compactification is introduced in order to obtain a complete cell state space which consists of a finite number of cells. The methodology of the simple cell mapping is then used to prepare a computer program. Using the program, all the singularities of the cell mapping and the domain of stability can be determined readily. The method is particularly attractive because it can easily take care of any non-linearity. Different systems can be treated by the same program by simply changing the form of the differential equation in the program and/or changing the system parameters. Three kinds of synchronous machine are studied i...

Singularities of n-dimensional cell functions and the associated index theory

Abstract In [1,2] a new and promising method for global analysis of non-linear systems has been presented in the form of cell-to-cell mappings. It is seen to be a very powerful method indeed. However, many fundamental questions about this new kind of mapping remain to be investigated. In this paper we present a theory of singularities for cell functions by introducing a system of singular entities. The key steps in the development are to introduce a simplicial structure to the cell space and to use simplicial mappings to create an associated continuous vector field for a given cell function. The construction of the associated vector field also allows us to develop a theory of index for cell functions. This theory of index then serves very nicely as a unifying force for the whole development reported in this paper.

Stability of saddle-like deformed configurations of plates and shallow shells

Abstract Studied in this paper are circular and square plates and shallow shells with circular and square bases subjected to certain transverse loadings which, according to the classical linear theory, will produce saddle-like deformed configurations. The analysis shows that the vastly different non-linear behavior is characterized by non-linear bifurcation, the existence of multiple equilibrium configurations and jump phenomenon in deformation, and it involves questions of stability. Moreover, the results demonstrate that when the deformation is appreciable the saddle-like configurations are unstable, and the square plates subjected to four corner forces and the circular plates under periodic edge loadings of cos 2θ type will take on configurations which are more or less cylindrical in shape. This paper is a sequel to [1] where much simpler problems of plates and shallow shells of infinite size are treated.

Simple example of digital control systems with chaotic rounding errors

Chaotic behaviour due to round-off characteristics in digital compensators is called a chaotic rounding error. First, we show sufficient conditions for the existence of chaotic rounding errors in digital control systems with first-order plants and first-order finite-wordlength compensators. Next, it is shown that a chaotic rounding error, which appears as a chaotic attractor in the state space, can be described by a Markov chain with an appropriate partition of the attractor. Finally, by analysing the Markov chain we calculate several statistical properties of the chaotic rounding error.

Characteristics of singular entities of simple cell mappings

For a given simple cell mapping C(z) there is a cell mapping increment function F(z, C K ) associated with the mapping C K (z). In Chapter 5 we have considered the singular entities of such cell functions. These include non-degenerate and degenerate singular k-multiplets m k , k ∈ N + 1, and cores of singular multiplets. We recall here that for dynamical systems governed by ordinary differential equations (Coddington and Levinson [1955], Arnold [1973]) and for point mapping dynamical systems (Hsu [1977], Bernussou [1977]), the singular points of the vector fields governing the systems can be further classified according to their stability character. In this spirit one may wish to classify singular entities of cell functions according to their “stability” character and to see how they influence the local and global behavior of the cell mapping systems. On this question a special feature of cell mappings immediately stands out. Since a cell function maps an N-tuple of integers into an N-tuple of integers, the customary continuity and differentiability arguments of the classical analysis cannot be used, at least not directly. Evidently, a new framework is needed in order to delineate various mapping properties of the singular entities of cell functions.

Cell simplex degeneracy, Liapunov function and stability of simple cell mapping systems

Abstract The mathematical structure governing simple cell mappings is further studied in this paper. The topics investigated are: the relationship between degeneracy of a cell simplex and the number of periodic orbits contained in the cell simplex, Liapunov functions for simple cell mappings, invariance principle and certain global stability questions. Unlike the algorithmic and computational approach to examine the behavior of simple cell mappings, the Liapunov function approach provides us with a different and yet instructive perspective.