Local Stable Manifolds Research Articles

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

Center stable manifold for ground states of nonlinear Schrödinger equations with internal modes

We study the dynamics of solutions of nonlinear Schrödinger equation (NLS) near unstable ground states. The existence of the local center stable manifold around ground states and the asymptotic stability for the solutions on the manifold is proved. The novelty of our result is that we allow the existence of internal modes.

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14]] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed.

Chaotic Dynamics of Spatially Homogeneous Spacetimes

In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.

On entropy and pressure via preimage structure for random dynamical systems

In this paper, entropy, pressure and equilibrium states are investigated for noninvertible random dynamical systems via the preimage structure. For a continuous random dynamical system F on a compact metric space and an F-invariant measure μ, the folding entropy Fμ(F) and the preimage entropy hm,μ(F) are introduced, and it is shown that these two measure-theoretic versions of entropies coincide with each other if F has uniform separation of preimages. Some fundamental properties for hm,μ(F) such as Shannon-McMillan-Breiman Theorem, Kolmogorov-Sinai Theorem and upper semi-continuity are established. From the topological viewpoint, the preimage pressures Pm(F,⋅), Pp(F,⋅) and Pp˜(F,⋅) (resp. topological preimage entropy hm(F), hp(F) and hp˜(F)) are introduced via the preimage structure of single points. Some basic properties are discussed, and particularly, variational principles are obtained. For a C1 nondegenerate random partially hyperbolic dynamical system F on a closed Riemannian manifold, we propose the notions of metric stable entropy hm,μs(F) and stable pressures Pms(F,⋅) and Pps(F,⋅) (resp. topological stable entropies hms(F) and hps(F)) via the preimage structure of local stable manifolds. Several basic properties are obtained and variational principles are established particularly. At the end, as an application of variational principles, p-equilibrium states (resp. s-equilibrium states) and p-tangent functionals (resp. s-tangent functionals) are studied.

Symplectic Algorithms for Stable Manifolds in Control Theory

In this note, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in [Sakamoto-van~der Schaft, IEEE Transactions on Automatic Control, 2008]. Our algorithm includes two key aspects. The first one is to prove a precise estimate for radius of convergence and the errors of local approximate stable manifolds. The second one is to extend the local approximate stable manifolds to larger ones by symplectic algorithms which have better long-time behaviors than general-purpose schemes. Our approach avoids the case of divergence of the iterative sequence of approximate stable manifolds, and reduces the computation cost. We illustrate the effectiveness of the algorithm by an optimal control problem with exponential nonlinearity.

Smooth stable manifolds for the non-instantaneous impulsive equations with applications to Duffing oscillators.

In this paper, we present a theory of smooth stable manifold for the non-instantaneous impulsive differential equations on the Banach space or Hilbert space. Assume that the non-instantaneous linear impulsive evolution differential equation admits a uniform exponential dichotomy, we give the conditions of the existence of the global and local stable manifolds. Furthermore, -smoothness of the stable manifold is obtained, and the periodicity of the stable manifold is given. Finally, an application to nonlinear Duffing oscillators with non-instantaneous impulsive effects is given, to demonstrate the existence of stable manifold.

Local manifolds for non-autonomous boundary Cauchy problems: existence and attractivity

In this work we establish the existence of local stable and local unstable manifolds for nonlinear boundary Cauchy problems. Moreover, we illustrate our results by an application to a non-autonomous Fisher–Kolmogorov equation.

Complexity in a Hybrid van der Pol System

In this work, we study, from a numerical point of view, the dynamics of a specific hybrid map which is that of the kicked van der Pol system. The dynamics of this system is generated by a two-stage procedure: the first stage is the time-[Formula: see text] map of the vector field associated with the van der Pol equation, and the second stage is a translation. We propose a numerical method for computing local (un)stable manifolds of a given fixed point, which leads to high order polynomial parameterization of the embedding. Such a representation of the dynamics on the manifold is obtained in terms of a simple conjugacy relation and by constructing two contracting operators. We illustrate our techniques by plotting (for specific values of the parameters) both stable and unstable manifolds displaying homoclinic intersection. Furthermore, our numerical study reveals the presence of a strange attractor included in the closure of the unstable manifold of the fixed point.

A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

We address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

Topological and geometric hyperbolicity criteria for polynomial automorphisms of

AbstractWe prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

More on knots in Robinson's attractor

In an earlier paper the second author made a study of the knotted periodic orbits in a strange attractor for a set of differential equations in a paper by Clark Robinson. The attractor is modeled by a Lorenz-like template. It was shown that the knots and links are positive but need not be positive braids. Here we show that they are fibered, have positive signature, and that each knot-type appears infinitely often. We then construct a zeta type function that counts periodic orbits by the twisting of the local stable manifolds.

Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

This work presents and investigates a new chaotic system with eight terms. By numerical simulation, the two-scroll chaotic attractor is found for some certain parameters. And, by theoretical analysis, we discuss the dynamical behavior of the new-type Lorenz-like chaotic system. Firstly, the local dynamical properties, such as the distribution and the local stability of all equilibrium points, the local stable and unstable manifolds, and the Hopf bifurcations, are all revealed as the parameters varying in the space of parameters. Secondly, by applying the way of Poincaré compactification inℝ3, the dynamics at infinity are clearly analyzed. Thirdly, combining the dynamics at finity and those at infinity, the global dynamical behaviors are formulated. Especially, we have proved the existence of the infinite heteroclinic orbits. Furthermore, all obtained theoretical results in this paper are further verified by numerical simulations.

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 ].

Minimal collision arcs asymptotic to central configurations

<p style='text-indent:20px;'>We are concerned with the analysis of finite time collision trajectories for a class of singular anisotropic homogeneous potentials of degree <inline-formula><tex-math id="M1">\begin{document}$ -\alpha $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M2">\begin{document}$ \alpha\in(0,2) $\end{document}</tex-math></inline-formula> and their lower order perturbations. It is well known that, under reasonable generic assumptions, the asymptotic normalized configuration converges to a central configuration. Using McGehee coordinates, the flow can be extended to the <i>collision manifold</i> having central configurations as stationary points, endowed with their stable and unstable manifolds. We focus on the case when the asymptotic central configuration is a global minimizer of the potential on the sphere: our main goal is to show that, in a rather general setting, the local stable manifold coincides with that of the initial data of minimal collision arcs. This characterisation may be extremely useful in building complex trajectories with a broken geodesic method. The proof takes advantage of the generalised Sundman's monotonicity formula.

Preimage pressure, stable pressure and equilibrium states

In this paper, the pressure of a potential via the dynamics of preimage structure are investigated for noninvertible systems and the properties for the related equilibrium states are considered. For a continuous map on a compact metric space, a notion of preimage pressure via the preimages of single points is introduced, and then some fundamental properties are considered and a variational principle is established for the system with uniform separation of preimages. For a C1-partially hyperbolic endomorphism on a closed Riemannian manifold, a notion of stable pressure via the preimages of local stable manifolds is introduced and a corresponding variational principle is established. Moreover, equilibrium states and tangent functionals for pointwise preimage pressures and stable pressures are also considered. By describing the set of equilibrium states and tangent functionals and a sense of continuous dependence of equilibrium states with respect to the potential, we prove that the continuous potentials with unique equilibrium state form a dense Gδ set. In addition, for any finite collection of ergodic measures, we can find a continuous potential such that its set of equilibrium states contains the given set.

Periodic Solutions to Partial Neutral Functional Differential Equations in Admissible Spaces on a Half-Line

We prove the existence and uniqueness of periodic solutions to partial neutral function differential equation of the form $\frac {\partial Fu_{t}}{\partial t}= A(t)Fu_{t} +g(t,u_{t}),~t\in [0,\infty )$, on a Banach space X, where the operator-valued function t↦A(t) and the nonlinear delay operator g(t,v) are both T-periodic with respect to t, whereas g is φ-Lipschitz with respect to v for φ belonging to an admissible space. Then, in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy, we apply our abstract results to study the existence, uniqueness of periodic solutions. We also prove the conditional stability, and the existence of local stable manifolds around such periodic solutions.

Validated computations for connecting orbits in polynomial vector fields

In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simultaneously solves for the local (un)stable manifolds and the orbit which connects these. We obtain explicit rigorous control on the distance between the numerical approximation and the heteroclinic orbit. Transversality of the stable and unstable manifolds is also proven.

Global center stable manifold for the defocusing energy critical wave equation with potential

In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi, 0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It is constructed in a first step locally around any solution scattering to $\phi$, which might be very far away from $\phi$ in the $\dot{H}^1\times L^2(\mathbb{R}^3)$ norm. In a second crucial step a no-return property is proved for any solution which starts near, but not on the local manifolds. This ensures that the local manifolds form a global one. Scattering to an unstable steady state is therefore a non-generic behavior, in a strong topological sense in the energy space. This extends our previous result [18] to the nonradial case. The new ingredients here are (i) application of the reversed Strichartz estimate from [3] to construct a local center stable manifold near any solution that scatters to $(\phi, 0)$. This is needed since the endpoint of the standard Strichartz estimates fails nonradially. (ii) The nonradial channel of energy estimate introduced by Duyckaerts-Kenig-Merle [14], which is used to show that solutions that start off but near the local manifolds associated with $\phi$ emit some amount of energy into the far field in excess of the amount of energy beyond that of the steady state $\phi$.