Nonautonomous Flows Research Articles

Universality of Euler flows and flexibility of Reeb embeddings

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao [38,39] launched a programme to address the global existence problem for the Euler and Navier Stokes equations based on the concept of universality. Inspired by this proposal, in this article we prove that the stationary Euler equations exhibit several universality features. More precisely, we show that any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension. The solutions we construct are of Beltrami type, and being stationary they exist for all time. Using this result, we establish the Turing completeness of the steady Euler flows, i.e., there exist solutions that encode a universal Turing machine and, in particular, these solutions have undecidable trajectories. Our proofs deepen the correspondence between contact topology and hydrodynamics, which is key to establish the universality of the Reeb flows and their Beltrami counterparts. An essential ingredient in the proofs, of interest in itself, is a novel flexibility theorem for embeddings in Reeb dynamics in terms of an h-principle in contact geometry, which unveils the flexible behavior of the steady Euler flows. These results can be viewed as lending support to the intuition that solutions to the Euler equations can be extremely complicated in nature.

An inverse boundary value problem for isotropic nonautonomous heat flows

We study an inverse boundary value problem on the determination of principal order coefficients in isotropic nonautonomous heat flows stated as follows; given a medium, and in the absence of heat sources and sinks, can the time-dependent thermal conductivity and volumetric heat capacity of the medium be uniquely determined from the Cauchy data of temperature and heat flux measurements on its boundary? We prove uniqueness in all dimensions under an assumption on the thermal diffusivity of the medium, which is defined as the ratio of the thermal conductivity and volumetric heat capacity. Our assumption on the thermal diffusivity is related to construction of certain families of exponential solutions to the heat equation.

The problem of acceleration in the dynamics of a double-link wheeled vehicle with arbitrarily directed periodic excitation

This study investigates the motion of a nonholonomic mechanical system that consists of two wheeled carriages articulated by a rigid frame. There is a point mass which oscillates at a given angle ?? to the main axis of one of the carriages. As a result, periodic excitation occurs in the system. The equations of motion in quasi-velocities are obtained. Eventually, the dynamics of a double-link wheeled vehicle is modeled by a system that defines a nonautonomous flow on a three-dimensional phase space. The behavior of integral curves at large velocities depending on the angle ?? is investigated. We use the generalized Poincar?e transformation and reduce the original problem to the stability problem for the system with a degenerate linear part. The proof of stability uses the restriction of the system to the central manifold and averaging by normal forms up to order 4. The range of values of ?? for which one of the velocity components increases indefinitely is found and asymptotics for the solutions of the initial dynamical system is determined.

The two-process model for sleep–wake regulation: A nonsmooth dynamics perspective

Since its inception four decades ago the two-process model introduced by Borbély has provided the conceptual framework to explain sleep–wake regulation across many species, including humans. At its core, high level notions of circadian and homeostatic processes are modelled with a low dimensional description in the form of a one dimensional nonautonomous and nonsmooth flow, with the rate of change of homeostatic sleep pressure switching at specific times. These events in time can be described by an implicit map from one switching time to another and have given rise to an elegant mathematical description of periodic orbits and their instabilities using the theory of iterated maps. In this paper we show that an equivalent description can be obtained from a direct analysis of the underlying nonsmooth flow. We further show how to construct the Lyapunov exponent of the nonsmooth flow and use this to uncover a more detailed picture of the Arnol’d tongue structure of the model.Given the growing interest in studying networks of sleepers, where interactions may occur continuously throughout the day–night cycle and not just at event times, we advocate for the future use of techniques from nonsmooth dynamical systems in studying networks of the two-process model.

Analysis of Driver Behavior in Mixed Autonomous and Non-autonomous Traffic Flows

Autonomous vehicles are expected to improve road safety and efficiency in future transportation systems. A driving simulator study was designed to identify driving styles and the cooperation between human drivers and other AVs. The study captured driver’s following behavior in a fully autonomous driving environment at unsignalized intersections. Participants were asked to make a series of maneuvers (straight through intersection, left turn, and right turn) in two different speed conditions (30, 40 mph) and two different traffic density conditions (with or without other traffic). Analysis of Variance showed that drivers had a significantly larger deviation (defined as the area between two trajectories) during left turn maneuvers when they were traveling at higher speeds. Moreover, the first turning operation had smaller deviation than the second turning operation. The findings have implications for the design of driver-assistance guidance systems in future mixed autonomous and non-autonomous traffic flows.

Determination of the 3D Navier-Stokes equations with damping

<abstract><p>This paper is concerned the determination of trajectories for the three-dimensional Navier-Stokes equations with nonlinear damping subject to periodic boundary condition. By using the energy estimate of Galerkin approximated equation, the finite number of determining modes and asymptotic determined functionals have been shown via the Grashof numbers for the non-autonomous and autonomous damped Navier-Stokes fluid flow respectively.</p></abstract>

Spectral Early-Warning Signals for Sudden Changes in Time-Dependent Flow Patterns

Lagrangian coherent sets are known to crucially determine transport and mixing processes in non-autonomous flows. Prominent examples include vortices and jets in geophysical fluid flows. Coherent sets can be identified computationally by a probabilistic transfer-operator-based approach within a set-oriented numerical framework. Here, we study sudden changes in flow patterns that correspond to bifurcations of coherent sets. Significant changes in the spectral properties of a numerical transfer operator are heuristically related to critical events in the phase space of a time-dependent system. The transfer operator approach is applied to different example systems of increasing complexity. In particular, we study the 2002 splitting event of the Antarctic polar vortex.

Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems.

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.

Local Nonautonomous Schrödinger Flows on Kähler Manifolds

In this paper, we prove that the nonautonomous Schrodinger flow from a compact Riemannian manifold into a Kahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher regularity.

Uniform attractors for vanishing viscosity approximations of non-autonomous complex flows

In this paper we prove the existence of uniform global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of non-autonomous complex fluids.

From Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data

One way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the “best” approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance—in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.

Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre

We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.

A Numerical Scheme for Computing Stable and Unstable Manifolds in Nonautonomous Flows

There are many methods for computing stable and unstable manifolds in autonomous flows. When the flow is nonautonomous, however, difficulties arise since the hyperbolic trajectory to which these manifolds are anchored, and the local manifold emanation directions, are changing with time. This article utilizes recent results which approximate the time-variation of both these quantities to design a numerical algorithm which can obtain high resolution in global nonautonomous stable and unstable manifolds. In particular, good numerical approximation is possible locally near the anchor trajectory. Nonautonomous manifolds are computed for two examples: a Rossby wave situation which is highly chaotic, and a nonautonomus (time-aperiodic) Duffing oscillator model in which the manifold emanation directions are rapidly changing. The numerical method is validated and analyzed in these cases using finite-time Lyapunov exponent fields and exactly known nonautonomous manifolds.

High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing

We study the high-frequency limit of non-autonomous gradient flows in metric spaces of energy functionals comprising an explicitly time-dependent perturbation term which might oscillate in a rapid way. On grounds of the existence results by Ferreira and Guevara (2015) on non-autonomous gradient flows (which we also extend to a broader range of energy functionals), we prove that the associated solution curves converge to a solution of the time-averaged evolution equation in the limit of infinite frequency. Under additional assumptions on the energy, we obtain an explicit rate of convergence. Furthermore, we specifically investigate nonlinear drift-diffusion equations with time-dependent drift which formally are gradient flows with respect to the L2-Wasserstein distance. We prove that a family of weak solutions obtained as a limit of the Minimizing Movements scheme exhibits the above-mentioned behavior in the high-frequency limit.

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.

Diagonalizing quadratic bosonic operators by non-autonomous flow equation

We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

The Tendency toward a Moving Equilibrium

The chasing problem of how the actual values chase a moving equilibrium was raised explicitly in 1965 by John Hicks. To address the issue, we formalize a framework of discrete nonautonomous dynamical systems associated with a moving equilibrium and develop analytical methods for analyzing the tendency toward the moving equilibrium. We seek to embody Marshall's economic tendency principle applied to the tendency toward a moving equilibrium in mathematical form. It establishes the equivalence that a nonautonomous flow chases a moving equilibrium if and only if the tendency is produced in a bounded sequence of periods. The equivalence assertion is key to resolving the chasing problem of economic significance. As a result, we construct a model of an evolving competitive economy underlying price adjustments to describe the supply, demand, and prices starting in the vicinity of a moving equilibrium remaining close to it.

On generalized donating regions: Classifying Lagrangian fluxing particles through a fixed curve in the plane

For a fixed simple curve in a nonautonomous flow, the fluxing index of a passively-advected Lagrangian particle is the total number of times it goes across the curve within a given time interval. Such indices naturally induce flux sets, equivalence classes of the particles at the initial time. This line of research mainly concerns donating regions, the explicit construction of the flux sets from a sufficiently continuous velocity field. In the author's previous paper (Zhang, 2013 [13]) the flux sets with indices ±1 were constructed from characteristic curves of the flow field. This work generalizes the notion of donating regions and shows that flux sets are index-by-index equivalent to the generalized donating regions for any finite integer index, provided that the two backward streaklines seeded at the two endpoints of the simple curve neither intersect nor self-intersect. All Lagrangian particles marked by their initial positions are thus classified by their fluxing indices.

Controlling the Unsteady Analogue of Saddle Stagnation Points

It is well known that saddle stagnation points are crucial flow organizers in steady (autonomous) flows due to their accompanying stable and unstable manifolds. These have been extensively investigated experimentally, numerically, and theoretically in situations related to macro- and micromixers in order to either restrict or enhance mixing. Saddle points are also important players in the dynamics of mechanical oscillators, in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior. The entity analogous to a saddle point in an unsteady (nonautonomous) flow is a time-varying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time. Within the context of nearly steady flows, the unsteady velocity perturbation required to ensure that such a hyperbolic (saddle) trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained via a heuristic approach. An expression for the error in the hyperbolic trajectory's motion is also derived. This provides a new tool for the control of both fluid transport and mechanical oscillators. The method is applied to two examples---a four-roll mill and a Duffing oscillator---and the performance of the control strategy is shown to be excellent in both instances.

A non-autonomous two-phase flow model with oscillating external force and its global attractor

In this article, we consider a non-autonomous diffuse interface model for an isothermal incompressible two-phase flow in a two-dimensional bounded domain. Assuming that the external force is singularly oscillating and depends on a small parameter ϵ , we prove the existence of the uniform global attractor A ϵ . Furthermore, using the method similar to that of Chepyzhov and Vishik (2007) [22] in the case of the two-dimensional Navier–Stokes systems, we study the convergence of A ϵ as ϵ goes to zero. Let us mention that the nonlinearity involved in the model considered in this article is slightly stronger than the one in the two-dimensional Navier–Stokes system studied in Chepyzhov and Vishik (2007) [22].