Stable Manifold Theorem Research Articles

Asymptotic stability of rarefaction wave and boundary layer for outflow problem on the viscous vasculogenesis model *

In this paper, we are concerned with the outflow problem on a simplified viscous vasculogenesis model in the half-line R+ . Firstly, we establish the global-in-time asymptotic stability of the rarefaction wave. Secondly, we obtain the unique existence and decay property of the boundary layer by using stable manifold theorem. Moreover, the asymptotic stability and convergence rates of solution towards boundary layer are obtained. The appearance of concentration makes the stationary problem more difficult than Navier–Stokes equations or Navier–Stokes–Poisson equations.

Distributed Gradient Flow: Nonsmoothness, Nonconvexity, and Saddle Point Evasion

The article considers distributed gradient flow (DGF) for multiagent nonconvex optimization. DGF is a continuous-time approximation of distributed gradient descent that is often easier to study than its discrete-time counterpart. The article has two main contributions. First, the article considers optimization of nonsmooth, nonconvex objective functions. It is shown that DGF converges to critical points in this setting. The article then considers the problem of avoiding saddle points. It is shown that if agents’ objective functions are assumed to be smooth and nonconvex, then DGF can only converge to a saddle point from a zero-measure set of initial conditions. To establish this result, the article proves a stable manifold theorem for DGF, which is a fundamental contribution of independent interest. In a companion article, analogous results are derived for discrete-time algorithms.

A Dynamical Theory for Singular Stochastic Delay Differential Equations I: Linear Equations and a Multiplicative Ergodic Theorem on Fields of Banach Spaces

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

Rotor imbalance suppression by optimal control

AbstractAn imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open‐loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. Asymptotic behavior of the control system is studied rigorously. By Łojasiewicz inequality, convergence of the optima as time towards a steady configuration is ensured. An explicit estimate of the convergence rate is given. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Moreover, a closed‐loop control strategy based on Reinforcement Learning is proposed. Numerical simulations have been performed, validating the theoretical results.

A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Building on results obtained in [ 21 ], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [ 20 ].

Some Qualitative Properties of Traveling Wave Fronts of Nonlocal Diffusive Competition-Cooperation Systems of Three Species with Delays

This paper is concerned with nonlocal diffusion systems of three species with delays. By modified version of Ikehara’s theorem, we prove that the traveling wave fronts of such system decay exponentially at negative infinity, and one component of such solutions also decays exponentially at positive infinity. In order to obtain more information of the asymptotic behavior of such solutions at positive infinity, for the special kernels, we discuss the asymptotic behavior of such solutions of such system without delays, via the stable manifold theorem. In addition, by using the sliding method, the strict monotonicity and uniqueness of traveling wave fronts are also obtained.

Analysis of Asymptotic Escape of Strict Saddle Sets in Manifold Optimization

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 22 October 2019Accepted: 18 June 2020Published online: 23 September 2020Keywordsmanifold optimization, projected gradient descent, strict saddles, stable manifold theorem, phase retrievalAMS Subject Headings58D17, 37D10, 65F10, 90C26Publication DataISSN (online): 2577-0187Publisher: Society for Industrial and Applied MathematicsCODEN: sjmdaq

Behavior of accelerated gradient methods near critical points of nonconvex functions

We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate. (The previous step encodes gradient information from earlier stages in the iterative process). We show by means of the stable manifold theorem that the heavy-ball method is unlikely to converge to strict saddle points, which are points at which the gradient of the objective is zero but the Hessian has at least one negative eigenvalue. We then examine the behavior of the heavy-ball method and other accelerated gradient methods in the vicinity of a strict saddle point of a nonconvex quadratic function, showing that both methods can diverge from this point more rapidly than the steepest-descent method.

Local stable manifold of Langevin differential equations with two fractional derivatives

In this paper, we investigate the existence of local center stable manifolds of Langevin differential equations with two Caputo fractional derivatives in the two-dimensional case. We adopt the idea of the existence of a local center stable manifold by considering a fixed point of a suitable Lyapunov-Perron operator. A local center stable manifold theorem is given after deriving some necessary integral estimates involving well-known Mittag-Leffler functions.

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.

Erratum to: Local stable manifold theorem for fractional systems

Erratum to: Local stable manifold theorem for fractional systems

Fractional dynamical systems: A fresh view on the local qualitative theorems

The aim of this work is to describe the qualitative behavior of the solution set of a given system of fractional differential equations and limiting behavior of the dynamical system or flow defined by the system of fractional differential equations. In order to achieve this goal, it is first necessary to develop the local theory for fractional nonlinear systems. This is done by the extension of the local center manifold theorem, the stable manifold theorem and the Hartman-Grobman theorem to the scope of fractional differential systems. These latter two theorems establish that the qualitative behavior of the solution set of a nonlinear system of fractional differential equations near an equilibrium point is typically the same as the qualitative behavior of the solution set of the corresponding linearized system near the equilibrium point. Furthermore, we discuss the stability conditions for the equilibrium points of these systems. We point out that, the fractional derivative in these systems is in the Caputo sense.

Center stable manifold for planar fractional damped equations

In this paper, we discuss the existence of a center stable manifold for planar fractional damped equations. By constructing a suitable Lyapunov–Perron operator via giving asymptotic behavior of Mittag–Leffler function, we obtain an interesting center stable manifold theorem. Finally, an example is provided to illustrate the result.

Integrability of continuous bundles

Abstract We give new sufficient conditions for the integrability and unique integrability of continuous tangent subbundles on manifolds of arbitrary dimension, generalizing Frobenius’ classical theorem for C 1 {C^{1}} subbundles. Using these conditions, we derive new criteria for uniqueness of solutions to ODEs and PDEs and for the integrability of invariant bundles in dynamical systems. In particular, we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

Stable manifolds results for planar Hadamard fractional differential equations

In this paper, we investigate local stable manifold for planar Hadamard fractional differential equations. By adopting Lyapunov–Perron operator approach and establishing new estimation of Mittag-Leffler function associated with Hadamard fractional derivative, we derive another interesting local stable manifold theorem for our problem. Finally, an example is given to illustrate our theoretical results.

Local stable manifold theorem for fractional systems

In this paper we prove existence of local stable manifold around a hyperbolic equilibrium point of a fractional system. We illustrate this result with an example and plot corresponding local stable manifold.

Contraction method and Lambda-Lemma

We reprove the $\lambda$-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf -- the local (un)stable manifold. These dynamical thickenings are key tools in our recent work [Web]. The present paper provides their construction.

Generating maps, invariant manifolds, conjugacy

The idea of generating functions and maps is presented, first in global symplectic geometry and then in the theory of invariant manifolds, as introduced by McGehee and Sander in 1996. Their result on the stable manifold theorem is generalised and simplified; the proofs no longer use any functional analysis. Then comes an original “non-autonomous” version of the previous results, yielding–besides Pesin’s invariant laminations–seemingly unrelated results on invariant manifolds and conjugacies, presented in the end after a basic example.

Generalized Nonuniform Dichotomies and Local Stable Manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.

On stable manifolds for planar fractional differential equations

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov–Perron operator. An example is provided to illustrate the result.