Theory Of Inertial Manifolds Research Articles

Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.

Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.

Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez–Schopf delayed oscillator and the irregularity of ENSO

We revisit the classical Suarez–Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical–numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited attracting periodic orbits. These subregions must be separated by a curve on which homoclinic “figure eights”, bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez–Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.

Frequency theorem for parabolic equations and its relation to inertial manifolds theory

We obtain a version of the Frequency Theorem (a theorem on solvability of certain operator inequalities), which allows to construct quadratic Lyapunov functionals for semilinear parabolic equations. We show that the well-known Spectral Gap Condition, which was used in the theory of inertial manifolds by C. Foias, R. Temam and G.R. Sell, is a particular case of some frequency inequality, which arises within the Frequency Theorem. In particular, this allows to construct inertial manifolds for semilinear parabolic equations (including also some non-autonomous problems) in the context of a more general geometric theory developed in our adjacent works. This theory is based on quadratic Lyapunov functionals and generalizes the frequency-domain approach used by R.A. Smith. We also discuss the optimality of frequency inequalities and its relationship with known old and recent results in the field.

Using approximate inertial manifold approach to model turbulent non-premixed combustion

The theory of inertial manifolds (IM) is used to develop reduced-order models of turbulent combustion. In this approach, the dynamics of the system are tracked in a low-dimensional manifold determined in situ without invoking laminar flame structures or statistical assumptions about the underlying turbulent flow. The primary concept in approximate IM (AIM) is that slow dominant dynamical behavior of the system is confined to a low-dimension manifold, and fast dynamics respond to the dynamics on the IM instantaneously. Decomposition of slow/fast dynamics and formulation of an AIM is accomplished by only exploiting the governing equations. Direct numerical simulation (DNS) of initially non-premixed fuel–air mixtures developing in forced isotropic turbulence have been carried out to investigate the proposed model. Reaction rate parameters are varied to allow for varying levels of extinction and reignition. The AIM performance in capturing different flame behaviors is assessed both a priori and a posteriori. It is shown that AIM captures the dynamics of the flames including extinction and reignition. Moreover, AIM provides scalar dissipation rate, mixing time for reactive scalars, and closures for nonlinear terms without any additional modeling. The AIM formulation is found promising and provides a new approach to modeling turbulent combustion.

Regularity of the Inertial Manifolds for Evolution Equations in Admissible Spaces and Finite-Dimensional Feedback Controllers

Our main aims in this paper are to investigate the regularity of inertial manifolds for non-autonomous semi-linear evolution equations and to give an application of inertial manifolds to a feedback control problem. We first prove that the inertial manifolds are smooth if the nonlinear term is smooth. Then, using the theory of inertial manifolds for non-autonomous semi-linear evolution equations, we construct a feedback controller for a class of control problems for the one-dimensional reaction-diffusion equations with the Lipschitz coefficient of the nonlinear term which may depend on time and belong to an admissible space.

Approximate inertial manifold-based order reduction of rigid-flexible coupling manipulator

An order reduction method for the flexible deformation response analysis of rigid flexible manipulators is proposed based on the approximate inertial manifold theory. This method allows a lower dimensional simplified model to be constructed from a subspace smaller than the entire state space. In this paper, truncated three-order modes are used to construct a first-order system of AIM. Compared with the traditional Galerkin method, the results show that the proposed method can reduce the degree of freedom of the system and improve the computational efficiency without obviously losing the precision of the solution, which is convenient for the subsequent vibration analysis and controller design of the system.

Globally Stabilizing Finite-Dimensional Damping Control for Dissipative PDE with Bounded Inputs

The main aim of this paper is to address the global asymptotic stabilization (GAS) of a class of dissipative partial differential equations (pdes) via finite-dimensional admissible (regular and bounded) damping (output) controls. To this end, we use the inertial manifold theory to derive infinite-dimensional dissipative control systems given by interconnection of a finite-dimensional control system on the inertial manifold plus an infinite-dimensional zero-input system on the complement. We show that the GAS of such systems is reduced to the finite-dimensional one. Then, we prove that the finite-dimensional systems which are zero-input point-dissipative (have global attractors K) and those which are B-strictly passive (passivity relative to bounded sets) are connected. Finally, we use the control Lyapunov functions (CLF) theory to design admissible feedback damping controls for the GAS of B-strictly passive systems.

Inertial manifolds and finite-dimensional reduction for dissipative PDEs

This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well as various attempts to generalize it based on the so-called Mañé projection theorems. The recent counter-examples showing that the underlying dynamics may in a sense be infinite dimensional if the spectral gap condition is violated, as well as a discussion of the most important open problems, are also included.

Lipschitz deviation and embeddings of global attractors

Hunt and Kaloshin (1999 Nonlinearity 12 1263–75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d < k into for any N > 2k + 1, using a linear map L whose inverse is Hölder continuous with exponent α < (N − 2d)/N(1 + τ(X)/2), where τ(X) is the ‘thickness exponent’ of X. More recently, Ott et al (2006 Ergod. Theory Dyn. Syst. 26 869–91) studied the effect of such embeddings on the Hausdorff dimension of X, and showed that for ‘most’ linear maps , dH(L(X)) ⩾ min(N, dH(X)/(1 + τ(X)/2)). They also conjectured that ‘many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero’. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and—appealing to results from the theory of approximate inertial manifolds—we prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, dev(X) = 0 for the attractor of the 2D Navier–Stokes equations with forcing f ∊ L2, while current results only guarantee that τ(X) = 0, when f ∊ C∞.

