Invariant Manifold Theorem Research Articles

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

Fast terminal sliding-mode control based on unknown input observer for the tracking control of underwater vehicles

The control of underwater vehicles is always challenging due to nonlinear dynamics, unmodeled dynamics, system uncertainty, and environmental disturbances. In this paper, a robust control scheme is designed for the tracking control of underwater vehicles with system uncertainty and time-varying environmental disturbances. First, the dynamic model of underwater vehicles is simplified to an explicit expression for the inertia matrix without the need for complex hydrodynamic coefficients. Then, according to the invariant manifold theorem, a single-parameter unknown input observer (UIO) is designed to estimate uncertainty. Based on the simplified model and UIO, a fast terminal sliding mode control with an inverse function (IFTSMC) is designed. The finite-time stability of the closed-loop system is proved. Finally, to evaluate the performance of the proposed control scheme, comparative numerical simulations are conducted with 20% dynamic uncertainty and time-varying interference. The results indicate that the proposed control scheme is effective and robust under dynamic uncertainty and time-varying disturbances.

Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.

Nonuniform (h,k,μ,ν)-dichotomy with applications to nonautonomous dynamical systems

The paper develops a more general notion of dichotomy, referred to as “nonuniform (h,k,μ,ν)-dichotomy”. The new notion unifies most versions of existing dichotomy in the literature, includes them in one comprehensive mathematics, and reveals more dichotomic behaviors of dynamical systems. Then we show that any linear nonautonomous system admits a nonuniform (h,k,μ,ν)-dichotomy if it has an (h,k) Lyapunov exponent with different signs or a Lyapunov function with different growth rates in the stable and unstable subspaces of solutions. This implies that the nonuniform (h,k,μ,ν)-dichotomy arises naturally and exists widely in the linear systems. Using the general notation, new versions of robustness, Hartman–Grobman theorem, and stable invariant manifold theorem for nonautonomous dynamical systems in Banach spaces will be presented. Especially, these new results not only generalize the corresponding ones in the literature, but also characterize the influences of different growth functions in the stable and unstable subspaces and in the uniform and nonuniform parts of a linear system for solving the above problems. This improves the application range of dichotomy in the nonautonomous systems.

Beyond the Melnikov method: A computer assisted approach

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough’, as is the case in the classical Melnikov approach.

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un) stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et. al. (submitted), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show tha the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.

INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

Invariant manifold theory via generating maps

We present a synthetic approach to invariant manifold theorems, based upon the notion of a generating map. To cite this article: M. Chaperon, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The Lipschitzian core of some invariant manifold theorems

AbstractIn the Michael Herman Memorial Volume, we stated and proved some rather simple invariant-manifold theorems, having many old and new applications. This paper presents the final state of the Lipschitzian part of the theory: the results are put into a more general framework and new properties are established. Both the hypotheses and the proofs of all our statements are quite simple. Smoothness will be treated as in Chaperon [Stable manifolds and the Perron–Irwin method. Ergod. Th. & Dynam. Sys.24 (2004), 1359–1394] in a forthcoming book.

Projected Runge-Kutta methods for index 3 differential-algebraic equations near equilibria, periodic orbits and attracting sets

In the present paper, we analyse the geometric properties of projected Runge-Kutta methods for the solution of index 3 differential-algebraic equations in the Hessenberg form. We show that the geometric phase portrait is well reproduced under discretization in the vicinity of equilibria, periodic orbits or asymptotically stable invariant sets. The main tools are embedding techniques and an invariant manifold theorem which allow a reduction of the problem to the classical ordinary differential equation case.

The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.

GENERALIZED SYNCHRONIZATION, FREQUENCY-LOCKING AND PHASE-LOCKING OF COUPLED SINE CIRCLE MAPS

Applying invariant manifold theorem, we study the existence of generalized synchronization of a coupled system, with local systems being different sine circle maps. We specify a range of parameters for which the coupled system achieves generalized synchronization. We also investigate the relation between generalized synchronization, predictability and equivalence of dynamical systems. If the parameters are restricted in the specified range, then all the subsystems are topologically equivalent, and each subsystem is predictable from any other subsystem. Moreover, these subsystems are frequency locked even if the frequencies are greatly different in the absence of coupling. If the local systems are identical without coupling, then the widths of the phase-locked intervals of the coupled system are the same as those of the individual map and are independent of the coupling strength.

Existence of Inertial Manifolds for Partly Dissipative Reaction Diffusion Systems in Higher Space Dimensions

In this paper, we prove the existence of inertial manifolds for a partly dissipative reaction diffusion system of the form[formula]whereΩis a rectangular domain in R2or a cubic domain in R3. The proof is based on an Abstract Invariant Manifold Theorem for semiflows in a Hilbert space, which is proved by using the graph transform method. The Principle of Spatial Averaging also plays an important role in the proof.

Inertial manifolds for reaction diffusion equations in higher space dimensions

In this paper we show that the scalar reaction diffusion equation \[ u t = ν Δ u + f ( x , u ) , u ∈ R {u_t} = \nu \Delta u + f(x,u),\qquad u \in R \] with x ∈ Ω n ⊂ R n ( n = 2 , 3 ) x \in {\Omega _n} \subset {R^n}\quad (n = 2,3) and with Dirichlet, Neumann, or periodic boundary conditions, has an inertial manifold when (1) the equation is dissipative, and (2) f f is of class C 3 {C^3} and for Ω 3 = ( 0 , 2 π ) 3 {\Omega _3} = {(0,2\pi )^3} or Ω 2 = ( 0 , 2 π / a 1 ) × ( 0 , 2 π / a 2 ) {\Omega _2} = (0,2\pi /{a_1}) \times (0,2\pi /{a_2}) , where a 1 {a_1} and a 2 {a_2} are positive. The proof is based on an (abstract) Invariant Manifold Theorem for flows on a Hilbert space. It is significant that on Ω 3 {\Omega _3} the spectrum of the Laplacian Δ \Delta does not have arbitrary large gaps, as required in other theories of inertial manifolds. Our proof is based on a crucial property of the Schroedinger operator Δ + υ ( x ) \Delta + \upsilon (x) , which is valid only in space dimension n ≤ 3 n \leq 3 . This property says that Δ + υ ( x ) \Delta + \upsilon (x) can be well approximated by the constant coefficient problem Δ + υ ¯ \Delta + \bar \upsilon over large segments of the Hilbert space L 2 ( Ω ) {L^2}(\Omega ) , where υ ¯ = ( vol Ω ) − 1 ∫ Ω υ d x \bar \upsilon = {({\text {vol}}\Omega )^{ - 1}}\int _\Omega {\upsilon \;dx} is the average value of υ \upsilon . We call this property the Principle of Spatial Averaging. The proof that the Schroedinger operator satisfies the Principle of Spatial Averaging on the regions Ω 2 {\Omega _2} and Ω 3 {\Omega _3} described above follows from a gap theorem for finite families of quadratic forms, which we present in an Appendix to this paper.