[Paper Review] On the representation of maps by Lie transforms

Antonio Giorgilli|arXiv (Cornell University)|Nov 24, 2012

Quantum chaos and dynamical systems7 references4 citations

TL;DR

This paper presents a direct method to represent perturbative maps—such as the Schrödinger–Siegel and Chirikov standard maps—as Lie transforms, enabling the straightforward application of normal form techniques used for differential equations. By expressing integrable maps via Lie series and perturbations via Lie transforms, the authors construct a formal normal form algorithm that generalizes well-established methods from flows to discrete maps in arbitrary dimensions.

ABSTRACT

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Scrhöder--Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

Motivation & Objective

• To address the challenge of applying normal form theory—well-established for differential equations—to discrete maps.
• To develop a systematic formalism that allows perturbative maps to be represented using Lie series and transforms.
• To extend the applicability of normal form methods to symplectic maps and integrable perturbations in higher-dimensional systems.
• To provide a constructive algorithm for normalizing maps by leveraging composition formulas of Lie transforms.
• To demonstrate the method on canonical examples: the Schrödinger–Siegel map and the Chirikov standard map in arbitrary dimensions.

Proposed method

• The paper uses Lie series and Lie transforms as the foundational formalism, generalizing time-evolution flows to discrete maps.
• It represents an integrable map as a Lie series generated by a vector field, and a perturbation as a Lie transform.
• A key composition formula for Lie transforms is derived, enabling the representation of a map as a composition of an integrable Lie series and a perturbative Lie transform.
• The method relies on a generating sequence $ X $ and the associated Lie transform $ T_X $, with recursive definitions of $ E^X_s $ for each order $ s $.
• The formalism uses the Lie derivative $ L_V $ and the identity $ E^X_s = \sum_{l=1}^s \frac{l}{s} L_{X_l} E^X_{s-l} + \text{higher-order terms} $, ensuring consistency in composition.
• The proof of the composition formula uses induction and Jacobi's identity for Lie brackets, establishing the equivalence between the composed transform and the combined generating sequence.

Experimental results

Research questions

• RQ1 Can a general class of maps, particularly perturbations of integrable maps, be represented in a form compatible with normal form theory for differential equations?
• RQ2 Is it possible to construct a normal form algorithm for maps using Lie series and transforms without relying on Poincaré section interpolation?
• RQ3 How can the composition of Lie transforms be systematically formalized to represent a map as a single Lie transform with a combined generating sequence?
• RQ4 What is the structure of the generating sequence $ Z $ for the composition $ T_X \circ T_Y $, and how does it relate to $ X $ and $ Y $?
• RQ5 Can this formalism be extended to arbitrary dimensions, as demonstrated in the Schrödinger–Siegel and Chirikov standard maps?

Key findings

• The paper establishes a direct correspondence between maps and Lie transforms, enabling the use of normal form techniques designed for differential equations in the context of discrete maps.
• A formal composition formula for Lie transforms is derived, showing that $ T_X \circ T_Y = T_Z $, where $ Z $ is a generating sequence built from $ X $ and $ Y $, with explicit recursive expressions.
• The method allows the construction of a normal form algorithm for maps by applying standard Lie-theoretic techniques, such as iterative elimination of non-resonant terms.
• The approach is validated on two key examples: the Schrödinger–Siegel map and the Chirikov standard map, both extended to arbitrary dimensions.
• The formalism avoids the need for interpolation via periodic flows and provides a systematic, algebraic framework for perturbation theory in maps.
• The proof of the composition rule relies on induction and the Jacobi identity, confirming the consistency of the Lie transform formalism in the context of map representation.