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Time-metric equivalence and dimension change under time reparameterizations.

Adilson E Motter¹, Katrin Gelfert

¹Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|August 8, 2009

Summary

Time reparameterizations affect dynamical systems, transforming Lyapunov exponents and generalized dimensions. The information dimension shows nontrivial behavior, impacting chaos characterization and the Kaplan-Yorke conjecture.

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Area of Science:

Dynamical Systems Theory
Chaos Theory
Differential Geometry

Background:

Understanding the behavior of dynamical systems under transformations is crucial for characterizing chaos, especially in relativistic contexts.
Invariance of dynamical quantities under time reparameterizations is a key question in physics.

Purpose of the Study:

To investigate the impact of time reparameterizations on dynamical systems.
To derive transformation rules for Lyapunov exponents and generalized dimensions.
To clarify the invariance properties of the information dimension (D1).

Main Methods:

Analyzing time transformations as local metric transformations on Riemannian phase spaces.
Deriving transformation rules for Lyapunov exponents.
Examining the behavior of generalized dimensions (Dq) under time transformations.

Main Results:

Time transformations are locally equivalent to metric transformations.
A general transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces was established.
The spectrum of generalized dimensions (Dq) is preserved, except for the information dimension (D1).
The information dimension (D1) exhibits nontrivial transformation behavior, contrary to previous assumptions of invariance.

Conclusions:

Time reparameterizations have a significant, non-trivial impact on specific dynamical invariants like the information dimension.
The discontinuous behavior of D1 at q=1 provides a new avenue for constraining and extending the Kaplan-Yorke conjecture.
This work deepens the understanding of chaos and invariance in dynamical systems under time transformations.