Do you want to publish a course? Click here

Local normal forms of dynamical systems with a singular underlying geometric structure

60   0   0.0 ( 0 )
 Added by Kai Jiang
 Publication date 2019
  fields
and research's language is English




Ask ChatGPT about the research

In this paper we prove the existence of a simultaneous local normalization for couples $(X,mathcal{G})$, where $X$ is a vector field which vanishes at a point and $mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $mathcal{G}$ is singular but $X$ does not vanish at the origin.



rate research

Read More

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small rescalings.
We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.
We study the period doubling renormalization operator for dynamics which present two coupled laminar regimes with two weakly expanding fixed points. We focus our analysis on the potential point of view, meaning we want to solve $$V=mathcal{R} (V):=Vcirc fcirc h+V circ h,$$ where $f$ and $h$ are naturally defined. Under certain hypothesis we show the existence of a explicit ``attracting fixed point $V^*$ for $mathcal{R} $. We call $mathcal{R}$ the renormalization operator which acts on potentials $V$. The log of the derivative of the main branch of the Manneville-Pomeau map appears as a special ``attracting fixed point for the local doubling period renormalization operator. We also consider an analogous definition for the one-sided 2-full shift $S$ (and also for the two-sided shift) and we obtain a similar result. Then, we consider global properties and we prove two rigidity results. Up to some weak assumptions, we get the uniqueness for the renormalization operator in the shift. In the last section we show (via a certain continuous fraction expansion) a natural relation of the two settings: shift acting on the Bernoulli space ${0,1}^mathbb{N}$ and Manneville-Pomeau-like map acting on an interval.
We introduce a new concept of finite-time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite-time version of Pesins entropy formula and also an explicit formula of finite-time entropy for $2$-D systems are derived. We also discuss about how to apply the finite-time entropy field to detect special dynamical structures such as Lagrangian coherent structures.
In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{e}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{e}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.
comments
Fetching comments Fetching comments
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا