Complete classification of two-dimensional algebras

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

Download Full-text

On non-hyperbolic algebraic automorphisms of a two-dimensional torus

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.23.202103.295-307 ◽

2021 ◽

Vol 23 (3) ◽

pp. 295-307

Author(s):

Sergey V. Sidorov ◽

Ekaterina E. Chilina

Keyword(s):

Sufficient Conditions ◽

Complete Classification ◽

Orientable Surface ◽

Topological Classification ◽

Topological Conjugacy ◽

Two Dimensional ◽

Dimensional Torus ◽

Algebraic Automorphism ◽

Necessary And Sufficient

Abstract. This paper contains a complete classification of algebraic non-hyperbolic automorphisms of a two-dimensional torus, announced by S. Batterson in 1979. Such automorphisms include all periodic automorphisms. Their classification is directly related to the topological classification of gradient-like diffeomorphisms of surfaces, since according to the results of V. Z. Grines and A.N. Bezdenezhykh, any gradient like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. J. Nielsen found necessary and sufficient conditions for the topological conjugacy of orientation-preserving periodic homeomorphisms of orientable surfaces by means of orientation-preserving homeomorphisms. The results of this work allow us to completely solve the problem of realization all classes of topological conjugacy of periodic maps that are not homotopic to the identity in the case of a torus. Particularly, it follows from the present paper and the work of that if the surface is a two-dimensional torus, then there are exactly seven such classes, each of which is represented by algebraic automorphism of a two-dimensional torus induced by some periodic matrix.

Download Full-text

Nonexistence of Subcritical Solitary Waves

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-021-01659-y ◽

2021 ◽

Author(s):

Vladimir Kozlov ◽

Evgeniy Lokharu ◽

Miles H. Wheeler

Keyword(s):

Solitary Waves ◽

Open Problem ◽

Water Waves ◽

Complete Classification ◽

Arbitrary Distribution ◽

Two Dimensional ◽

Wave Speeds ◽

Nonexistence Result ◽

Sign Conditions

AbstractWe prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.

Download Full-text