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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 > 0 h > 0 approaches 0.

Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross-sectional resultants of force and moment. Using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite-dimensional manifold. In particular, the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations. Moreover, the set of semi-inverse solutions can be analyzed locally.

The unstable manifold of a saddle point of the Henon mapping is constructed analytically via a contraction mapping, for a range of parameter values where the second fixed point is a stable node. One invariant piece of this manifold connects the saddle with the second fixed point. Rigorous error bounds are derived for the each step of the iterative procedure. It is demonstrated that an algebraic approximation with known accuracy can be given of the unstable manifold.

We use the Leray-Schauder Fixed Point Theorem to prove the existence of an analytic fixed point for the period doubling accumulation renormalization operator. Our argument does not, however, show that the linearization of the renormalization operator at this fixed point is hyperbolic.

Sullivan's geometric measure on a geometrically finite hyperbolic manifold is shown to satisfy a mean ergodic theorem on horospheres and through this that the geodesic flow is Bernoulli.

In this article we prove, for a differentiable vector field or a diffeomorphism on a smooth manifold, that the set of points such that the semitrajectories issuing from them approach a particular semitrajectory at a given exponential rate, constitute a differentiable submanif old, provided the differential of the flow has a certain similar behavior on that trajectory. (See Theorem 1 below, for a precise statement). In particular, the stable manifold theorem for hyperbolic sets ([3], [6, XI]) follows as a corollary. Although we only consider the Crease, the same methods, which are essentially classical ([2, Ch. XIII]), could be applied to obtain higher differentiability properties. Since I have not seen in the literature this type of results for points which are neither equilibrium nor periodic points, and on account of [6, XI-8], I thought that their publication might not be entirely devoid of interest. 1* Terminology and notation are standard. If X is a differentiable vector field on a smooth manifold M, will always denote the corresponding flow, and φt the diffeomorphism x —> φ(x, t), xeM, te R. For brevity, we shall sometimes write x(t) or y(t) instead of φ(x, t) or φ(y, t) respectively.

We study the behavior of solutions of the perturbed system \[({\text{N}})\qquad z'(t)Az(t) + B * z(t) + Hz(t),\quad 0 \leqq t 0$, $1 \leqq N < \infty $, where each root $s_j $ has multiplicity $m_j $, $j = 1, \cdots ,N$, and (E) has no roots $s_j $ with $\operatorname{Re} s_j = 0$. The principal result is a stable manifold theorem for (N...

The Generalized Ginzburg-Landau equations, introduced by one of us (H.H.), are considered in a simplified version to clarify their relation to the center manifold theorem.

We have obtained escape-speed diagrams for a conservative one-parameter quadratic mapping of the plane to itself (the Cremona Map). The escape speed of orbits appears to have a fractal distribution (in the plane) which is related to the eigenvectors of the mapping in the vicinity of the unstable fixed point. We suggest that this phenomenon is due to stability enhancements caused by the stable manifold as it traverses the plane. Since our charts allow classification of regions in the plane by their escape speed, the method actually provides a means of tracking the stable manifold directly. Numerical results indicate that the sections of the manifolds from longer period points extending into the escaping region accumulate onto the stable manifold of the unstable fixed point. An overall stability analysis as the parameter of the mapping is varied shows a remarkable resemblance to the corresponding diagram for the Restricted Three Body Problem, which we discuss briefly.

In this paper we prove a stable manifold theorem for noncompact solution curves of an ordinary differential equation and apply the theorem to establish the existence of two-dimensional stable and unstable manifolds for a particular solution curve of the Rikitake equations. Some information on the configuration of these surfaces in phase space is obtained yielding a rough description of the oscillatory nature of typical solutions.

We develop a global perturbation technique similar to that of Melnikov [1963] for detecting the presence of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations. Our techniques are more general in that they apply to systems undergoing large amplitude excitation at low frequencies and to systems undergoing quasiperiodic excitation. We discuss the dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and the recent results concerning orbits homoclinic to tori obtained by Wiggins [1986] and Meyer and Sell [1986] . We illustrate the method with an example of a parametrically excited pendulum. Additionally, we show how our results are complementary to a result of Arnold [1962] concerning the infinite time behavior of adiabatic invariants in Hamiltonian systems. We illustrate this by using our method to prove the existence of Smale horseshoes for a pendulum whose length undergoes a slow periodic variation.

SynopsisIt is established that under certain restrictions the solution u of the characteristic initial value problem uxy+g(x, y)u = 0, u(x, 0) = p(x) and u(0, y) = q(y), where p(x) > 0 and q(y) > 0, in [0, ∞) x [0, ∞) changes sign along a monotonic decreasing curve which is asymptotic to the axes.

