We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature sa...

In this paper, we construct (a) a pair of two regular Cantor sets in higher dimension which exhibits $C^1$-stable intersection and (b) hyperbolic basic set $C^2$-robust homoclinic tangency the largest codimension for any dimensional manifold, using blenders. The former implies that an analog Moreira's theorem on real line does not hold dimension. latter solves question posed by Barrientos A.Rai...

On a compact Kähler manifold X with a holomorphic 2-form α, there is an almost complex structure associated with α. We show how this implies vanishing theorems for the Gromov-Witten invariants ofX . This extends the approach, used in [LP] for Kähler surfaces, to higher dimensions. Let X be a Kähler surface with a non-zero holomorphic 2-form α. Then α is a section of the canonical bundle and its...

Studying a manifold with a certain geometric structure becomes an interesting topic in differential topology when such a structure has some stability. The local stability of symplectic and contact structures, the Darboux theorem, is a fundamental property for symplectic and contact topology. There is a well-known global stability theorem for contact structures, the Gray theorem (see [3]): a one...

This paper, according to the process of capital return, establishes a differential dynamics model investment with two time delays. When both delays are zero, it is proved that positively invariant, uniformly bounded, and globally asymptotically stable by using comparison principle Bendixson–Dulac theorem. at least one delay not Hopf bifurcation theorem, conditions local asymptotic stability exi...

The C-structurally stable diffeomorphims of a compact manifold are those that satisfy Axiom A and the strong transversality condition (AS). We generalize the concept of AS from diffeomorphisms to invariant compact subsets. Among other properties, we show the structural stability of the AS invariant compact sets K of surface diffeomorphisms f . Moreover if f̂ is the dynamics of a compact manifold...

Let M be a closed oriented 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straighten...

In this paper, the bifurcation analysis for Burgers mapping has been studied. The existence and stability of the fixed points of this map was derived. The conditions of existence for pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. The control of the map around stable NeimarkSacker bifurcation has been ac...

We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the Stable Manifold...