We present explicit high order composition methods based on a second order symmetric method to numerically integrate Hill's lunar problem. The linear/nonlinear splitting of the non-separable Hamiltonian allows us to build a class of integrators that are simple to use and efficient in comparison with other standard symplectic methods. Our numerical results show that the methods preserve the energy very well in long time integration.

In this talk I will analyse the activity of a neural eld model of Wilson- Cowan or Amari type. This model typically takes the form of an integro- dierential equation dened on a spatially extended domain. Assuming a ring rate given by the Heaviside function, I will review the techniques used for establishing the existence and stability of stationary solutions and show that this depends on the coupling function being symmetric. I will then derive conditions for the existence of travelling wave solutions for the case of asymmetric coupling functions. This work is ongoing and extends on previous results.

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse or instability point) is one of the key problems in power system operations and planning such an index indicates the severity of a power system with regard to voltage collapse or instability. While much work has been done on the former problem, the latter problem poses a more severe challenge both to the engineering and mathematical communities. This paper first presents a review of the main methods for detecting Hopf bifurcation in the general field of bifurcation theory and then attempts to generalize the previously studied fold detection and index methods to the Hopf case. The proposed Hopf test functions should be of wide interest while the proposed test function based index method gives an indication of the distance of the current operating point from the Hopf instability. Promising numerical results have been obtained using some standard dynamical test systems.

Simple performance indices for detecting and predicting stability problems such as the voltage collapse are useful analysis tools for planning and operating engineers in, power utilities. We propose a fast guided continuation method to predict or compute a reliable performance index that takes proper care of limits of reactive power resources.. The main feature is the novel combination of a Q limits index and a performance index to achieve computational efficieny and index reliability. Numerical experiments using standard IEEE 9-, 14-, 30-, 57-, 118-, 300-bus systems and a NGC 40-bus system show that the pro, posed method can predict or compute the voltage collapse point much faster than. the standard continuation.

Voltage collapse in a power system can occur following a progressive decline in voltage magnitude at the system buses and, mathematically, this phenomenon is associated with a fold bifurcation point occurring in the nonlinear algebraic equations used to model the power system. In this paper, we first discuss some of the methods used to speed up the process of detecting a fold bifurcation, focussing on designing test function methods to predict a performance index. We then discuss some iterative methods that can be used to improve the Newton iterations. In particular, we present new and efficient preconditioners of the two-level type for the Jacobian matrix. Numerical results are given using standard IEEE test bus systems.

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practicalThis paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.

Test function methods are well known techniques to monitor a continuation method to determine if a fold (saddle-point) bifurcation point has been reached. Two such test functions due to [Abbott, 1978] and [Seydel, 1979] are considered here in a unified framework and their connection to the tangent vector index method from power system community is shown. The challenge in power systems applications is to adapt a test function method to produce a performance index that approximates the fold parameter λ* without the need of following the solution path to the fold point (x*, λ*). We demonstrate that the tangent vector index method, due to its connection with the mathematically justifiable test function methods, can lead to a useful performance index.

One of the important planning and operations problems in electrical power systems is to determine the distance of a loading parameter from the voltage collapse point-the so-called performance index. Such an index indicates the severity of a power network with regard to voltage collapse. This paper investigates a class of test function methods due to Seydel (1979) that may be adapted to produce a performance index. Experiments have shown that an arbitrary choice of the parameters in the method leads to unreliable indexes. We first present a result that gives an equivalent formula, involving determinants, for the Seydel test function. We then analyse the behaviour of such determinants in order to determine what parameters are suitable to produce reliable indexes. For typical power systems with reactive power changes, it turns out that only a restricted set of parameters can ensure a reliable index by the Seydel method; for other cases, a general modification is proposed. Promising numerical results using several standard IEEE test examples are presented.

In this talk I will analyse the activity of a one-dimensional neural field model developed by Amari (1). His approach was to analyse a ho- mogeneous neural field with a symmetric connection function of lateral inhibition type resulting in stationary pulse solutions. Here I will show that if the connection function is asymmetric, then the neural fieled may exhibit travelling pulse solutions. I will construct travelling pulse solu- tions for the case of a Heaviside step function for which I will derive the shape and velocity of the pulses. I will further determine the necessary conditions for the stability of the travelling pulses using Evans function techniques.