The unstable manifold of the origin becomes homoclinic orbit when r = rho. For r > rho, the two-dimensional stable manifold of the origin develops an inhomogeneous structure in local regions of the phase space. Due to a convoluted structure of only one half of this stable manifold, the set remains R2 in some local regions, whereas in other regions it locally becomes R2 × S(r), where S(r) is an uncountable, but non-Cantor set. S(r) can be put into correspondence with the infinite set of unstable periodic points of the logistic map. The open intervals between the points of S(r) are connected to form the basins of attraction, B±, of the stable fixed points C±. The future itinerary of the trajectories which start in these intervals can be put into correspondence with any semi-infinite Bernoulli sequence. The topological structure of the basins B± is determined, including the spatial ordering of the Bernoulli sequences. This is illustrated in more detail for a particular value of r. It is shown that the spatial ordering of these itineraries can be produced by a series of maps, for all 0 < r < rt (where C± become unstable). The α-limit set (i.e., the limit set of (x(t), y(t), z(t)) as t → – ∞) of points on the stable manifold of the origin, Ws, contains an infinite number of unstable periodic orbits. As r approaches a value rhe 24.06 (in the case σ = 10, b = 8/3; Kaplan and Yorke [2]) the unstable manifold of the origin, Wu, becomes heteroclinic to the simplest of these unstable periodic orbits (the ω-limit sets of Wu± are the periodic orbits nearest C±). The topology of Ws is studied only for r < rhe in the present study.

where d is any metric on M compatible with its topology. If x is a hyperbolic periodic point of f, the stable manifold theorem[7] implies that then W”(x) and W”(x) are 1-1 immersed Euclidean spaces. An f f Diff M is Morse-Smale iff: (1) Q(f) is finite, and hence equal to the set of periodic points. (2) The periodic points are hyperbolic. (3) (Transversal intersection condition) For each pair of periodic points p, 4, W’(p) and W”(q) have transversal intersection. We will need the following special case of the fin-stability theorem due to J. Palis [8].

The stable theory (which allows connected sums with S 2 × S 2 ) is unified and extended using current 4-manifold techniques. Principal new results are a stable 5-dimensional s -cobordism theorem, and the fact that 1-connected smooth 4-manifold pairs stably have handle decompositions with no 1-handles.

The dynamical systems herein are Morse-Smale diffeomorphisms and flows on C ∞ {C^\infty } compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form K exp ⁡ ( − D N ) K\;\exp ( - DN) , where D D may be any number smaller than C = min p { 1 / m log Jac D P f m | W u ( P ) } C = {\min _p}\{ 1/m\;\log \;\operatorname {Jac}\;{D_P}\,{f^m}|{W^u}(P)\} . Here the minimum is taken over all nonsink P P in the nonwandering set of f f , and m m is the period of P P . We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.

Special functions are indicated that describe the smoothness asymptotics of the fundamental solution of a strictly hyperbolic equation in a neighborhood of the wave front set (assuming finite multiplicity of the critical points of the generating function). The special functions are given in the form of integrals of closed differential forms over (generally speaking, relative) homology classes of a complex manifold that depends on a point outside the wave front set. Bibliography: 24 titles.

Detailed structure of the attracting set of the piecewise linear Henon mapping (x,y)→(1−a|x|+by,x) with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that (1) the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and (2) the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A. Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.

Iff is a C1 + ɛ diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products.

For a manifold which is moving and changing with time, consider some numerical property which at each instant is equal to an integral over the manifold. We derive a general expression for the time rate of change of this integral. Corollaries include a precise general form of Crofton's boundary theorem, de Hoff's interface displacement equations (with some new extensions) and a theorem in fluid mechanics.

It is not known whether there exist oscillatory and capture orbits in the planar 3-body problem. Sitnikov [6] proved such orbits exist for the restricted 3-body problem. Alekseev [I] extended this work and related it to the existence of homoclinic orbits. McGehee and Easton ]3] studied a model problem closely related to the planar 3-body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3-sphere. The goal of this paper is to construct an invariant 3-sphere “at infinity” in the planar 3-body problem and to prove that the stable and unstable manifolds of this sphere are Lipschitz manifolds. To complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3-sphere intersect each other. This is a difficult question which is not treated here. However, using methods of Melnikov and using one of the masses as a perturbation parameter such an investigation might be possible. The stable manifold theorem which is proven here is not standard since the invariant 3-sphere is not hyperbolic. Geometric methods of McGehee [ 4 ] are extended and applied to obtain the theorem. A special case of the system of equations that we study has the form J, =x: dY)lX, + E,(x, Y)L f, = 4 dY>l-x* + E,(x, Y)L (0.1) 4’ =

In applications of the Internal Model Principle of control theory to nonlinear differential systems, a semistable-center-unstable invariant manifold plays an important role. The existence of this manifold has been known for some time: in this paper its smoothness is related to the location of the spectrum of the linearization of the system at the critical element.