Solitary waves for Korteweg–de Vries equation with small delay

This paper deals with the existence of solitary waves for Korteweg–de Vries equation with time delay. Based upon the inertial manifold theory and differential manifold geometric theory, the existence of solitary wave solution is proved when the delay is small enough. Up to now, studies on solitary wave for such delay differential equation are not available, so the results of this paper are new.

Feedback control via inertial manifolds for nonautonomous evolution equations

In this paper we extend a method to control the dynamicsof evolution equations by finite dimensional controllers which wassuggested by Brunovsky [3] to nonautonomous evolutionequations using nonautonomous inertial manifold theory.

Finite-dimensional description of the long-term dynamics for the 2D Doi-Hess model for liquid crystalline polymers in shear flow

The existence of inertial manifolds for a Smoluchowski equation arising in the 2D Doi-Hess model for liquid crystalline polymers subjected to a shear flow is investigated. The presence of a non-variational drift term dramatically complicates the long-term dynamics from the variational gradient case, in which it is solely characterized by the steady states. Several transformations are used in order to transform the equation into a form suitable for application of the standard theory of inertial manifolds. A nonlinear and nonlocal transformation developed in Inertial manifolds for a Smoluchowski equation on a circle and Inertial manifolds for a Smoluchowski equation on the unit sphere,, to appear, is used to eliminate the first-order derivative from the micro-micro interaction term. A traveling wave transformation eliminates the first-order derivative from the non-variational term, transforming the equation into a nonautonomous one for which the theory of nonautonomous inertial manifolds applies.

Optimal dynamic inversion control design for a class of nonlinear distributed parameter systems with continuous and discrete actuators

Combining the principles of dynamic inversion and optimisation theory, two stabilising state-feedback control design approaches are presented for a class of nonlinear distributed parameter systems. One approach combines the dynamic inversion with variational optimisation theory and it can be applied when there is a continuous actuator in the spatial domain. This approach has more theoretical significance in the sense that it does not lead to any singularity in the control computation and the convergence of the control action can be proved. The other approach, which can be applied when there are a number of discrete actuators located at distinct places in the spatial domain, combines dynamic inversion with static optimisation theory. This approach has more relevance in practice, since such a scenario appears naturally in many practical problems because of implementation concern. These new techniques can be classified as 'design-then-approximate' techniques, which are in general more elegant than the 'approximate-then-design' techniques. However, unlike the existing design-then-approximate techniques, the new techniques presented here do not demand involved mathematics (like infinite-dimensional operator theory, inertial manifold theory and so on). To demonstrate the potential of the proposed techniques, a real-life temperature control problem for a heat transfer application is solved, first assuming a continuous actuator and then assuming a set of discrete actuators.

Chaos synchronization for continuous chaotic systems by inertial manifold approach

Abstract In this paper, we are concerned with the complete and generalized synchronization problems for chaotic ODE systems. Some general results are obtained by means of inertial manifold theory. Some examples, supported by numerical simulation, are given to illustrate the conciseness and effectiveness of the method.

Dichotomy property of solutions of quasilinear equations in problems on inertial manifolds

Exponential dichotomy properties are studied for non-autonomous quasilinear partial differential equations that can be written as an ordinary differential equation in a Hilbert space . It is assumed that the non-linear function is essentially subordinated to the linear operator ; namely, the gap property from the theory of inertial manifolds must hold. Integral manifolds and attracting at an exponential rate an arbitrary solution of this equation as and , respectively, are constructed. The general results established are applied to the study of the dichotomy properties of solutions of a one-dimensional reaction-diffusion system and of a dissipative hyperbolic equation of sine-Gordon type.

Exact Finite Dimensional Feedback Control via Inertial Manifold Theory with Application to the Chafee–Infante Equation

A class of nonlinear parabolic partial differential equations is considered, and an exact finite dimensional feedback control law is designed in order to force the systems to behave in a prescribed way. The feedback law is obtained via inertial manifold theory by reducing the system to finite dimensions. The control achieved is exact, as opposed to approximate, as obtained in a previous work. The result is applied to the Chafee–Infante equation, a one-dimensional scalar reaction-diffusion equation, with distributed control.

Methods for dimension reduction and their application in nonlinear dynamics

We compare linear and nonlinear Galerkin methods in their efficiency to reduce infinite dimensional systems, described by partial differential equations, to low dimensional systems of ordinary differential equations, both concerning the effort in their application and the accuracy of the resulting reduced system. Important questions like the choice of the form of the ansatz functions (modes), the choice of the number m of modes and, finally, the construction of the reduced system are addressed. For the latter point, both the linear or standard Galerkin method making use of the Karhunen Loeve (proper orthogonal decomposition) ansatz functions and the nonlinear Galerkin method, using approximate inertial manifold theory, are used. In addition, also the post-processing Galerkin method is compared with the other approaches.

Wavelet-like collocation method for finite-dimensional reduction of distributed systems

A wavelet-like collocation method is proposed to approach the reduction of dissipative distributed systems, expressed by means of partial differential equations, applying the methods of inertial manifold theory. The collocation method proposed, based on localized trial functions, provides a convenient numerical framework to develop approximate inertial manifolds in the case of partial differential problems (e.g. reaction/diffusion models) containing nonpolynomial nonlinearities. The collocation method is based on the interpolation of concentration/temperature fields by means of Gaussian–sinc functions. As model systems, we consider reaction diffusion schemes such as the non-isothermal model for stockpile ignition and the Elezgaray–Arneodo diffusion model.

Three counterexamples in the theory of inertial manifolds

An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no finite-dimensionalC1 invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Ω ⊂ ℝm without inertial manifolds\(\mathcal{M} \subset L^{\text{2}} (\Omega )\) with the property of absolute normal hyperbolicity on the setE of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property of normal hyperbolicity onE. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found.