We consider a three-dimensional slow-fast system with quadratic nonlinearity and additive noise. The associated deterministic system of this stochastic differential equation (SDE) exhibits a periodic orbit and a slow manifold. The deterministic slow manifold can be viewed as an approximate parameterization of the fast variable of the SDE in terms of the slow variables. In other words the fast variable of the slow-fast system is approximately "slaved" to the slow variables via the slow manifold. We exploit this fact to obtain a two dimensional reduced model for the original stochastic system, which results in the Hopf-normal form with additive noise. Both, the original as well as the reduced system admit ergodic invariant measures describing their respective long-time behaviour. We will show that for a suitable metric on a subset of the space of all probability measures on phase space, the discrepancy between the marginals along the radial component of both invariant measures can be upper bounded by a constant and a quantity describing the quality of the parameterization. An important technical tool we use to arrive at this result is Girsanov's theorem, which allows us to modify the SDEs in question in a way that preserves transition probabilities. This approach is then also applied to reduced systems obtained through stochastic parameterizing manifolds, which can be viewed as generalized notions of deterministic slow manifolds.
A theory of Ruelle–Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal’s oscillatory components manifested by peaks in the PSD. It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V. These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V, and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V, is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V, i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak. The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous “signature” of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane–Zebiak model of El Niño-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.
A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics' slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated --- in the context of the Kuramoto-Sivashinsky turbulence --- as providing efficient closures without slaving for a cutoff scale kc placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.
The problem of emergence of fast gravity-wave oscillations in rotating, stratified flow is reconsidered. Fast inertia-gravity oscillations have long been considered an impediment to initialization of weather forecasts, and the concept of a “slow manifold” evolution, with no fast oscillations, has been hypothesized. It is shown on a reduced Primitive Equation model introduced by Lorenz in 1980 that fast oscillations are absent over a finite interval in Rossby number but they can develop brutally once a critical Rossby number is crossed, in contradistinction with fast oscillations emerging according to an exponential smallness scenario such as reported in previous studies, including some others by Lorenz. The consequences of this dynamical transition on the closure problem based on slow variables is also discussed. In that respect, a novel variational perspective on the closure problem exploiting manifolds is introduced. This framework allows for a unification of previous concepts such as the slow manifold or other concepts of “fuzzy” manifold. It allows furthermore for a rigorous identification of an optimal limiting object for the averaging of fast oscillations, namely the optimal parameterizing manifold (PM). It is shown through detailed numerical computations and rigorous error estimates that the manifold underlying the nonlinear Balance Equations provides a very good approximation of this optimal PM even somewhat beyond the emergence of fast and energetic oscillations.
This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an L2-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes. These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller u_R* and the optimal controller u* are derived. These estimates demonstrate that the closeness of u_R* to u*? is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with u_R* and u*; and (ii) the energy kept in the high modes of the PDE solution either driven by u_R* or u* itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results. The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.
This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.
In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.