Finite-dimensional Invariant Manifolds Research Articles

Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds

The accurate simulation of complex dynamics in fluid flows demands a substantial number of degrees of freedom, i.e. a high-dimensional state space. Nevertheless, the swift attenuation of small-scale perturbations due to viscous diffusion permits in principle the representation of these flows using a significantly reduced dimensionality. Over time, the dynamics of such flows evolves towards a finite-dimensional invariant manifold. Using only data from direct numerical simulations, in the present work we identify the manifold and determine evolution equations for the dynamics on it. We use an advanced autoencoder framework to automatically estimate the intrinsic dimension of the manifold and provide an orthogonal coordinate system. Then, we learn the dynamics by determining an equation on the manifold by using both a function-space approach (approximating the Koopman operator) and a state-space approach (approximating the vector field on the manifold). We apply this method to exact coherent states for Kolmogorov flow and minimal flow unit pipe flow. Fully resolved simulations for these cases require $O(10^3)$ and $O(10^5)$ degrees of freedom, respectively, and we build models with two or three degrees of freedom that faithfully capture the dynamics of these flows. For these examples, both the state-space and function-space time evaluations provide highly accurate predictions of the long-time dynamics in manifold coordinates.

Invariant Manifolds for the Thin Film Equation

The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document} is examined. We investigate the higher order asymptotics of solutions converging towards self-similar Smyth–Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.

Fractional term structure models: No-arbitrage and consistency

In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White models. It turns out that similar to the Brownian case such a family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family.

Optimal control of tokamak and stellarator plasma behaviour

The control of plasma transport, laminar and turbulent, is investigated, using the methods of scaling, optimal control and adaptive Monte Carlo simulations. For this purpose, the asymptotic behaviour of kinetic equation is considered in order to obtain finite-dimensional invariant manifolds, and in this way the finite-dimensional theory of control can be applied. We imagine the labyrinth of open doors and after applying self-similarity, the motion moved through all the desired doors in repeatable ways as Brownian motions. We take local actions for each piece of contractive ergodic motion, and, after self-organization in adaptive invariant measures, the optimum movement of particles is obtained according to the principle of maximum entropy. This is true for deterministic and stochastic cases that serve as models for plasma dynamics.

Equivariant versal unfoldings for linear retarded functional differential equations

We continue our investigation of versality for parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds. In this paper, we consider RFDEs equivariant with respect to the action of a compact Lie group. In a previous paper (Buono and LeBlanc, <em>J. Diff. Eqs.</em>, <strong>193 </strong>, 307-342 (2003)), we have studied this question in the general case (i.e. no <em>a priori</em> restrictions on the RFDE). When studying the question of versality in the equivariant context, it is natural to want to restrict the range of possible unfoldings to include only those which share the same symmetries as the original RFDE, and so our previous results do not immediately apply. In this paper, we show that with appropriate projections, our previous results on versal unfoldings of linear RFDEs can be adapted to the case of linear equivariant RFDEs. We illustrate our theory by studying the linear equivariant unfoldings at double Hopf bifurcation points in a $\mathbb D_3$-equivariant network of coupled identical neurons modeled by delay-differential equations due to delays in the internal dynamics and coupling.

Versal unfoldings for linear retarded functional differential equations

We consider parametrized families of linear retarded functional differential equations (RFDEs) projected onto finite-dimensional invariant manifolds, and address the question of versality of the resulting parametrized family of linear ordinary differential equations. A sufficient criterion for versality is given in terms of readily computable quantities. In the case where the unfolding is not versal, we show how to construct a perturbation of the original linear RFDE (in terms of delay differential operators) whose finite-dimensional projection generates a versal unfolding. We illustrate the theory with several examples, and comment on the applicability of these results to bifurcation analyses of nonlinear RFDEs.

Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2

We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R 2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.

An Lp Theory of Invariant Manifolds for Parabolic Partial Differential Equations on [formula omitted

We study the problem about the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form∂u/∂τ=▵u+F(u, ∇u)onRd×[1, ∞). We show that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite dimensional invariant manifolds which control the long time asymptotics of solutions. We consider the problem for these equations in the framework of weighted Sobolev spaces of Lp type. The Lp theory of this problem gives the L∞ estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term F and their initial data.

Integrable vs. nonintegrable geodesic soliton behavior

We study confined solutions of certain evolutionary partial differential equations (PDE) in 1+1 space–time. The PDE we study are Lie–Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler–Poincaré equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L 2 norm of the velocity. These PDE possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call “pulsons”. We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. The results for head-on antisymmetric collisions of pulsons tend to be singularity formation. Numerical simulations of these PDE show that their evolution by geodesic dynamics for confined (or compact) initial conditions in various nonintegrable cases possesses the same type of multi-soliton behavior (elastic collisions, asymptotic sorting by pulse height) as the corresponding integrable cases do. We conjecture this behavior occurs because the integrable two-pulson interactions dominate the dynamics on the invariant pulson manifold, and this dynamics dominates the PDE initial value problem for most choices of confined pulses and initial conditions of finite extent.

Invariant Manifolds and Long-Time Asymptotics for the Vlasov--Poisson--Fokker--Planck Equation

We study the large time behavior of small solutions to the Cauchy problem for the Vlasov--Poisson--Fokker--Planck equation, which is a degenerate parabolic equation with nonlocal nonlinearity. We construct finite dimensional invariant manifolds in a neighborhood of the origin in polynomially weighted Sobolev spaces, which enables us to compute systematically the long-time asymptotics for small solutions. To construct invariant manifolds, we make use of the similarity variables transformation as in C. E. Wayne's work in 1997, where invariant manifolds for parabolic equations in unbounded domains are constructed.

Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation $$$$ where Ω⊂ℝ n , n∈{1,2,3 }, is a bounded domain with sufficiently smooth boundary, and f is cubic-like, for example f(u) =u−u 3. Based on the results of [26] the nonlinear Cahn-Hilliard equation will be discussed. This equation generates a nonlinear semiflow in certain affine subspaces of H 2(Ω). In a neighborhood U e with size proportional to e n around the constant solution \(\), where μ lies in the spinodal region, we observe the following behavior. Within a local inertial manifold \(\) containing \(\) there exists a finite-dimensional invariant manifold\(\) which dominates the behavior of all solutions starting with initial conditions from a small ball around \(\) with probability almost 1. The dimension of \(\) is proportional to e− n and the elements of \(\) exhibit a common geometric quantity which is strongly related to a characteristic wavelength \(\) proportional to e.

Equilibria with Many Nuclei for the Cahn–Hilliard Equation

Let f be a bistable nonlinearity such as u−u3. We consider multi-peaked stationary solutions to the Cahn–Hilliard equation ut=−Δ(ε2Δu+f(u)) in Ω, ∂u/∂n=∂Δu/∂n=0 on ∂Ω, with the average value of u in the metastable region. By “multi-peaked” we mean states which, as ε→0, tend to a constant value everywhere except for a finite number of points, which we call nuclei, in Ω, where the states tend to a different constant value. For any N we find such solutions with N peaks located at certain geometrically identified points. The proof is based on a dynamical systems viewpoint where the stationary solutions being sought are equilibrium points on a finite-dimensional invariant manifold of multi-peaked states. In addition to the existence of these solutions we also discuss their strong instability, justifying the name nuclei for the points of concentration.

Invariant Manifolds for Parabolic Partial Differential Equations on Unbounded Domains

In this paper finite-dimensional invariant manifolds for nonlinear parabolic partial differential equations of the form are constructed. Such results are somewhat surprising because of the continuous spectrum of the linearized equation. These manifolds control the long-time behavior of solutions of these equations and can be used to construct systematic, rigorous expansions of the long-time asymptotics in inverse powers of . They also give a new perspective on the change in the long-time asymptotics of the equation with nonlinear term , when passes through the critical value .

Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation

In this paper we describe invariant geometrical ~structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.

Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation

The paper addresses, for retarded functional differential equations (FDEs) with parameters, the computation of normal forms associated with the flow on a finite-dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity. The analysis is based on the theory previously developed for FDEs without parameters by considering an extension of a phase space appropriate to the computation of these normal forms. As an application, the generic Hopf bifurcation for scalar retarded FDEs is considered.

Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity

The paper addresses, for retarded functional differential equations (FDEs), the computation of normal forms associated with the flow on a finite-dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity. A phase space appropriate to the computation of these normal forms is introduced, and adequate nonresonance conditions for the computation of the normal forms are derived. As an application, the general situation of Bogdanov-Takens singularity and its versal unfolding for scalar retarded FDEs with nondegeneracy at second order is considered, both in the general case and in the case of differential-delay equations of the form ẋ( t) = ƒ( x( t), x( t−1)).

Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

On a completely integrable nonlinear hyperbolic variational equation

We show that the nonlinear partial differential equation, ( u t +uu x ) xx =1/2( u x 2 ) x , is a completely integrable, bi-variational, bi-Hamiltonian system. The corresponding equation for w=u xx belongs to the Harry Dym hierarchy. This equation arises in two different physical contexts in two nonequivalent variational forms. It describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field, and it is the high-frequency limit of the Camassa-Holm equation, which is an integrable model equation for shallow water waves. Using the bi-Hamiltonian structure, we derive a recursion operator, a Lax pair, and an infinite family commuting Hamiltonian flows, together with the associated conservation laws. We also give the transformation to action-angle coordinates. Smooth solutions of the partial differential equation break down because their derivative blows up in finite time. Nevertheless, the Hamiltonian structure and complete integrability appear to remain valid globally in time, even after smooth solutions break down. We show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions. We compute the bi-Hamiltonian structure on this invariant manifold which is obtained by restricting the bi-Hamiltonian structure of the partial differential equation.

On a completely integrable nonlinear hyperbolic variational equation

We show that the nonlinear partial differential equation, ( u t+uu x) xx =1/2( u x 2) x , is a completely integrable, bi-variational, bi-Hamiltonian system. The corresponding equation for w=u xx belongs to the Harry Dym hierarchy. This equation arises in two different physical contexts in two nonequivalent variational forms. It describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field, and it is the high-frequency limit of the Camassa-Holm equation, which is an integrable model equation for shallow water waves. Using the bi-Hamiltonian structure, we derive a recursion operator, a Lax pair, and an infinite family commuting Hamiltonian flows, together with the associated conservation laws. We also give the transformation to action-angle coordinates. Smooth solutions of the partial differential equation break down because their derivative blows up in finite time. Nevertheless, the Hamiltonian structure and complete integrability appear to remain valid globally in time, even after smooth solutions break down. We show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions. We compute the bi-Hamiltonian structure on this invariant manifold which is obtained by restricting the bi-Hamiltonian structure of the partial differential equation.