Differentiable conjugacies of Morse-Smale diffeomorphisms

Stable and unstable manifolds for hyperbolic fixed points and sets have been well studied, see for instance [26]. Pesin developed a general theory of stable and unstable manifolds for diffeomorphisms corresponding to its non-vanishing Lyapunov exponents [62]. His work translated the linear theory of Lyapunov exponents (multiplicative ergodic theorem) into the non-linear theory of stable and unstable invariant manifolds. It was demonstrated that this theory provides a fundamental tool for the development of the smooth ergodic theory. Some extensions of Pesin’s result had been given [33, 79], and even in infinite dimensional spaces [56, 80]. More recently, Liu and Qian developed a rigorous theory of stable and unstable manifolds for ran-dom diffeomorphisms [51].There have been some partial results concerning invariant manifolds for endo-morphisms [69, 83, 80]. However, those results are not enough for the study of smooth ergodic theory of endomorphisms. In order to study the relationship between entropy and Lyapunov exponents, we will need more subtle results. The purpose of this chapter is to an unstable manifold theorem of this kind for non-invertible differentiable maps of finite dimensional manifolds by borrowing the techniques given in [51]. Readers familiar with [51] can easily see that when the inverse limit space is introduced, some situations are very close to those in [51]. Nevertheless, for the convenience of a larger group of readers who may not wish to involve any randomness, we still give a detailed presentation. The source of this chapter is mainly [100] except Section 6-9 which follow from [51, Chapter III].KeywordsLyapunov ExponentInvariant ManifoldUnstable ManifoldStable ManifoldAbsolute ContinuityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Two-manoeuvres transfers between LEOs and Lissajous orbits in the Earth–Moon system

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

Computer assisted error bounds for linear approximation of (un)stable manifolds and rigorous validation of higher dimensional transverse connecting orbits

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

We study the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter $$\nu $$. The eigenvalues of the linearized system are pure imaginary for $$\nu 0$$ (these are the same basic assumptions as found in the Hamiltonian–Hopf bifurcation theorem of the authors). For $$ \nu > 0 $$ the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for $$ \nu < 0 $$ there are no longer stable and unstable manifolds attached to the equilibrium. We study the evolution of these manifolds as the parameter is varied. If the sign of a certain term in the normal form is positive then for small positive $$\nu $$ the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. This happens at the Lagrange equilibrium point $${\mathcal {L}}_4$$ of the restricted three-body problem at the Routh critical value $$\mu _1$$. On the other hand if the sign of this term in the normal form is negative then for $$\nu =0$$ the stable and unstable manifolds persists and then as $$\nu $$ decreases from zero they detach from the equilibrium to follow a hyperbolic periodic solution.

In this chapter we want to describe and motivate some aspects of the theory of invariant manifolds which we will explore throughout the rest of this book, as well as give a brief survey of the field and the breadth of applications of invariant manifold methods and ideas throughout science and engineering. Roughly speaking, an invariant manifold is a surface contained in the phase space of a dynamical system that has the property that orbits starting on the surface remain on the surface throughout the course of their dynamical evolution in one or both directions of time; i.e., an invariant manifold is a collection of orbits which form a surface. If the surface has a boundary, then trajectories can leave the surface by crossing the boundary. Additionally, under certain conditions, the set of orbits which approach or recede from an invariant manifold M asymptotically in time are also invariant manifolds which are called the stable and unstable manifolds, respectively, of the invariant manifold. Moreover, the stable and unstable manifolds may be foliated or fibered by lower dimensional submanifolds having the property that trajectories starting on these fibers satisfy certain asymptotic growth rate conditions. Many of these features persist under perturbation, a property which is useful for the development of many types of global perturbation methods. We begin by considering areas of applications where ideas from invariant manifold theory have played an important role.

We establish rather general and simple theorems implying, among other things, the pseudo-(un)stable manifold theorem, Sternberg's theorem on smooth conjugacy between hyperbolic germs of maps or vector fields, and results of Fenichel, Hirsch, Pugh and Shub on existence, uniqueness and structural stability of stable or unstable manifolds at compact invariant manifolds. The Perron–Irwin approach via sequence spaces plays a crucial role.

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton--Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

The evolution of invariant manifolds in Hamiltonian–Hopf bifurcations

Arthroscopic and open management of dynamic scaphoid instability

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

The detailed dynamics of F+H2, and to a lesser extent H+H2, in the presence of laser fields are examined in order to establish the relationship between field-free unstable trajectories and the effect of external perturbations. Weak fields are found to alter the dynamics of individual unstable trajectories and, in particular, to induce transitions amongst the unstable trajectory manifold. As a result, the field has little effect, on the average, on gross collision properties. Intense fields substantially alter the qualitative stable/ unstable structure of phase space and hence the reaction probability. An interesting high field limit is also discussed.

The invariant manifold structures of the libration points for restricted three-body problem will provide the framework for understanding complex dynamical phenomena. These stable and unstable invariant manifold tubes associated to libration point orbits can be used to construct new low-energy spacecraft trajectories. So, in this paper, we will introduce the libration point and its invariant manifolds structure. First, we introduce the restricted three-body problem, libration point and Halo orbit. Then, we discuss and provide the stable and unstable invariant manifold tubes associated to libration point orbits based on manifolds and dynamical system theory. Finally, we apply these dynamical properties to design the capture trajectory in planetary exploration.

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