Linearization of Hyperbolic Finite-Time Processes

We adapt the notion of processes to introduce an abstract framework for dynamics in finite time, i.e.\ on compact time sets. For linear finite-time processes a notion of hyperbolicity namely exponential monotonicity dichotomy (EMD) is introduced, thereby generalizing and unifying several existing approaches. We present a spectral theory for linear processes in a coherent way, based only on a logarithmic difference quotient, prove robustness of EMD with respect to a suitable (semi-)metric and provide exact perturbation bounds. Furthermore, we give a complete description of the local geometry around hyperbolic trajectories, including a direct and intrinsic proof of finite-time analogues of the local (un)stable manifold theorem and theorem of linearized asymptotic stability. As an application, we discuss our results for ordinary differential equations on a compact time-interval.Comment: 32 page

Transport in Transitory Dynamical Systems

We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the "action" of relevant heteroclinic orbits at the intersection of invariant manifolds of "forward" and "backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure

On a characterization of blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity

We derive characterizations of blow-up behavior of solutions of ODEs by means of dynamics at infinity with complex asymptotic behavior in autonomous systems, as well as in nonautonomous systems. Based on preceding studies, a variant of closed embeddings of phase spaces and the time-scale transformation determined by the structure of vector fields at infinity reduce our characterizations to unravel the structure of local stable manifolds of invariant sets on the horizon, the corresponding geometric object of the infinity in the embedded manifold. Geometric and dynamical structure of normally hyperbolic invariant manifolds (NHIMs) on the horizon induces blow-up solutions with the specific blow-up rates. Using the knowledge of NHIMs, blow-up solutions in nonautonomous systems can be characterized in a similar way.Comment: 40 pages, no figure

Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of "transport barriers" in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterizing transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate, in a concrete manner, the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of "flow transition" which occurs when finite-time hyperbolicity is lost, or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing, and important, new area of dynamical systems theory

Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold

Local stable and unstable manifolds and their control in nonautonomous finite-time flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear Science