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Measuring Delay Discounting in Humans Using an Adjusting Amount Task

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Universal Dichotomy for Dynamical Systems with Variable Delay.

Andreas Otto¹, David Müller¹, Günter Radons¹

¹Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany.

Physical Review Letters

|February 11, 2017

Summary

This summary is machine-generated.

The functional form of time-dependent delays significantly impacts system dynamics, revealing two distinct universality classes. One class mirrors constant delay systems, while the novel second class exhibits mode-locking and linear Lyapunov spectrum scaling.

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Area of Science:

Dynamical Systems and Chaos Theory
Nonlinear Dynamics
Mathematical Physics

Background:

Time-dependent delay differential equations are crucial in modeling complex phenomena.
Understanding the influence of the delay's functional form on system dynamics is an open challenge.
Previous studies often simplified delay functions, potentially missing key dynamic behaviors.

Purpose of the Study:

To investigate how the functional form of the retarded argument in time-dependent delay systems affects their dynamics.
To identify and characterize distinct universality classes based on delay functional forms.
To establish a theoretical framework for analyzing these systems using iterated maps and Koopman operators.

Main Methods:

Development of an iterated map, termed the access map, associated with the retarded argument.
Application of Koopman operator theory to analyze the system's dynamics.
Characterization of universality classes based on the properties of the access map and Lyapunov spectrum scaling.

Main Results:

Identification of two universality classes for systems with time-dependent delays.
The first class behaves analogously to systems with constant delays.
The second, novel class exhibits mode-locking in access maps and asymptotically linear Lyapunov spectrum scaling, unlike the typical logarithmic scaling.

Conclusions:

The functional form of the retarded argument is a fundamental determinant of system dynamics, not merely a secondary parameter.
A new universality class, characterized by specific spectral properties and mode-locking, emerges from specific delay functional forms.
System behavior depends fractally on delay parameters, highlighting the intricate relationship between delay structure and dynamic outcomes.