Classification of nonsingular four-dimensional flows with a untwisted saddle orbit

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott–Ruelleresonant states associated with Anosov and Morse–Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse–Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

CONTENTS Introduction Chapter I. Development of the Poincaré-Bendixson theory for flows on closed two-dimensional manifolds § 1. Possible types of trajectories. Singular and non-singular trajectories § 2. Classification of the connected components filled out by the non-singular trajectories Chapter II. Topological classification of flows S 3. The scheme of a flow on a sphere. Topological classification of Morse-Smale flows on two-dimensional manifolds § 4. The Poincaré rotation number of a flow on a torus and its properties. Transitive and singular flows on a torus § 5. The rotation homotopy class of half-trajectories of flows on closed orientable two-dimensional manifolds and its properties § 6. Necessary and sufficient conditions for the topological equivalence of transitive flows on closed orientable two-dimensional manifolds § 7. Necessary and sufficient conditions for the topological equivalence of non-trivial minimal sets of flows on closed orientable two-dimensional manifolds. Realization of non-trivial minimal sets by geodesic lines References

Any link in a 3-manifold is the closed orbits of a non-singular Morse-Smale flow after taking the split sum with the unlink and the connected sum with S2 × S1s.

The classical Lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants. When applied to continuous systems, one obtains the Hopf formula relating the singularities of a flow to the Euler characteristic of the underlying manifold. The main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold. There are serious obstruction to finding such a formula for flows. The first is that there are flows on any closed manifold of dimension > 4 of zero Euler characteristic with no periodic orbits whatsoever [27]. These flows, however, are unstable by Pugh's C a closing lemma [19]. So it is natural to restrict attention to flows that satisfy Axiom A and the No Cycles property, since Smale has shown that these flows are O-stable [23]. Secondly there are gradient flows on any closed manifold that are Axiom A-No Cycles and have no closed orbits [24]. So it is natural to study only nonsingular flows. The possibility that a Lefschetz type formula for such flows existed was suggested to us by R. Bowen. Even nonsingular Morse-Smale flows, however, are difficult to deal with since Asimov's geometric index is not an invariant of the nonsingular homotopy class [2]. We must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold. We define in Sect. 2 the class of homology proper flows. Such a flow has the property that closed orbits of large period lie in large homology classes. In Sect. 7 we show that a homology proper Axiom A-No Cycles flow does satisfy a Lefschetz-type formula that relates invariants of the closed orbits to a manifold invariant. This formula is closely related to Franks' study of Smale flows on S 3 in which the linking numbers of closed orbits are used to compute the Alexander polynomial of a certain link. Indeed the manifold invariant used in Sect. 7 is a close analogue of an Alexander polynomial that we call the Alexander

Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds

Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.  &nbspWe recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map.   &nbspIn the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariantmanifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.  &nbspIn the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.  &nbspScattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.

Feedback equivalence of constant linear systems

Non-singular Morse–Smale flows on 3-dimensional manifolds

The delay differential equation, $$\dot x$$ (t)=−μx(t)+f(x(t−1)), with μ>0 and a real function f satisfying f(0)=0 and f′>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C=C([−1, 0], ℝ) of initial data for solutions of the equation. In this paper, for a set of μ and f which include examples from neural network theory, we show that this three-dimensional set is the global attractor, i.e., the compact invariant set which attracts all bounded subsets of C. The proof involves, among others, results on uniqueness and absence of periodic orbits.

The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.

Projective equivalence of homogenous binary polynomials

We develop a general compactification framework to facilitate analysis of nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded but open dimension and then extended at one or both ends by gluing in flow-invariant subspaces that carry autonomous dynamics of the limit systems from infinity. We derive the weakest decay conditions possible for the compactified system to be continuously differentiable on the extended phase space. This enables us to use equilibria and other compact invariant sets of the limit systems from infinity to analyze the original nonautonomous problem in the spirit of dynamical systems theory. Specifically, we prove that solutions of interest are contained in unique invariant manifolds of saddles for the limit systems when embedded in the extended phase space. The uniqueness holds in the general case, that is even if the compactification gives rise to a centre direction and the manifolds become centre or centre-stable manifolds. A wide range of problems including pullback attractors, rate-induced critical transitions (R-tipping) and nonlinear wave solutions fit naturally into our framework, and their analysis can be greatly simplified by the compactification.

Let ξ = (E, p,B, T k) be a smooth principal bundle with projection p : E → B and structure group T k = (R/Z)k. Consider a finite group ∆, right action R∆ : B ×∆ → B, and an atlas A(U) of the bundle ξ associated with a covering U consisting of invariant open sets. The existence of such an atlas follows from Theorem 6.1 ([1], Chap. III). Note that elements of the covering U need not to be connected subsets of the manifold B. For charts ξU , ξV ∈ A(U), V ∩U = ∅, we have a transition function ξV U : V ∩U → T k. Denote by θ the canonical 1-form on the group T k with values in its Lie algebra t. The composition θ ◦ ξV U will be called the transition form from the chart ξU to the chart ξV . Assume that the transition forms θ ◦ ξV U are invariant with respect to the action R∆ for all U, V ∈ U , V ∩ U = ∅. In this case, A(U) is called an almost ∆-atlas, its equivalence class A an almost∆-structure on ξ, and the pair ρ = (ξ,A) an almost∆-bundle. For each chart ξU of an almost ∆-atlas A(U), points b ∈ U , and elements t ∈ T k and δ ∈ ∆, we let R∆(ξU (b, t), δ) = ξU (R∆(b, δ), t). This defines a collection R∆ = {R∆|U ∈ U} of actions R∆ : EU ×∆ → EU , EU = p−1(U), which will be called a multivalued action of ∆ on the space E of the bundle ξ. Consider principal bundles ξ and ξ′ with projections p : E → B and p′ : E′ → B and structure group T k. Let f : E → E′ be a smooth mapping such that p = p′ ◦ f and f(v · t) = f(v) · t for all v ∈ E and t ∈ T k. Then f : ξ → ξ′ is a morphism of principal bundles over B. Assume that the bundles ξ and ξ′ possess almost ∆-structures A and A′. Choose atlases A(U) ∈ A, A′(U) ∈ A′ and sets U, V ∈ U , V ∩ U = ∅. To these sets there correspond transition function ζV U : V ∩ U → T k and transition form θ ◦ ζV U from the chart ξU to the chart ξ′ V under the morphism f . If the forms θ ◦ ζV U are invariant with respect to the action R∆ for all ξU ∈ A(U) and ξ′ V ∈ A′(U), then f is called a morphism of almost ∆-bundles ρ = (ξ,A) and ρ′ = (ξ′,A′). Almost∆-bundles overB with structure groupT k and theirmorphisms form a categoryK(B,T k, R∆). The set B(B,T k, R∆) of equivalence classes of objects of this category is a group with respect to the operation induced by the addition of transition functions.

On local invariants of singular symplectic forms