Approximate Invariant Manifolds Research Articles

Model reduction of rotor-foundation systems using the approximate invariant manifold method

Model reduction of rotor-foundation systems using the approximate invariant manifold method

Slow manifold reduction as a systematic tool for revealing the geometry of phase space

Many non-dissipative reduced plasma models can be derived from more fundamental non-dissipative models by restricting to an approximate invariant manifold. I present a general systematic procedure for finding the Hamiltonian formulation of a plasma model that can be derived in this manner. Several illustrative examples are considered in detail.

Nonlinear Normal Modes in a Two-Stage Isolator Using a Modified Finite-Element Galerkin Method

A modified Galerkin method is proposed to approximate the nonlinear normal modes in a new type of a two-stage isolator. Besides the displacement of payload and the force transmissibility of this typical nonlinear dynamic system, the nonlinear normal modes defined as invariant manifolds can provide more information about the nonlinear coupling between the system components when periodic motions corresponding to the normal modes of the system occur. The presented approach applies a combination of finite-element discretization and Fourier series expansion for the approximate invariant manifolds. A Galerkin projection of the governing equations for the approximate invariant manifolds yields a set of nonlinear algebraic equations in expansion coefficients, which can be solved numerically with a general choice of zero as initial guess for the cases in this work. The resultant approximate solutions for the invariant manifolds can accurately describe the nonlinear interactions between system components in periodic motions of the specific nonlinear normal modes. In addition, one can solve the invariant manifolds for an annular domain of interest directly by this method, without considering other domain that includes the origin of phase space.

Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow

In this article, we aim to classify the dynamic transitions and bifurcations for a family of axisymmetric geophysical fluid problems of a generic fourth-second order structure. A transition theorem is established by reducing the governing partial differential equations to a complex-valued ordinary differential equation, derived by employing approximate invariant manifolds. We develop an algorithm for the numerical determination of the transition/bifurcation types. Finally we apply the transition theorem and algorithm to examine the baroclinic instability in a two-layer quasi-geostrophic system in an annular channel and with different bathymetry profiles. Our numerical results show that with concave bathymetry the transition (bifurcation) is always continuous (supercritical Hopf bifurcation), whereas for convex bathymetry a jump transition (subcritical Hopf bifurcation) may occur in the basic azimuthal currents that rotate in the same direction.

Metastability for scalar conservation laws in a bounded domain

The initial-boundary-value problem for a viscous scalar conservation law in a bounded interval I = ( − l,l ) is considered, with emphasis on the metastable dynamics , whereby the time-dependent solution develops internal transition layers that approach their steady state in an asymptotically exponentially long time interval as the viscosity coefficient e > 0 goes to zero. We describe such behavior by deriving an ODE for the position ξ of the internal interface. The main tool of our analysis is the construction of a one-parameter family of approximate stationary solutions {Ue e(.;ξ)}ξ∈I , parametrized by the location of the shock layer ξ , to be considered as an approximate invariant manifold for the problem. By using the properties of the linearized operator at U e , we estimate the size of the layer location.

Centre manifold reduction approach for the lubrication equation

The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness ε undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when ε → 0.The approach is inspired by the paper by Mielke and Zelik (2009 Mem. Am. Math. Soc. 198 1–97), where the centre manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDEs such as lubrication equations.While it has previously been shown by Glasner and Witelski (2003 Phys. Rev. E 67 016302), that the system of ODEs governing the coarsening dynamics can be obtained via formal asymptotic methods, the centre manifold reduction approach presented here pursues the rigorous justification of this asymptotic limit.

Approximate invariant manifolds up to exponentially small terms

This paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces E 0 ( dim m 0 ), E 1 ( dim m 1 ). Under light Diophantine conditions on the linear part, we prove that there is a polynomial change of coordinate in E 1 allowing to eliminate, in the E 1 component of the vector field, all terms depending only on the coordinate u 0 ∈ E 0 , up to an exponentially small remainder. This main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite-dimensional vector fields. In the elliptic case, our results also proves, with very light assumptions on the linear part in E 1 , that for initial data very close to a certain analytic manifold, the solution stays very close to this manifold for a very long time, which means that the modes in E 1 stay very small.

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

The renormalization group (RG) method is one of the singular perturbation methods which is used in searching for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, and inheritance of symmetries from those for the original equation to those for the RG equation are proved. Further, it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper)normal forms theory, the center manifold reduction, the geometric singular perturbation method, and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.

Nonlinear normal modes of structural systems via asymptotic approach

An asymptotic approach, based on the method of multiple scales, is employed to construct the nonlinear normal modes (NNM's) of self-adjoint structural systems with arbitrary linear inertia and elastic stiffness operators, general cubic inertia and geometric nonlinearities. The methodology employed for constructing the approximate invariant manifolds of individual NNM's––away from internal resonances––and of the resonant modes––near three-to-one internal resonances––attempts to generalize previous studies based on asymptotic techniques. The theory is applied to a hinged–hinged uniform elastic beam carrying a lumped mass and undergoing axis stretching. Depending on the lumped mass relative to the beam mass and on its position along the span, different classes of nonlinear normal modes and their stability are investigated.

Invariant grids for reaction kinetics

In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based on a formulation of the condition of invariance as an equation, and its solution by Newton iterations. A grid-based version of MIM is developed (the method of invariant grids). We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appear only in the calculation of the tangent spaces. This fact determines high computational efficiency of the method of invariant grids. The method is illustrated by two examples: the simplest catalytic reaction (Michaelis–Menten mechanism), and the hydrogen oxidation. The algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics.

THE DOUBLE-CUSP MAP FOR THE FORCED LORENZ SYSTEM

It is found that adding constant forcing terms to the Lorenz equations breaks the (x,y)→(-x,-y) symmetry of the system. As a result of the symmetry breaking the consecutive z max return map (the cusp map) gets split into a double-cusp map. We investigate the double-cusp map for the forced Lorenz system using approximate invariant manifolds about the fixed points and use the double-cusp map to understand the stability of the attractor.

Global bifurcations in Rayleigh-Bénard convection. Experiments, empirical maps and numerical bifurcation analysis

We use nonlinear signal processing techniques, based on artificial neural networks, to construct an empirical mapping from experimental Rayleigh-Bénard convection data in the quasiperiodic regime. The data, in the form of a one-parameter sequence of Poincaré sections in the interior of a mode-locked region (resonance horn), are indicative of a complicated interplay of local and global bifurcations with respect to the experimentally varied Rayleigh number. The dynamic phenomena apparent in the data include period doublings, complex intermittent behavior, secondary Hopf bifurcations, and chaotic dynamics. We use the fitted map to reconstruct the experimental dynamics and to explore the associated local and global bifurcation structures in phase space. Using numerical bifurcation techniques we locate the stable and unstable periodic solutions, calculate eigenvalues, approximate invariant manifolds of saddle type solutions and identify bifurcation points. This approach constitutes a promising data post-processing procedure for investigating phase space and parameter space of real experimental systems; it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings.

The Utility of an Invariant Manifold Description of the Evolution of a Dynamical System

The long-time evolution of a physical system may be dominated by a small number of modes. An invariant manifold description of the asymptotic evolution, briefly discussed here in two simple examples, can give a powerful and useful view of the physical system. The relationship between invariant manifolds and centre manifolds is also discussed, as is the relationship between invariant manifolds and modal numerical approximations. Many classical approximations in fluid mechanics can be viewed as approximating the evolution on an approximate invariant manifold.