Fast Manifolds Research Articles

Nonlinear Model Reduction for Slow–Fast Stochastic Systems Near Unknown Invariant Manifolds

Nonlinear Model Reduction for Slow–Fast Stochastic Systems Near Unknown Invariant Manifolds

Reduction of the optimal tracking problem in the presence of noise

In this paper, the decomposition method based on the theory of fast and slow integral manifolds is used to analyze the optimal tracking problem. We consider a singularly perturbed optimal tracking problem with a given reference trajectory in the case of incomplete information about the state vector in the presence of random external perturbations.

Comparative analysis of two flow decomposition approaches and their applications in diagnosing water vapor in extreme precipitation events over Xinjiang province, China

Water vapor flux divergence under the terrain-following coordinate system is decomposed into slow and fast manifolds with the adoption of the Barnes filter method and the synoptic approach based on the Helmholtz theorem in this study. The performances and applications of both methods are evaluated in precipitating events over the Xinjiang province in China when traditional analysis fails to capture the evolutions of extreme precipitations caused by multi-scale systems and complicated terrains. A blizzard with a short duration under weak moisture environments and a persistent torrential rainfall event with abundant water vapor in the summer are selected as test examples for comparative analysis of the relationships between water vapor divergence and precipitation characteristics with the adoptions of both traditional synoptic diagnosis and two decomposition methods. It is found that divergent areas of the total water vapor flux divergence at the middle and high levels are blocked by false convergence at the low levels as the precipitation weakens in both cases, which leads to the failure of traditional synoptic diagnoses. Meanwhile, the two decomposition methods can provide superior indicators for extreme precipitation, especially before rainfall decreases or terminates. The synoptic approach performs better than the Barnes filter method as the latter sharply decreases in accuracy as the complexity of the flow fields and moisture conditions increase. Remedies for relieving this problem are proposed before being used for further applications.

Critical traveling wave solution in the Meinhardt type model

The traveling waves problem for the singularly perturbed semilinear parabolic equations is considered in the paper. It is shown that the corresponding problem for a singularly perturbed ODE system can be reduced to a certain problem of lower dimension using a splitting transformation based on the technique of slow and fast integral manifolds. The Meinhardt model is considered as an example.

Boundary Orders and Geometry of the Signed Thom–Smale Complex for Sturm Global Attractors

We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE *ut=uxx+f(x,u,ux)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t = u_{xx} + f(x,u,u_x) \\end{aligned}$$\\end{document}on the unit interval 0< x<1 with Neumann boundary conditions. We assume f to be dissipative with N hyperbolic equilibria vin {mathcal {E}}. The global attractor {mathcal {A}} of (*), also called Sturm global attractor, consists of the unstable manifolds of all equilibria v. As cells, these form the Thom–Smale complex{mathcal {C}}. Based on the fast unstable manifolds of v, we introduce a refinement {mathcal {C}}^s of the regular cell complex {mathcal {C}}, which we call the signed Thom–Smale complex. Given the signed cell complex {mathcal {C}}^s and its underlying partial order, only, we derive the two total boundary orders h_iota :{1,ldots , N}rightarrow {mathcal {E}} of the equilibrium values v(x) at the two Neumann boundaries iota =x=0,1. In previous work we have already established how the resulting Sturm permutation σ:=h0-1∘h1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sigma :=h_{0}^{-1} \\circ h_1, \\end{aligned}$$\\end{document}conversely, determines the global attractor {mathcal {A}} uniquely, up to topological conjugacy.

Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate

This paper presents the study on the chaotic wave and chaotic dynamics of the nonlinear wave equations for a simply supported truss core sandwich plate combined with the transverse and in-plane excitations. Based on the governing equation of motion for the simply supported sandwich plate with truss core, the reductive perturbation method is used to simplify the partial differential equation. According to the exact solution of the unperturbed equation, two different kinds of the topological structures are derived, which one structure is the resonant torus and another structure is the heteroclinic orbit. The characteristic of the singular points in the neighborhood of the resonant torus for the nonlinear wave equation is investigated. It is found that there exists the homoclinic orbit on the unperturbed slow manifold. The saddle-focus type of the singular point appears when the homoclinic orbit is broken under the perturbation. Additionally, the saddle-focus type of the singular point occurs when the resonant torus on the fast manifold is broken under the perturbation. It is known that the dynamic characteristics are well consistent on the fast and slow manifolds under the condition of the perturbation. The Melnikov method, which is called the first measure, is applied to study the persistence of the heteroclinic orbit in the perturbed equation. The geometric analysis, which is named the second measure, is used to guarantee that the heteroclinic orbit on the fast manifold comes back to the stable manifold of the saddle on the slow manifold under the perturbation. The theoretical analysis suggests that there is the chaos for the Smale horseshoe sense in the truss core sandwich plate. Numerical simulations are performed to further verify the existence of the chaotic wave and chaotic motions in the nonlinear wave equation. The damping coefficient is considered as the controlling parameter to study the effect on the propagation property of the nonlinear wave in the sandwich plate with truss core. The numerical results confirm the validity of the theoretical study.

