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The stable and unstable manifolds associated with a saddle point in two-dimensional non-areapreserving flows under general time-aperiodic perturbations are examined. An improvement to existing geometric Melnikov theory on the normal displacement of these manifolds is presented. A new theory on the previously neglected tangential displacement is developed. Together, these enable locating the perturbed invariant manifolds to leading order. An easily usable Laplace transform expression for the location of the perturbed time-dependent saddle is also obtained. The theory is illustrated with an application to the Duffing equation. 1. Introduction. Invariant manifolds are important entities in continuous dynamical systems, forming crucial flow organizers. Their movement under perturbations can alter the global flow structure. The original results of Melnikov [40] relate to the normal displacement of stable and unstable manifolds in a homoclinic structure in two-dimensional area-preserving flow, under a time-sinusoidal perturbation. The transverse zeroes of the so-called Melnikov function identify when the perturbed invariant manifolds intersect, leading to chaos via the Smale-Birkhoff theorem [26, 4, 57]. Extensions of the Melnikov method to higher dimensions [25, 44, 60, 54], time-aperiodicity and/or finite-time [41, 44, 58, 62, 8], subharmonic bifurcations [40, 57, 61, 59], nonhyperbolicity [54, 60, 63], and non-area-preservation [32] are available.While Melnikov methods can be used to determine how invariant manifolds move normal to the original manifolds, there has been no method in the literature in which the tangential movement is characterized. This study addresses this issue, arriving at a Melnikov-like function for the tangential displacement, under general time-dependent perturbations. The original two-dimensional flow is assumed to contain a saddle structure but need not be areapreserving. The displacement is expressed as a function of the original position p on the manifold and the time-slice t. Along the way, a similar quantification for the normal displacement is obtained, in which potential divergence issues in the Melnikov function and the legitimacy of ignoring higher-order terms are explicitly addressed. The normal and tangential results together permit the locating of the perturbed stable and unstable manifolds of the

A theory for the fluid flux generated across heteroclinic separatrices under the influence of time-aperiodic perturbations is presented. The flux is explicitly defined as the amount of fluid transferred per unit time, and its detailed timedependence monitored. The perturbations are allowed to be significantly discontinuous in time, including for example impulsive (Dirac delta type) discontinuities. The flux is characterized in terms of time-varying separatrices, with easily computable formulae (directly related to Melnikov functions) provided.

The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.

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