Homoclinic tangencies versus uniform hyperbolicity for conservative 3-flows

We prove that a volume-preserving three-dimensional flow can be C¹-approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C¹-topology.     

How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency

Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S², in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191-202; S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9-18; S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 50 (1979) 101-151]. It turns out that in the space of Cr smooth diffeomorphisms Diffr(M) of a compact surface M there is an open set U such that a Baire generic diffeomorphism f∈U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism have infinitely many sinks.” Our main result roughly says that with probability one for any positive D a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis' Conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have probability zero.One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp.].     

Twists and Gromov hyperbolicity of riemann surfaces

The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.     

Global dominated splittings and the Newhouse phenomenon

We prove that given a compact -dimensional boundaryless manifold , , there exists a residual subset of the space of diffeomorphisms such that given any chain-transitive set of , then either admits a dominated splitting or else is contained in the closure of an infinite number of periodic sinks/sources. This result generalizes the generic dichotomy for homoclinic classes given by Bonatti, Diaz, and Pujals (2003).

It follows from the above result that given a -generic diffeomorphism , then either the nonwandering set may be decomposed into a finite number of pairwise disjoint compact sets each of which admits a dominated splitting, or else exhibits infinitely many periodic sinks/sources (the `` Newhouse phenomenon"). This result answers a question of Bonatti, Diaz, and Pujals and generalizes the generic dichotomy for surface diffeomorphisms given by Mañé (1982).

     

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

In this paper we obtain the equivalence of the Gromov hyperbolicity between an extensive class of complete Riemannian surfaces with pinched negative curvature and certain kind of simple graphs, whose edges have length 1, constructed following an easy triangular design of geodesics in the surface.     

On critical orbits and sectional hyperbolicity of the nonwandering set for flows

In this note, we introduce the notion of nonuniformly sectional hyperbolic set and use it to prove that any C¹-open set which contains a residual subset of vector fields with nonuniformly sectional hyperbolic critical set also contains a residual subset of vector fields with sectional hyperbolic nonwandering set. This not only extends Theorem A of Castro [11], but using suspensions we recover it.     

Existence and decay of solutions to the dissipative quasi-geostrophic equation with delays

We consider the dissipative quasi-geostrophic equation with forcing and advective terms containing delays. We prove the existence of weak solutions and algebraic decay of the energy, provided the initial data _θ0${\theta }_0$ is in L^p∩L²,1≤p<2$L^p\cap L^2,1\le p<2$ and the forcing term satisfies some mild conditions on its long time behavior.     

General functional response and recruitment in a predator–prey system with capture on both species

This work provides a mathematical model for a predator‐prey system with general functional response and recruitment, which also includes capture on both species, and analyzes its qualitative dynamics. The model is formulated considering a population growth based on a general form of recruitment and predator functional response, as well as the capture on both prey and predators at a rate proportional to their populations. In this sense, it is proved that the dynamics and bifurcations are determined by a two‐dimensional threshold parameter. Finally, numerical simulations are performed using some ecological observations on two real species, which validate the theoretical results obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Successional stability of vector fields in dimension three

A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the Whitney topology by a successionally stable vector field.

     

On genericity of shadowing and periodic shadowing property

Let M be a smooth compact manifold (possibly with boundary) or compact manifold with dimM?3 or dimM?6. We prove that generic homeomorphism on M has the periodic shadowing property.     

Existence and stability of spatially localized patterns

Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at $\lambda =0$ and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results.     

1:1内共振条件下矩形薄板的全局分叉和多脉冲混沌动力学

首次利用广义Melnikov方法研究了一个四边简支矩形薄板的全局分叉和多脉冲混沌动力学．矩形薄板受面外的横向激励和面内的参数激励．利用von Krmn模型和Galerkin方法得到一个二自由度非线性非自治系统用来描述矩形薄板的横向振动．在1∶1内共振条件下,利用多尺度方法得到一个四维的平均方程．通过坐标变换把平均方程化为标准形式,利用广义Melnikov方法研究该系统的多脉冲混沌动力学,并且解释了矩形薄板模态间的相互作用机理．在不求同宿轨道解析表达式的前提下,提供了一个计算Melnikov函数的方法．进一步得到了系统的阻尼、激励幅值和调谐参数在满足一定的限制条件下,矩形薄板系统会存在多脉冲混沌运动．数值模拟验证了该矩形薄板的确存在小振幅的多脉冲混沌响应．     

Universally meager sets and principles of generic continuity and selection in Banach spaces

In this paper we use large cardinals to address some problems about generic continuity and generic selection that occur in the study of fragmentability and differentiability in the context of Banach spaces.     

空间同宿环和异宿环的稳定性

关于平面同（异）宿环的稳定性已有不少文献讨论过，但关于空间同（异）宿环的稳定性尚没有任何结果．本文在可定义回复映射的条件下给出了同（异）宿环在其部分邻域中是渐近稳定的判据．这些结果在某种意义下是平面系统相应结果的推广，包括并推广了［２］，［３］的结果．本文最后讨论了Ｌｏｒｅｎｚ系统同宿环和三种群竞争系统异宿环的稳定性，所得结果和Ｓｐａｒｒｏｗ与Ｍａｙ等的数值结果相吻合．     

Winning controllers and regions in competitive nonlinear dynamic games with two qualitative objectives and uncertain information

The Liapunov technique has been introduced in differential games in a sequence of works over the last decade. We discuss application of this technique to the interface between two competitive semi-games, each with different qualitative objective. The feedback information for controllers is provided from designed, analytically integrable state predictors with reduced dynamics, which considerably shortens the computing time. The case is illustrated on the scenario of target reaching before being intercepted, and intercepting before target is reached.Deceased.     

Noninvertibility and the structure of basins of attraction in a model adaptive control system

Summary We present a two-dimensional, nonlinear map, arising from a simple adaptive control problem, which exhibits disconnected boundaries separating the basins of attraction of its coexisting attractors. We perform a detailed study of the relation between this phenomenon and the noninvertible nature of the map and demonstrate how the complex basin structure is caused by a change in the number of preimages of points along a stable manifold.     

Existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system

This paper investigates the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibro-impact system. The criterion for existence of grazing period-n motion is presented. A local analysis based on the discontinuity-mapping approach is employed to derive a normal form Poincaré mapping near the grazing trajectory. Based on the above approach, a condition of stability can be formulated, such that a grazing trajectory will be discontinuous if the condition is unfulfilled. The predicted grazing bifurcations are in agreement with the numerical results. In particular, comparison of the grazing bifurcation diagrams of the normal form Poincaré mapping and the simulation diagrams of the original differential equation illustrates the validity of the discontinuity-mapping approach.     

Bifurcations and chaos control in a discrete-time predator-prey system of Leslie type

We investigate the dynamics of a discrete-time predator-prey system of Leslie type. We show algebraically that the system passes through a flip bifurcation and a Neimark-Sacker bifurcation in the interior of $\R^{2}_+$ using center manifold theorem and bifurcation theory. Numerical simulations are implimented not only to validate theoretical analysis but also exhibits chaotic behaviors, including phase portraits, period-11 orbits, invariant closed circle, and attracting chaotic sets. Furthermore, we compute Lyapunov exponents and fractal dimension numerically to justify the chaotic behaviors of the system. Finally, a state feedback control method is applied to stabilize the chaotic orbits at an unstable fixed point.