Saddle Slow Manifolds and Canard Orbits in mathbb{R}^{4} and Application to the Full Hodgkin\u2013Huxley Model

Many physiological phenomena have the property that some variables evolve much faster than others. For example, neuron models typically involve observable differences in time scales. The Hodgkin–Huxley model is well known for explaining the ionic mechanism that generates the action potential in the squid giant axon. Rubin and Wechselberger (Biol. Cybern. 97:5–32, 2007) nondimensionalized this model and obtained a singularly perturbed system with two fast, two slow variables, and an explicit time-scale ratio ε. The dynamics of this system are complex and feature periodic orbits with a series of action potentials separated by small-amplitude oscillations (SAOs); also referred to as mixed-mode oscillations (MMOs). The slow dynamics of this system are organized by two-dimensional locally invariant manifolds called slow manifolds which can be either attracting or of saddle type.In this paper, we introduce a general approach for computing two-dimensional saddle slow manifolds and their stable and unstable fast manifolds. We also develop a technique for detecting and continuing associated canard orbits, which arise from the interaction between attracting and saddle slow manifolds, and provide a mechanism for the organization of SAOs in mathbb{R}^{4}. We first test our approach with an extended four-dimensional normal form of a folded node. Our results demonstrate that our computations give reliable approximations of slow manifolds and canard orbits of this model. Our computational approach is then utilized to investigate the role of saddle slow manifolds and associated canard orbits of the full Hodgkin–Huxley model in organizing MMOs and determining the firing rates of action potentials. For ε sufficiently large, canard orbits are arranged in pairs of twin canard orbits with the same number of SAOs. We illustrate how twin canard orbits partition the attracting slow manifold into a number of ribbons that play the role of sectors of rotations. The upshot is that we are able to unravel the geometry of slow manifolds and associated canard orbits without the need to reduce the model.

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, \begin{document}$u_t = u_{xx} + f(x, u, u_x)\, , $ \end{document} on the unit interval \begin{document}$0 with Neumann boundary conditions. Equilibria \begin{document}$v_t = 0$\end{document} are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} as 3-cell templates \begin{document}$\mathcal{C}$\end{document} . The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries \begin{document}$x = 0$\end{document} and \begin{document}$x = 1$\end{document} , respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.

Folded Saddles and Faux Canards

We study the two parameter family of faux canards associated with the folded saddle singularity within the folded singularity normal form. Recently, rotational behavior of folded saddle faux canards has been reported by Vo and Wechselberger in [SIAM J. Math. Anal., 47 (2015), pp. 3235--3283] where they studied examples of systems close to a folded saddle-node type I limit. This is a surprising observation and merits a closer look at faux canards which have been somewhat neglected in the literature. We address this gap in canard knowledge and provide a comprehensive analysis of folded saddle faux canards, both numerically and analytically. We show that for certain values of $\mu$---the eigenvalue ratio of the associated folded singularity within the reduced flow---faux canards may possess rotations about the primary faux canard, and that the stable and unstable fast manifolds (i.e., the nonlinear stable and unstable fast fiber bundles) of the primary faux canard form the boundaries of sets of solutions wit...

Steady state likelihood ratio sensitivity analysis for stiff kinetic Monte Carlo simulations.

Kinetic Monte Carlo simulation is an integral tool in the study of complex physical phenomena present in applications ranging from heterogeneous catalysis to biological systems to crystal growth and atmospheric sciences. Sensitivity analysis is useful for identifying important parameters and rate-determining steps, but the finite-difference application of sensitivity analysis is computationally demanding. Techniques based on the likelihood ratio method reduce the computational cost of sensitivity analysis by obtaining all gradient information in a single run. However, we show that disparity in time scales of microscopic events, which is ubiquitous in real systems, introduces drastic statistical noise into derivative estimates for parameters affecting the fast events. In this work, the steady-state likelihood ratio sensitivity analysis is extended to singularly perturbed systems by invoking partial equilibration for fast reactions, that is, by working on the fast and slow manifolds of the chemistry. Derivatives on each time scale are computed independently and combined to the desired sensitivity coefficients to considerably reduce the noise in derivative estimates for stiff systems. The approach is demonstrated in an analytically solvable linear system.

