Local Stable Manifolds Research Articles

Invariant manifolds and foliations for random differential equations driven by colored noise

In this paper, we prove the existence of local stable and unstable invariant manifolds for a class of random differential equations driven by nonlinear colored noise defined in a fractional power of a separable Banach space. In the case of linear noise, we show the pathwise convergence of these random invariant manifolds as well as invariant foliations as the correlation time of the colored noise approaches zero.

Existence of local stable manifolds for some nondensely defined nonautonomous partial functional differential equations

In this paper, we establish the existence of local stable manifolds for a semi-linear differential equation, where the linear part is a Hille–Yosida operator on a Banach space and the nonlinear forcing term [Formula: see text] satisfies the [Formula: see text]-Lipschitz conditions, where [Formula: see text] belongs to certain classes of admissible function spaces. The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.

Local stable and unstable manifolds for Anosov families

Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. They are time-dependent dynamical systems with hyperbolic behavior. In addition to presenting several properties and examples of Anosov families, in this paper we build local stable and local manifolds for such families.

Local stable manifolds for nonlinear planar fractional differential equations with order 1<α<2

This paper reports the local stable manifolds near hyperbolic equilibria for nonlinear planar fractional differential equations of order 1<α<2. By using several useful estimates of Mittag‐Leffler function and fractional calculus technique, we construct two suitable Lyapunov‐Perron operators and set up their fixed points as the desired stable manifolds. We further present a specific example to compute explicitly the corresponding stable manifold as the application.

On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

We first study the Massera problem for the existence of a τ−periodic solution for some nondensely defined partial differential equation, where the autonomous linear part satisfies the Hille‐Yosida condition and the delayed nonlinear part satisfies a locally Lipschitz condition. Second, inspired by an existing study, we prove in the dichotomic case, for τ=1, the existence‐uniqueness and conditional stability of the periodic solution. Moreover, we show the existence of a local stable manifold around such solution. Our theoretical results are finally illustrated by an application.

Local integral manifolds for nonautonomous and ill-posed equations with sectorially dichotomous operator

We show the existence and $ C^{k, \gamma} $ smoothness of local integral manifolds at an equilibrium point for nonautonomous and ill-posed equations with sectorially dichotomous operator, provided that the nonlinearities are $ C^{k, \gamma} $ smooth with respect to the state variable. $ C^{k, \gamma} $ local unstable integral manifold follows from $ C^{k, \gamma} $ local stable integral manifold by reversing time variable directly. As an application, an elliptic PDE in infinite cylindrical domain is discussed.

Principle of linearized stability and instability for parabolic partial differential equations with state-dependent delay

In this paper, the stability properties of a parabolic partial differential equation with state-dependent delay are investigated by the heuristic approach. The previous works [1,2] obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions, and resolved the problem of linearization for this equation. Here, we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation, and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution. By a biological application, we finally verify all hypotheses for an age structured diffusive model with state-dependent delay and consider its stability behavior.

Solutions to Partial Functional Differential Equations with Infinite Delays: Periodicity and Admissibility

Under some appropriate conditions, we prove the existence and uniqueness of periodic solutions to partial functional differential equations with infinite delay of the form $\dot {u}=A(t)u+g(t,u_{t})$ on a Banach space X where A(t) is 1-periodic, and the nonlinear term g(t, ϕ) is 1-periodic with respect to t for each fixed ϕ in fading memory phase spaces, and is φ(t)-Lipschitz for φ belonging to an admissible function space. We then apply the attained results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds

We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilibrium. An important component of our method is that we obtain mathematically rigorous lower bounds on the size of this neighborhood, as well as validated a posteriori error bounds for the polynomial approximation. In the second step we use a rigorous numerical integrating scheme to propagate the boundary of the local stable/unstable manifold as long as possible, i.e., as long as the integrator yields validated error bounds below some desired tolerance. The procedure exploits adaptive remeshing strategies which track the growth/decay of the Taylor coefficients of the advected curve. In order to highlight the utility of the procedure, we study the embedding of some...

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.

Disintegration of invariant measures for hyperbolic skew products

We study hyperbolic skew products and the disintegration of the SRB measure into measures supported on local stable manifolds. Such a disintegration gives a method for passing from an observable v on the skew product to an observable $$\overline v $$ on the system quotiented along stable manifolds. Under mild assumptions on the system we prove that the disintegration preserves the smoothness of v, first in the case where v is Holder and second in the case where v is $${\mathcal{C}^1}$$ .

High-order parameterization of (un)stable manifolds for hybrid maps: Implementation and applications

In this work we study, from a numerical point of view, the (un)stable manifolds of a certain class of dynamical systems called hybrid maps. The dynamics of these systems are generated by a two stage procedure: the first stage is continuous time advection under a given vector field, the second stage is discrete time advection under a given diffeomorphism. Such hybrid systems model physical processes where a differential equation is occasionally kicked by a strong disturbance. We propose a numerical method for computing local (un)stable manifolds, which leads to high order polynomial parameterization of the embedding. The parameterization of the invariant manifold is not the graph of a function and can follow folds in the embedding. Moreover we obtain a representation of the dynamics on the manifold in terms of a simple conjugacy relation. We illustrate the utility of the method by studying a planar example system.

Erratum to: Local stable manifold theorem for fractional systems

Erratum to: Local stable manifold theorem for fractional systems

Validated Computation of Heteroclinic Sets

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following an argument built on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map, and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of $\mathbb{R}^3$.

Dichotomy and periodic solutions to partial functional differential equations

We establish a sufficient condition for existence and uniqueness of periodic solutions to partial functional differential equations of the form $\dot{u}=A(t)u+F(t)(u_t)+g(t,u_t)$ on a Banach space $X$ where the operator-valued functions $t\mapsto A(t)$ and $t\mapsto F(t)$ are $1$ -periodic, the nonlinear operator $g(t,φ)$ is $1$ -periodic with respect to $t$ for each fixed $φ∈ {\mathcal{C}}:=C([-r,0],X)$ , and satisfying $\|g(t,φ_1)-g(t,φ_2)\|≤\varphi(t)\|φ_1-φ_2\|_C$ for $φ_1, φ_2∈ {\mathcal{C}}$ with $\varphi$ being a positive function such that $\sup_{t≥0}∈t_{t}^{t+1}\varphi(τ)dτ < ∞$ . We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family $(A(t))_{t≥ 0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

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.

On equilibrium states for partially hyperbolic horseshoes

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

Local Stable and Unstable Manifolds and Their Control in Nonautonomous Finite-Time Flows

It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-intervals is considered. With microfluidic flow control in mind, a methodology for manipulating these directions in any prescribed time-varying fashion by applying a local velocity shear is developed. The results are verified for both smoothly and discontinuously time-varying directions using finite-time Lyapunov exponent fields, and excellent agreement is obtained.

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.

Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh–Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.