Given a family of flows parametrized by an interval and a Morse decomposition which continues across the interval, a procedure is devised to detect connecting orbits at various parameter values. This is done by putting a small drift on the parameter space and considering the flow on the product of the phase space and the parameter interval. The Conley index and connection matrix are used to analyse the flow on the product space, then the drift is allowed to go to zero to obtain information about the original family of flows. This method can be used to detect connections between rest points of the same index for example.

The Conley index of an isolated invariant set is defined only for flows; we construct an analogue called the ‘shape index’ for discrete dynamical systems. It is the shape of the one-point compactification of the unstable manifold of the isolated invariant set in a certain topology which we call its ‘intrinsic’ topology (to distinguish it from the ‘extrinsic’ topology which it inherits from the ambient space). Like the Conley index, it is invariant under continuation. A key point is the construction of a certain ‘index category’ associated with the isolated invariant set; this construction works equally well for flows or discrete time systems, and its properties imply the basic properties of both the Conley index and the shape index.

This paper applies the Melnikov method to autonomous perturbations of completely integrable Hamiltonian systems. The forcing of the perturbed system is caused by internal oscillations which are not necessarily decoupled. A unified treatment is presented which relates some results of Holmes and Marsden with a result of Lerman and Umanskii. It is also shown that two forms of the Melnikov function by integrals are in fact equal.