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Discrete averaging for discrete time dynamical systems

Vassili Gelfreich, Arturo Vieiro

Abstract

In this paper we develop the theory of discrete averaging designed to study discrete time dynamical systems defined by iterates of a map. The discrete averaging uses weighted averages over a segment of trajectory to find an autonomous vector field that approximates the original map. The method provides a simple and effective tool for finding adiabatic invariants, both numerically and analytically. It is capable of strengthening various theorems of the classical averaging theory because it eliminates two intermediate steps used in the classical averaging: the suspension procedure that assigns a rapidly oscillating flow to the map and time-dependent coordinate changes that eliminate the dependence on time. We discuss two applications of the discrete averaging - to the dynamics of a near-identity map and to the dynamics of a map in a neighbourhood of a resonant fixed point. We show that the discrete averaging provides explicit uniform bounds for approximation errors. We also show that the discrete averaging can be used to establish domain of validity of adiabatic approximations in numerical experiments.

Discrete averaging for discrete time dynamical systems

Abstract

In this paper we develop the theory of discrete averaging designed to study discrete time dynamical systems defined by iterates of a map. The discrete averaging uses weighted averages over a segment of trajectory to find an autonomous vector field that approximates the original map. The method provides a simple and effective tool for finding adiabatic invariants, both numerically and analytically. It is capable of strengthening various theorems of the classical averaging theory because it eliminates two intermediate steps used in the classical averaging: the suspension procedure that assigns a rapidly oscillating flow to the map and time-dependent coordinate changes that eliminate the dependence on time. We discuss two applications of the discrete averaging - to the dynamics of a near-identity map and to the dynamics of a map in a neighbourhood of a resonant fixed point. We show that the discrete averaging provides explicit uniform bounds for approximation errors. We also show that the discrete averaging can be used to establish domain of validity of adiabatic approximations in numerical experiments.
Paper Structure (12 sections, 7 theorems, 108 equations, 2 figures)

This paper contains 12 sections, 7 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Discrete averaging
  3. Classical averaging vs discrete averaging
  4. Interpolating vector fields
  5. An example: the Hénon map near a resonance
  6. Approximation by autonomous flows
  7. Uniform bounds in the space of analytic diffeomorphisms
  8. Application to tangent-to-identity maps
  9. Dynamics near a resonant equilibrium
  10. Stability of a fully resonant elliptic fixed point
  11. Hidden symmetries of interpolating flows
  12. Conclusions

Key Result

Theorem 1

If a tangent to identity family $F_\varepsilon\in C^{n+1}\bigl(D\times [-\varepsilon_0,\varepsilon_0]\bigr)$ for some $\varepsilon_0>0$, then there is a unique polynomial in $\varepsilon$ vector field $g_\varepsilon(x)=\sum_{k=0}^{n-1}\varepsilon^k g_k(x)$ such that uniformly on every compact subset of $D$. Moreover, for $1\le k\le m \le n$ where $X_m$ is an interpolating vector field of order $m

Figures (2)

  • Figure 1: Iterates of the Henon map \ref{['henon']} (left) and level lines of the Taylor polynomial of the adiabatic invariant $h_2$ of order 6 in $x,y$ and one in $\varepsilon$ (right) for $c=1+\varepsilon$ with $\varepsilon=10^{-3}$.
  • Figure 2: Domain of validity of discrete averaging for the Hénon map with $\varepsilon=-10^{-3}$ (left column) and $\varepsilon=10^{-3}$ (right column). In the top row, the colors indicate the optimal $n$ values that minimize $G_{(x,y)}(n)= |X_{2n}(x,y) - X_{2n+2}(x,y)|_2$. The bottom row plots exhibit colors representing the decimal logarithm of the minimum values of $G_{x,y}(n)$ over $n$. For reference, orbits of the Hénon map are shown in light gray in all plots.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Remark 2: Computing the formal interpolating flow using discrete averaging
  • Theorem 3
  • proof
  • Remark 4: Symmetric interpolation schemes
  • Theorem 5: Interpolating vector fields near the fixed point
  • Lemma 6: Escape times
  • proof
  • Lemma 7: Interpolating vector fields
  • ...and 6 more