On presente une famille de solutions homothetiques de l'equation pour une courbe planaire ∂X/∂τ=KN et on demontre l'existence de varietes non lineaires stables et instables autour de telles solutions

The deficiencies of the theories that characterize the maximal compact invariant set of T as asymptotically stable, and that some iterate of T has a fixed point are discussed. It is shown that this fixed point condition is always satisfied for condensing and local dissipative T. Applications are given to a class of neutral functional differential equations.

Perturbations of a dynamical system are defined and the behavior of compact asymptotically stable sets under these perturbations is determined. The occurrence of critical points in a perturbed planar dynamical system is also investigated. In [1] it is shown that if C is an asymptotically stable cycle of a planar dynamical system 7r and if 7ri is a net of planar dynamical systems which converges to 7r, then there are limit cycles Ci of ri such that Ci-*C. This paper presents a similar result in a more general setting for perturbed asymptotically stable sets. If irj is a net of dynamical systems which converges to a dynamical system rr and if M is a compact asymptotically stable set of vr, then eventually there are asymptotically stable sets Mi of 7r arbitrarily close to M. Moreover, if M is invariant with respect to all ri, then Mi-*M. R, R+, and Rwill denote the reals, nonnegative reals, and nonpositive reals respectively. A dynamical system vr on a topological space X is a mapping of Xx R onto X which satisfies the following three conditions (where xITt= ir(x, t)): (i) 7r is continuous in the product topology. (ii) x7rO=x for each x E X. (iii) (x7rt)vrs=xr(t+s) for each x E X and s, t E R. If A c X and B cR, then ATrB will denote the set {x7rr: x E A, t E B}. L+(x) and L-(x) will denote the positive limit set of x and the negative limit set of x respectively. A subset M of X is called an (negative) attractor if there is a neighborhood U of M such that (L-(x)) L+(x) c M for every x E U. If M is an (negative) attractor, then (A-(M)) A+(M) will denote the largest such neighborhood. A subset S of X is called a section with respect to I7 iff (S7Tt) nS= 0 for all t$O. In a topological space X it is possible to define limits of nets of subsets Received by the editors April 12, 1971 and, in revised form, January 7, 1972. AMS 1970 subject classifications. Primary 34C35, 34D10; Secondary 34C05.

In [l], Hirsch and Pugh announce a theorem on the stable manifold of a “nearly fixed” point of a hyperbolic map of a Banach space into itself. (Details will be given in [2].) The object of this note is to give a simple proof of their theorem, following the proof in [3, Exercise 5.4, pp. 241, 566-5671, for the case of a fixed point in Euclidean space. By weakening “hyperbolicity” slightly, it will be seen that there is no loss of generality in supposing that the “nearly fixed” point is actually a “fixed” point. This weakening of hyperbolicity also facilitates the handling of parameters and proofs of differentiability. Our results concern, not the iterates T 0 ... 0 T of a fixed map T but, more generally, the products T, 0 T,,-, o .*. 0 Tl of a given sequence of maps Tl , T, ,... . This generalization also has methodical advantages. The “existence and continuity” proof is suggested by a procedure of Hartman and Wintner [4, 51 in the asymptotic theory of ordinary differential equations, and employed by Coffman [6] in dealing with difference equations in a context similar to the situation here. “ Differentiability” will be deduced from “continuity and continuous dependence on parameters,” as in Hadamard’s proof for the smooth dependence of solutions of ordinary differential equations on initial conditions and parameters; cf. [3, pp. 95-IOO]. The main results are stated in Section 3; see also Section 7. An appendix deals with stable manifolds of Anosov diffeomorphisms.

Some two-dimensional nonlinear systems with an irregular singularity at infinity are investigated. Properties of their solutions on and outside a one-dimensional stable manifold are studied. Representations for solutions on the manifold are derived in the form of one-parameter exponential series. It is shown how solutions not tending to zero at infinity deviate from the stable manifold.

Proof of Kupka and Smale approximation theorem concerning differential equations defined on compact manifold