Clark Robinson

The C 1 Closing Lemma, including Hamiltonians

An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C 1 diffeomorphisms to C 1 Hamiltonian vector fields.

Stability theorems and hyperbolicity in dynamical systems

We present an approach to proving structural stability and semi-stability theorems for diffeomorphisms and flows using the idea of shadowing an e-chain. We treat the cases when the whole manifold is hyperbolic and when the chain recurrent set is hyperbolic and the strong transversality condition is satisfied. The last two sections discuss the progress made on the converse to the structural stability theorem and a criterion for hyperbolicity. 0. Introduction. The first section presents the shadowing and stability results for a diffeomorphism near a single hyperbolic set. This section is very much in the spirit of C. Conley, [4] and [5]. The second section shows how these results can be carried over to several hyperbolic sets when the strong transversality condition is satisfied. The semi-stability theorem for diffeomorphisms is discussed in some detail in § 2. With this new proof the result carries over to flows as mentioned in § 3. The fourth section reviews the results of R. Mane and V. Pliss on the converse to the structural stability theorem. Finally, the last section discusses a criterion for hyperbolicity; namely, a linear bundle isomorphism is hyperbolic if the map on the base space is chain-recurrent and the zero section is an isolated invariant set. The results of § 1 through § 3 assume the sets are compact. We really only need uniform hyperbolicity and uniformly continuous derivatives. J. Hale has been asking how much of this theory carries over to functional differential equations and flows on infinite dimensions, [12]. In this context, see the papers of W. Oliva [24], D. Henry [ 13], and J. Montgomery [ 19]. An invariant set A C M is called hyperbolic for f: M-* M if there exist a continuous invariant splitting TM | A = E

Structural stability of C1 diffeomorphisms

Abstract In this paper we prove that if f is a C 1 diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C 2 diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C 1 flows in another paper.

Bifurcation to infinitely many sinks

This paper considers one parameter families of diffeomorphisms {Ft} in two dimensions which have a curve of dissipative saddle periodic pointsPt, i.e.Ftn(Pt)=Pt and |detDFtn(Pt)|<1. The family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofPt as the parameter varies throught0. Gavirlov and Silnikov proved that if the new homoclinic intersections are created nondegenerately att0, then there is an infinite cascade of periodic sinks, i.e. there are parameter valuestn accumulating att0 for which there is a sink of periodn [GS2, Sect. 4]. We show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately. We give computer evidence to show that this latter result is probably applicable to the Hénon map forA near 1.392 andB equal −0.3.Newhouse proved a related result which showed the existence of infinitely many periodic sinks for a single diffeomorphism which is a perturbation of a diffeomorphism with a nondegenerate homoclinic tangency. We give the main geometric ideas of the proof of this theorem. We also give a variation of a key lemma to show that the result is true for a fixed one parameter family which creates a nondegenerate tangency. Thus under the nondegeneracy assumption, not only is there a cascade of sinks proved by Gavrilov and Silnikov, but also a single parameter valuet* with infinitely many sinks.

Homoclinic bifurcation to a transitive attractor of Lorenz type

The author proves that a cubic differential equation in three dimensions has a transitive attractor similar to that of the geometric model of the Lorenz equations. In fact, what is proved is that such an attractor results if a double homoclinic connection of a fixed point with a resonance condition among the eigenvalues is broken in a careful way.

Sustained Resonance for a Nonlinear System with Slowly Varying Coefficients

J. Kevorkian [SIAM J. Appl. Math., 20 (1971), pp. 364–373; 26 (1974), pp. 638–669] studied resonance for a spinning reentry vehicle using a model system of ordinary differential equations with slowly varying coefficients. He and L. Lewin [SIAM J. Appl. Math., 35 (1978), pp. 738–754] gave formal multiple-time-scale expansions and numerical results to give a description of a mechanism for capture in sustained resonance. J. Sanders [SIAM J. Math. Anal., 10 (1979), pp. 1220–1243] studied these equations more rigorously using the method of averaging, but still could not prove the existence of sustained resonance. This paper continues the study of these equations using higher order averaging and the Melnikov method and shows rigorously that capture in sustained resonance does take place for some initial conditions. The Melnikov method measures the opening of a saddle connection for a small perturbation in terms of an integral. Since it has usually been applied to perturbations which depend periodically on time,...

