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Inhomogeneities in non-hyperbolic one-dimensional invariant sets

Brian E. Raines (2004)

Fundamenta Mathematicae

The topology of one-dimensional invariant sets (attractors) is of great interest. R. F. Williams [20] demonstrated that hyperbolic one-dimensional non-wandering sets can be represented as inverse limits of graphs with bonding maps that satisfy certain strong dynamical properties. These spaces have "homogeneous neighborhoods" in the sense that small open sets are homeomorphic to the product of a Cantor set and an arc. In this paper we examine inverse limits of graphs with more complicated bonding...

Initial boundary value problem for the mKdV equation on a finite interval

Anne Boutet de Monvel, Dmitry Shepelsky (2004)

Annales de l’institut Fourier

We analyse an initial-boundary value problem for the mKdV equation on a finite interval ( 0 , L ) by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k -plane. This RH problem is determined by certain spectral functions which are defined in terms of the initial-boundary values at t = 0 and x = 0 , L . We show that the spectral functions satisfy an algebraic “global relation” which express the implicit relation between all boundary values in terms of spectral...

Initial data stability and admissibility of spaces for Itô linear difference equations

Ramazan Kadiev, Pyotr Simonov (2017)

Mathematica Bohemica

The admissibility of spaces for Itô functional difference equations is investigated by the method of modeling equations. The problem of space admissibility is closely connected with the initial data stability problem of solutions for Itô delay differential equations. For these equations the p -stability of initial data solutions is studied as a special case of admissibility of spaces for the corresponding Itô functional difference equation. In most cases, this approach seems to be more constructive...

Instability of equilibria in dimension three

Marco Brunella (1998)

Annales de l'institut Fourier

In this paper we show that if v is an analytic vector field on ℝ 3 having an isolated singular point at 0, then there exists a trajectory of v which converges to 0 in the past or in the future. The proof is based on certain results concerning desingularizaton of vector fields in dimension three and on index-type arguments à la Poincaré-Hopf.

Integrable analytic vector fields with a nilpotent linear part

Xianghong Gong (1995)

Annales de l'institut Fourier

We study the normalization of analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flows of any analytic vector fields with a nilpotent linear part.

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