Stability and oscillations in a slow-fast flexible joint system with transformation delay

Flexible joints are usually used to transfer velocities in robot systems and may lead to delays in motion transformation due to joint flexibility. In this paper, a link-rotor structure connected by a flexible joint or shaft is firstly modeled to be a slow-fast delayed system when moment of inertia of the lightweight link is far less than that of the heavy rotor. To analyze the stability and oscillations of the slow-fast system, the geometric singular perturbation method is extended, with both slow and fast manifolds expressed analytically. The stability of the slow manifold is investigated and critical boundaries are obtained to divide the stable and the unstable regions. To study effects of the transformation delay on the stability and oscillations of the link, two quantitatively different driving forces derived from the negative feedback of the link are considered. The results show that one of these two typical driving forces may drive the link to exhibit a stable state and the other kind of driving force may induce a relaxation oscillation for a very small delay. However, the link loses stability and undergoes regular periodic and bursting oscillation when the transformation delay is large. Basically, a very small delay does not affect the stability of the slow manifold but a large delay affects substantially.

Fast and slow invariant manifolds in chemical kinetics

In the framework of the classical theory of Singularly Perturbed Systems (SPS) there exist two main types of invariant manifolds, i.e. fast and slow manifolds with their own intrinsic dynamics. There are many different methods for dimension and complexity reduction for mechanisms of chemical kinetics, based on the concept of slow invariant manifolds. For instance, most known and popular techniques such as Quasi-Steady State (QSSA) and Partial Equilibrium Approaches (PEA) as well as recently developed Intrinsic Low Dimensional Manifolds (ILDM) and Computational Singular Perturbation (CSP) methods have shown a potential to treat automatically very complex mechanisms comprising hundreds of chemically reacting species. In this respect, the importance of slow invariant manifolds has become more or less clear. However, the importance of fast invariant manifolds is much less transparent. Though without an accurate knowledge about fast manifolds (fast fibers in other terminology) slow manifolds cannot be appropriately identified.In this paper we shall discuss a concept of Singularly Perturbed Vector Fields (SPVF) that is a coordinate free theory of singularly perturbed systems with the main emphasis made on fast invariant manifolds and on the method which allows identification and evaluation of these objects. The importance of these objects in model reduction of mechanisms of chemical kinetics will be addressed. Motivation and application of the method will be demonstrated by a simple model example and by analysis of conventional QSSA and PEA methods.

Simplified CSP analysis of a stiff stochastic ODE system

We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.

Reaction-Diffusion Manifolds and Global Quasi-linearization: Two Complementary Methods for Mechanism Reduction

The paper outlines the current state in the model reduction of systems governing reacting flows by manifold methods. The main idea of such approaches is based on the fact that any reduced model defines a manifold of low dimen- sion imbedded in the system composition/state space. In this respect the decomposition into relatively fast and slow mo- tions due to multiple time scales present in the system is a crucial property of the reacting system. It allows the application of the geometrical framework of slow and fast invariant manifolds to model reduction. Recently developed approaches, namely, the so-called Reaction-Diffusion Manifolds (REDIMs) and Global-Quasi Linearization (GQL) are in the focus of this work. The methods extend and follow the well known ILDM method. The paper discusses both the theoretical basis of the approaches and detailed implementation schemes for studying, reducing and simulating the reacting flows systems. Simple yet containing all features of the reacting flows models of n-heptane/air and syngas/air systems are used to illus- trate and verify the methods.

The family of anisotropically scaled equatorial waves

[1] In the present work we introduce the family of anisotropic equatorial waves. This family corresponds to equatorial waves at intermediate states between the shallow water and the long wave approximation model. The new family is obtained by using anisotropic time/space scalings on the linearized, unforced and inviscid shallow water model. It is shown that the anisotropic equatorial waves tend to the solutions of the long wave model in one extreme and to the shallow water model solutions in the other extreme of the parameter dependency. Thus, the problem associated with the completeness of the long wave model solutions can be asymptotically addressed. The anisotropic dispersion relation is computed and, in addition to the typical dependency on the equivalent depth, meridional quantum number and zonal wavenumber, it also depends on the anisotropy between both zonal to meridional space and velocity scales as well as the fast to slow time scales ratio. For magnitudes of the scales compatible with those of the tropical region, both mixed Rossby-gravity and inertio-gravity waves are shifted to a moderately higher frequency and, consequently, not filtered out. This draws attention to the fact that, for completeness of the long wave like solutions, it is necessary to include both the anisotropic mixed Rossby-gravity and inertio-gravity waves. Furthermore, the connection of slow and fast manifolds (distinguishing feature of equatorial dynamics) is preserved, though modified for the equatorial anisotropy parameters used δ ∈ 〈 1]. New possibilities of horizontal and vertical scale nonlinear interactions are allowed. Thus, the anisotropic shallow water model is of fundamental importance for understanding multiscale atmosphere and ocean dynamics in the tropics.

