Local Stable Manifolds Research Articles

Invariants of twist-wise flow equivalence

Twist-wise flow equivalence is a natural generalization of flow equivalence that takes account of twisting in the local stable manifold of the orbits of a flow. Here we announce the discovery of two new invariants in this category.

An invariant of basic sets of Smale flows

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.

Stochastic stability of Bernoulli toral linked twist maps of finite and infinite entropy

AbstractWe construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system

This is a further study of the set of homoclinic solutions (i.e., nonzero solutions asymptotic to 0 as ¦x¦→∞) of the reversible Hamiltonian systemuiv +Pu″ +u−u2=0. The present contribution is in three parts. First, rigorously for P≤ −2, it is proved that there is a unique (up to translation) homoclinic solution of the above system, that solution is even, and on the zero-energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifolds. WhenP=−2 the origin is a node on its local stable and unstable manifolds, and whenP∈(−2,2) it is a focus. Therefore we can infer, rigorously, from the discovery by Devaney of a Smale horseshoe in the dynamics on the zero energy set, there are infinitely many distinct infinite families of homoclinic solutions forP∈(−2, −2+e) for somee>0. Buffoni has shown globally that there are infinitely many homoclinic solutions for allP∈(−2,0], based on a different approach due to Champneys and Toland. Second, numerically, the development of the set of symmetric homoclinic solutions is monitored asP increases fromP=−2. It is observed that two branches extend fromP=−2 toP=+2 where their amplitudes are found to converge to 0 asP ↗ 2. All other symmetric solution branches are in the form of closed loops with a turning point betweenP=−2 andP=+2. Numerically it is observed that each such turning point is accompanied by, though not coincident with, the bifurcation of a branch of nonsymmetrical homoclinic orbits, which can, in turn, be followed back toP=−2. Finally, heuristic explanations of the numerically observed phenomena are offered in the language of geometric dynamical systems theory. One idea involves a natural ordering of homoclinic orbits on the stable and unstable manifolds, given by the Horseshoe dynamics, and goes some way to accounting for the observed order (in terms ofP-values) of the occurrence of turning points. The near-coincidence of turning and asymmetric bifurcation points is explained in terms of the nontransversality of the intersection of the stable and unstable manifolds in the zero energy set on the one hand, and the nontransversality of the intersection of the same manifolds with the symmetric section in ℝ4 on the other. Some conjectures based on present understanding are recorded.

A Bernoulli toral linked twist map without positive Lyapunov exponents

The presence of positive Lyapunov exponents in a dynamical system is often taken to be equivalent to the chaotic behavior of that system. We construct a Bernoulli toral linked twist map which has positive Lyapunov exponents and local stable and unstable manifolds defined only on a set of measure zero. This is a deterministic dynamical system with the strongest stochastic property, yet it has positive Lyapunov exponents only on a set of measure zero. In fact we show that for any map g in a certain class of piecewise linear Bernoulli toral linked twist maps, given any epsilon > 0 there is a Bernoulli toral linked twist map g' with positive Lyapunov exponents defined only on a set of measure zero such that g' is within epsilon of g in the d metric.

On the local qualitative behavior of differential-algebraic equations

A theoretical framework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equilibrium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a given equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that these theorems are stated in terms of the given problem and not in terms of the corresponding vector field.

Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < ∞, whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincare section on the energy surface. In general 0 ≤l ≤n of the separatrices will be available to the Poincare section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.

The Behavior of Finite Element Solutions of Semilinear Parabolic Problems Near Stationary Points

The qualitative behavior of spatially semidiscrete finite element solutions of a semilinear parabolic problem near an unstable hyperbolic equilibrium $\overline u $ is studied. It is shown that any continuous trajectory is approximated by an appropriate discrete trajectory, and vice versa, as long as they remain in a sufficiently small neighborhood of $\overline u $. Error bounds of optimal order in the $L_2 $ and $H^1 $ norms hold uniformly over arbitrarily long time intervals. In particular, the local stable and unstable manifolds of the discrete problem converge to their continuous counterparts. Therefore, the discretized dynamical system has the same qualitative behavior near $\overline u $ as the continuous system.

