Related Experiment Video
Updated: Apr 15, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Detection of bifurcations in noisy coupled systems from multiple time series
Mark S Williamson1, Timothy M Lenton1
1Earth System Science Group, College of Life and Environmental Sciences, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, United Kingdom.
Abstract:
We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the system's fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.
Related Concept Videos
BIBO stability of continuous and discrete -time systems
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....
Linear time-invariant Systems
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...
Transmission-Line Differential Equations
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured...
Detection of Black Holes
Their closest cousins are neutron stars, which are composed almost entirely of neutrons packed against each other, making them extremely dense. A neutron star has the same mass as the Sun but its diameter is only a few kilometers. Therefore, the escape velocity from their surface is close to the speed of light.
Not until the 1960s, when the first neutron...
Multi-input and Multi-variable systems
In the absence of...
Classification of Signals
A continuous-time signal holds a value at every instant in time, representing information seamlessly. In contrast, a discrete-time signal holds values only at specific moments, often denoted as x(n), where...