Computing Slow Manifolds of Saddle Type

Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm.

Chaotically Spiking Canards in an Excitable System with 2D Inertial Fast Manifolds

We introduce a new class of excitable systems with two-dimensional fast dynamics that includes inertia. A novel transition from excitability to relaxation oscillations is discovered where the usual Hopf bifurcation is followed by a cascade of period doubled and chaotic small excitable attractors and, as they grow, by a new type of canard explosion where a small chaotic background erratically but deterministically triggers excitable spikes. This scenario is also found in a model for a nonlinear Fabry-Perot cavity with one pendular mirror.

Slow Manifold Structure in Explosive Kinetics. 1. Bifurcations of Points-at-Infinity in Prototypical Models

This article analyzes in detail the global geometric properties (structure of the slow and fast manifolds) of prototypical models of explosive kinetics (the Semenov model for thermal explosion and the chain-branching model). The concepts of global or generalized slow manifolds and the notions of heterogeneity and alpha-omega inversion for invariant manifolds are introduced in order to classify the different geometric features exhibited by two-dimensional kinetic schemes by varying model parameters and to explain the phenomena that may occur in model reduction practice. This classification stems from the definition of suitable Lyapunov-type numbers and from the analysis of normal-to-tangent stretching rates. In the case of the Semenov model, we show that the existence of a global slow manifold and its properties are controlled by a transcritical bifurcation of the points-at-infinity, which can be readily identified by analyzing the Poincaré projected system. The issue of slow manifold uniqueness and the implications of the theory with regard to the practical definition of explosion limits are thoroughly addressed.

Slow and fast invariant manifolds, and normal modes in a two degree-of-freedom structural dynamical system with multiple equilibrium states

Many problems in structural dynamics involve coupling between a stiff (high frequency) linear structure and a soft (low frequency) non-linear structure with multiple static equilibrium states. In this work, we analyze the slow and fast motions of a conservative structural system consisting of a non-linear oscillator with three equilibrium states coupled to a stiff linear oscillator. We combine analysis (singular perturbations) with geometry (manifolds) and computation to show that the system possesses invariant manifolds supporting either slow or fast motions. In particular, under appropriate conditions, a global slow invariant manifold passes through the three static equilibrium states of the system. The slow manifold is non-linear, orbitally stable, and it carries a continuum of in-phase periodic motions, including a homoclinic motion. We generalize the classical notion of vibrations-in-union to include systems with multiple equilibria, and thus identify the slow invariant manifold with a slow, non-linear normal mode of vibration.

Efficient and uniform production of III-nitride films by multiwafer MOVPE

We present results on the growth of Ga–In–Al–N films in multiwafer Planetary Reactors® with 7×2 in. wafer capacity. The unique design of these reactors allows us to combine high efficiency (Ga efficiency in GaN around 30%) and excellent uniformity. Thickness uniformities better than ±5% on a 2 in. wafer are routinely obtained. Compositional uniformity of GaInN is better than 2 nm. Wafer-to-wafer and run-to-run reproducibility concerning thickness and composition is within the 1% range. Besides this, the material grown also has a high purity. Hall mobilities higher than 500 cm 2 V −1 s −1 are obtained for GaN. Furthermore, the design of the reactors is dedicated to the growth of complex nitride heterostructures. Fast switching manifolds for various group III precursors and different carrier gases allow a flexible design of the growth runs including nucleation, bulk layers (doped/undoped) and quantum wells. The extremely low thermal mass of the reactor permits very rapid changes of the deposition temperature between room temperature and 1300°C (optional: 1600°C). Of course p- and n-type doping is also possible. Doping uniformities are below 1% indicating extremely uniform wafer temperatures. The results demonstrate that Planetary Reactors® are reliable and efficient tools for the mass production of blue-green optoelectronic devices.