Intersection properties of invariant manifolds in certain twist maps

We consider the spaceN ofC 2 twist maps that satisfy the following requirements. The action is the sum of a purely quadratic term and a periodic potential times a constantk (hereafter called the nonlinearity). The potential restricted to the unit circle is bimodal, i.e. has one local minimum and one local maximum. The following statements are proven for maps inN with nonlinearityk large enough. The intersection of the unstable and stable invariant manifolds to the hyperbolic minimizing periodic points contains minimizing homoclinic points. Consider two finite pieces of these manifolds that connect two adjacent homoclinic minimizing points (hereafter called fundamental domains). We prove that all such fundamental domains have precisely one point in their intersection (the Single Intersection theorem).In addition, we show that limit points of minimizing points are recurrent, which implies that Aubry Mather sets (with irrational rotation number) are contained in diamonds formed by local stable and unstable manifolds of nearby minimizing periodic orbits (the Diamond Configuration theorem). Another corollary concerns the intersection of the minimax orbits with certain symmetry lines of the map.

Asymptotic Behavior of Stable Manifolds

The relation between local stable manifolds of an ordinary differ- ential equation and its discretization is studied. We show that a local stable manifold of a hyperbolic fixed point of an ordinary differential equation is the limit of local stable manifolds of the same fixed point of its discretizations as the discretization parameter h > 0 approaches 0.

Discretization in the method of averaging

Let f : R × R m ¯ × R → R m ¯ , f = f ( ε , x , t ) f:R \times {R^{\overline m }} \times R \to {R^{\overline m }},f = f(\varepsilon ,x,t) be a C 2 {C^2} -mapping 1 1 -periodic in t t having the form f ( 0 , x , t ) = A x + o ( | x | ) f(0,x,t) = Ax + o(|x|) as x → 0 x \to 0 where A ∈ L ( R m ¯ ) A \in \mathcal {L}({R^{\overline m }}) has no eigenvalues with zero real parts. We study the relation between local stable manifolds of the equation \[ x ′ = ε f ( ε , x , t ) , ε &gt; 0 is small x’ = \varepsilon f(\varepsilon ,x,t),\varepsilon &gt; 0{\text {is}}\;{\text {small}} \] and of its discretization \[ x n + 1 = x n + ( ε / m ) f ( ε , x n , t n ) , t n + 1 = t n + 1 / m , {x_{n + 1}} = {x_n} + (\varepsilon /m)f(\varepsilon ,{x_n},{t_n}),{t_{n + 1}} = {t_n} + 1/m, \] where m ∈ { 1 , 2 , … } = N m \in \{ 1,2, \ldots \} = \mathcal {N} . We show behavior of these manifolds of the discretization for the following cases: (a) m → ∞ , ε → ε ¯ &gt; 0 m \to \infty ,\varepsilon \to \overline \varepsilon &gt; 0 , (b) m → ∞ , ε → 0 m \to \infty ,\varepsilon \to 0 , (c) m → k ∈ N , ε → 0 m \to k \in \mathcal {N},\varepsilon \to 0 .

Asymptotic behavior of stable manifolds

The relation between local stable manifolds of an ordinary differential equation and its discretization is studied. We show that a local stable manifold of a hyperbolic fixed point of an ordinary differential equation is the limit of local stable manifolds of the same fixed point of its discretizations as the discretization parameter h &gt; 0 h &gt; 0 approaches 0.

Orbites h�t�rocliniques au voisinage d'une bifurcation de Poincar� d�g�n�r�e pour les champs de vecteurs de ?2

We consider a ℝ2-ordinary differential equation, where the fixed point (0, 0) presents a degenerate Poincare-bifurcation of resonancek(=2k′) and dominanced(=k−1). We prove the existence of a 2-dimensional linear manifoldV in the parameter space. Λ, on which the perturbed dominant differential system (SD) possesses heteroclinic orbits between fixed points. The numerical continuation of the local stable or unstable manifolds of the saddle fixed points shows that for any neighborhood, in Λ, of a point ofV corresponding to a saddle heteroclinic orbit, there exists only one stable (resp. unstable) periodic orbit close to the stable — in the Andronov sense [1]-(resp. unstable) heteroclinic orbit. Applications are given fork=4 andk=6.

