Related Experiment Video
Updated: May 11, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Discontinuous attractor dimension at the synchronization transition of time-delayed chaotic systems
Steffen Zeeb1, Thomas Dahms, Valentin Flunkert
1Institute of Theoretical Physics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany. steffen.zeeb@physik.uni-wuerzburg.de
Summary
Investigating attractor dimensions in synchronized chaotic networks reveals a key discontinuity. This finding impacts understanding synchronization transitions in complex systems with time delays.
Area of Science:
- Complex Systems
- Nonlinear Dynamics
- Network Science
Background:
- Synchronization is crucial in many complex systems, but transitions can be abrupt.
- Time-delayed couplings introduce complexity in analyzing synchronization dynamics.
- Attractor dimension quantifies the complexity of chaotic systems.
Purpose of the Study:
- To investigate the behavior of attractor dimensions during the transition to complete synchronization in networks with time-delayed couplings.
- To determine the Kaplan-Yorke dimension and compare it with the correlation dimension at this transition.
- To analyze the scaling of these dimensions with network size and time delay.
Main Methods:
- Calculating the Kaplan-Yorke dimension from Lyapunov exponents for iterated maps and coupled semiconductor lasers.
- Comparing the Kaplan-Yorke dimension with the correlation dimension.
- Analyzing the magnitude of discontinuity in the Kaplan-Yorke dimension as a function of network size for Bernoulli map networks.
- Investigating the scaling of the Kaplan-Yorke dimension and Kolmogorov entropy with system size and time delay.
Main Results:
- The Kaplan-Yorke dimension is shown to be discontinuous at the transition to complete synchronization.
- A jump in the correlation dimension was observed in a system of Bernoulli maps.
- The magnitude of the discontinuity in the Kaplan-Yorke dimension was calculated for Bernoulli unit networks based on network size.
- The scaling of the Kaplan-Yorke dimension and Kolmogorov entropy with system size and time delay was investigated.
Conclusions:
- The transition to complete synchronization in chaotic networks with time-delayed couplings exhibits a discontinuous change in attractor dimension.
- These findings provide insights into the fundamental properties of synchronization transitions in complex dynamical systems.
- The study highlights the importance of considering attractor dimension discontinuities for a comprehensive understanding of network dynamics.
Related Concept Videos
01:31Classification of Systems-II
Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
01:19Limits with Oscillating Discontinuities
An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...
01:28Stability
The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
01:24BIBO stability of continuous and discrete -time systems
System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
01:23Linear time-invariant Systems
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
01:27State Space Representation
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Consider an RLC circuit, a...