Compounds containing 1,3-oxazepine and 1,3,4-oxadiazole (5)- (8) were synthesized by microwave assisted organic synthesis method through the condensation reaction of compounds (1 to 4) with phthalic anhydride respectively. The structure of the synthesized compounds were characterized by spectroscopic methods ( 1 H NMR, FTIR). The synthesized compounds were screened for their antibacterial activity disc diffusion method.

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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.

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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.