Formation of a fractal basin boundary in a forced oscillator

The question of how a fractal basin boundary arises in the sinusoidally forced Duffing's equation is considered. The author describes how the backwards system flow deforms a local stable manifold into the fractal boundary. Parts of the boundary are labeled in a way related to their time of formation. The truncated fractal boundary produced by a burst of sinusoidal forcing is briefly considered. The approach supplements the insights provided by the usual Poincare map techniques.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Linearization principle for a toroidal Hall current plasma with viscosity and resistivity

In this paper the stability analysis of an incompressible toroidal Hall Current plasma with resistivity and viscosity on the basis of the linearization of the governing equations and boundary condition is rigorously justified. A nonlinear local existence theorem for an initial-boundary value problem is first proved, and local stable and unstable invariant manifolds of a nonlinear resolving operator are then constructed. It is shown that linear stability implies nonlinear stability and global existence, and linear instability implies nonlinear instability in some sense.

Inclination lemmas with dominated convergence

Inclination lemmas (λ-lemmas) serve as tools for the investigation of dynamical systems in the neighborhood of saddle points. They assert convergence of inclinations (slopes of tangents) when the system shifts a given transversal to the stable or unstable manifold towards equilibrium. We derive estimates of the speed of this convergence, for preimagesg −k (H), k∈N, of a transversalH to the unstable manifold,g is assumed to be aC 2-map in a Banach space, not necessarily reversible, with a saddle pointx=0 and already normalized so that local stable and unstable manifolds are contained in linear spaces. The estimates are needed particularly in a study of nonlocal bifurcation-from heteroclinic to periodic solutions of the second kind-for parameterized functional differential equations $$\dot x(t) = ah(x(t - 1))$$ which describe phase-locked loops

Stable manifolds for hyperbolic sets

turns out that they STABLE MANIFOLDS AND HYPERBOLIC SETS &#39; MORRIS W. I-{HIRSCH AND CHARLES c. PUGH - 0. Introduction. Let U be an open set in a smooth manifold M and f &#39;.U —» M a C‘ map. A fixed point x of f is hyperbolic if the derivative ’I;f:M,, —g M, is an isomorphism and its spectrum is separated by the unit circle. If &#39;I&#39;= 7}}; this means that M, has a unique splitting E, x E, under Tsuch that T|E, is expanding and T|Ez is contracting. That is, for suitable equivalent norms on E 1 and E2, m3q{llT_&#39;lE1ll- qTlE2q} l 1- . The classical stable manifold theory says that this convenient behavior of T; f is reflected in the behavior off in a neighborhood Vof x: there is a submanifold W‘ of M tangent to E, at x such that , - W’n V= {ye I/[lim(f|V)qy = x},_ llq® there is also a submanifold W” tangent to E, such that W‘ n V= {ye V|lim(f|V)“&#39;y = x}. See for example Kelley [1, Appendix], [15] and [14], which contains further references. _ We call W‘ and W‘ local stable and unstable manifolds off at x, respectively. It enjoy the same diflerentiability as j; and if f is C‘ they depend continuously on f in the C‘ topologies. &#39; &#39; . _ For technical reasons we allow M to be an infinite dimensional manifold modelled on a Banach space. _ i &#39; _ The notion of hyperbolic fixed point can be generalized to that of a hyperbolic set A c: U. ‘This means that f (A) = A, and TAM (the tangent bundle of M over A) has an invariant splitting E1 GB B, such that &#39;If|E, is expanding and &#39;.l]&#39;|E1 is contracting. (For this theory M is assumed finite dimensional and A compact, although generalizations are possible.) In Smale’s theory of Q-stability, and related topics [12], [13], “generalized stable manifold theoremq plays a key role: there is a neighborhood Vol‘ A, and submanifolds W’(x), Wq(x) tangent to E,(x) and E ,(x) respectively for each x e A, such that W‘(x) = {ye Vllim dmvry. (flV)‘x)‘= 0}. lqII) Wq(x) = {ye Vl1imd((f|V)q‘y.(flV)q&#39;fc)_= 0}- l“Q l3J