Homoclinic bifurcation to a transitive attractor of Lorenz type, II

In this paper it is proven that there is a codimension two bifurcation of a double homoclinic connection of a fixed point with a resonance condition among the eigenvalues to a transitive attractor that is like that of the geometric model of the Lorenz equations. The two key parameters are the variation of the eigenvalues from resonance and the amount that the homoclinic connection is broken. Because of the need to work near resonance of two of the eigenvalues, one of the key steps in the proof is to calculate the Poincare–Dulac map past a fixed point in this situation. Also indicated is how bifurcation is realized for a specific cubic differential equation introduced by Rychlik, which is closely related to the Lorenz equations.

Homoclinic orbits and oscillation for the planar three-body problem

Sitnikov proved the existence of oscillation and capture for a special motion of the restricted three-body problem, m, = 0 [ 141. Alekseev made a systematic study of this situation [ 11. He also proved the same result for all nonzero masses. (See Section 5 below for a discussion of this case.) A good exposition of this example following lectures of Conley is contained in [9] using the stable manifold result for a degenerately hyperbolic closed orbit of McGehee [ 81. Easton and McGehee proposed a planar three-body example with negative energy which could (possibly) exhibit similar oscillation and capture [3]. While Sitnikov’s example has two degrees of freedom, this planar example, after all the integrals and symmetries are removed, has three degrees of freedom. They studied a model example which completely decouples when the third body is near infinity and proved oscillation and capture exist for this model example. Their work left the following three steps undone: (1) show the parabolic orbits form a submanifold for the real equations, (2) show the a-parabolic orbits and w-parabolic orbits are transverse in a strong sense (see the definition of a hyperbolic homoclinic orbit defined below), and (3) show that symbolic dynamics can be used to show oscillation and capture exist using only the fact that the binary asymptotically decouples and not that it completely decouples when the third body is near infinity. For negative energy h, as the third particle goes to infinity parabolically, the asymptotic motion of q = r2 rl is that of a two-body problem with energy h. After regularization the set of all two-body motions with energy h < 0 is the Hopf flow on the three sphere, S3. Let IVs(S3) (resp. wU(S”)) be the set of w-parabolic orbits (resp. w-parabolic orbits). Easton has now shown that Ws(S3) and wl((S3) are Lipschitz manifolds [2]. This paper proves they are real analytic manifolds and “Cm at infinity” (at S”)

TOPOLOGICALLY CROSSING HETEROCLINIC CONNECTIONS TO INVARIANT TORI

We consider transition tori of Arnold which have topologically crossing heteroclinic connections. We prove the existence of shadowing orbits to a bi-inflnite sequence of tori, and of symbolic dynamics near a flnite collec- tion of tori. Topological crossing intersections of stable and unstable manifolds of tori can be found as non-trivial zeroes of certain Melnikov functions. Our treatment relies on an extension of Eastons method of correctly aligned win- dows due to Zgliczy¶

Structural stability on manifolds with boundary

Abstract In application of dynamical systems, we are often interested in the motion in a compact subset of Euclidean space or possibly of a torus cross a Euclidean space. The natural way to study these problems is in a manifold with boundary, e.g., a disk. This paper gives sufficient conditions for a C1 flow ƒ t on a compact manifold with boundary, M, to be weakly structurally stable. In particular, if ƒ has a hyperbolic chain-recurrent set (Axiom A) and satisfies the transversality condition then it is weakly structurally stable. Since we impose no assumptions on the type of tangencies of trajectories with the boundary of M, we need to weaken the notion of conjugacy to allow a homeomorphism h from M into a collared manifold M′ which contains M in its interior. Such a homeomorphism allows us to bypass the singularity theory used by Percell and Sotomayor. A similar result for diffeomorphisms of manifolds with boundary is obtained. If the flow satisfies a quadratic external boundary condition, then it is possible to demand that the conjugacy preserve the boundary. These results also allow an infinite chain-recurrent set rather than the finite set of periodic orbits allowed in the previous results on manifolds with boundary.