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Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations. [ABSTRACT FROM AUTHOR]

Abstract: We provide a lower bound on the number of closed characteristics on singular energy levels of second-order Lagrangian systems in the presence of saddle-focus equilibria. The hypotheses on the Lagrangian are mild, and the bound is given in terms of the number of saddle-foci and a potential function determined by the Lagrangian. The method of proof is variational, combining techniques to minimize near a saddle-focus and an analog of the method of broken geodesics. [Copyright &y& Elsevier]