Nonautonomous control of stable and unstable manifolds in two-dimensional flows

Abstract

We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical example.

Accurate control of hyperbolic trajectories in any dimension

“…While the hyperbolic trajectory described above is exterior to the droplet, many droplet models in the literature themselves possess on the droplet's surface a saddle-like hyperbolic trajectory [26][27][28][29][30][31][32][33][34][35][36] whose motion influences intradroplet chaotic transport because its attached stable and unstable manifolds undergo nontrivial motion. There is currently only one method in the scientific literature which can control manifold paths [37], but it is limited to two-dimensional nearly steady flows with one-dimensional stable and unstable manifolds. In the quest for controlling manifolds in n dimensions, a first step would be to force the hyperbolic trajectory (to which these manifolds are attached) to follow a prescribed motion.…”

Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

“…Extending the results to higher dimensions, in particular three [65], would also be beneficial, since fluid transport across time-varying two-dimensional surfaces has a profound impact on geophysical and microfluidic mixing. The question as to whether, analogous to recent work [9,10,11], a Melnikov approach can be used to control stable and unstable manifolds but in a discontinuous fashion, is another future direction of research. The ability to reformulate the results for impulses which are randomly chosen (e.g., a randomly kicked Duffing oscillator), to extend to a countable number of impulses, or to formulate the problem for general δ-families, would also be of interest.…”

Local Stable and Unstable Manifolds and Their Control in Nonautonomous Finite-Time Flows

“…Now, it is easily verified that a solution to (15) which takes the value y(t) at time t can be stated as y(t) = Y (t)y(0) in terms of the fundamental matrix solution. If y(0) is in the range of P , then one can easily use the first exponential dichotomy condition to show that |y(s)| ≤ K s e λss |y(0)| for s ≥ 0 ;…”

“…first result on controlling stable and unstable manifolds [15]. Since Theorem 2.4 provides the methodology of pointing stable and unstable manifolds in user-desired time-varying directions, this provides insight into how best to focus energy in the most relevant areas, in order to obtain desired mixing characteristics.…”