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Mixing Rates of the Geometrical Neutral Lorenz Model.

Henk Bruin¹, Hector Homero Canales Farías¹

¹Faculty of Mathematics, University of Vienna, Vienna, Austria.

Journal of Statistical Physics

|December 11, 2023

Summary

This study modifies the Lorenz flow by replacing its hyperbolic saddle with a neutral saddle. This alteration results in polynomial decay of correlations, offering new insights into chaotic systems.

Keywords:

Mixing Neutral fixed point Neutral geometrical Lorenz flow Polynomial decay of correlations

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Area of Science:

Dynamical Systems
Chaos Theory
Mathematical Physics

Background:

The classical Lorenz flow exhibits exponential decay of correlations.
Hyperbolic fixed points are common in chaotic systems.
Understanding correlation decay is crucial for analyzing system long-term behavior.

Purpose of the Study:

To investigate polynomial decay of correlations in a modified Lorenz-like flow.
To replace the hyperbolic saddle at the origin with a neutral saddle.
To analyze the impact of this modification on correlation decay rates.

Main Methods:

Geometric construction of the Lorenz flow.
Modification of the linearized vector field near the origin to a neutral vector field.
Application of Araújo and Melbourne's methods for correlation decay analysis.

Main Results:

The modification results in polynomial tails for Dulac times.
Polynomial upper bounds for the decay of correlations are achieved for the modified flow.
This contrasts with the exponential mixing observed in the classical Lorenz flow.

Conclusions:

Replacing a hyperbolic saddle with a neutral saddle fundamentally alters correlation decay properties.
The study demonstrates a method to achieve polynomial decay of correlations in Lorenz-like systems.
This work contributes to the understanding of chaotic dynamics and their statistical properties.