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Fixed-point densities for a quasiperiodic kicked-oscillator map.

J. H. Lowenstein¹

¹Department of Physics, New York University, New York, New York 10003.

Chaos (Woodbury, N.Y.)

|September 1, 1995

Summary

This study analyzes a kicked harmonic oscillator, revealing a classification of fixed points based on residue R. It calculates the density function rho(R) and proves an equality for fixed point densities using decagonal symmetry.

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Area of Science:

Nonlinear Dynamics
Classical Mechanics
Chaos Theory

Background:

Investigates a one-dimensional harmonic oscillator subjected to periodic, position-dependent kicks.
Examines the stroboscopic phase space, revealing periodic/quasiperiodic fixed points and chaotic orbits.

Purpose of the Study:

Classify fixed points of a quasiperiodic kicked harmonic oscillator (q=5) based on local linear behavior.
Calculate the average density function rho(R) of fixed points within a given residue range.
Analyze singularities and discontinuities of rho(R) and prove an equality for fixed point densities.

Main Methods:

Utilizes a five-dimensional embedding to compute the density function rho(R).
Employs transcendental equations for efficient numerical determination of rho(R).
Applies decagonal symmetry and integral representation to prove density equality.

Main Results:

Fixed points are classified by a single variable, the residue R.
The density function rho(R) is calculated, with singularities and discontinuities identified.
An exact equality is proved for the densities of positive-R and negative-R fixed points.

Conclusions:

The study provides a method for efficient numerical determination of fixed point densities.
Decagonal symmetry leads to an equality between densities of stable and unstable fixed points below the period-doubling threshold.
Understanding fixed point behavior is crucial for analyzing complex dynamical systems.

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