Detection of bifurcations in noisy coupled systems from multiple time series

Abstract

Bifurcations in nonstationarity noise dynamic systems: The basins of attraction and the problems of predictability of final states AIP Conf. Proc. 502, 655 (2000) 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. V C 2015 AIP Publishing LLC. Abrupt changes in complex systems such as the human body, ecosystems, or the climate system can have unwelcome and expensive consequences. Ideally, given one or more time series of some measurable variables, one would like an early warning method to indicate whether an abrupt change is imminent. Abrupt changes can result from crossing a bifurcation in the underlying dynamical system. Recently, there has been a burgeoning body of work on the detection of approaching bifurcation in noisy systems whose deterministic dynamics are assumed to be approximated by a first order ordinary differential equation (ODE) of one dynamical variable. These methods work by looking for the system's recovery from noisy perturbations becoming more sluggish, a phenomenon known as critical slowing down. This is a generic feature of approaching bifurcations in systems modelled by autonomous first order differential equations of one dynamical variable with additive white noise, where the behaviour is decay toward a fixed point. However, it is not always possible to approximate parts of a complex system in such reduced manner. For example, the climate system can display coupled, oscillatory, or chaotic behaviour. Here, we make a direct g...

Generalized early warning signals in multivariate and gridded data with an application to tropical cyclones

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“…In Ref. 22, a method is introduced to anticipate bifurcations in dynamical systems with more than one variable. The method is intentioned as a higher-dimensional analogue of the one-dimensional ACF1 indicator.…”

“…Using the method in Ref. 22, we estimate the first eigenvalue of the Jacobian in a sliding window of 100 points. The mean (with error bars of one standard deviation) of 40 such experiments is show in Fig.…”

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“…For many systems of interest one or more of the above assumptions may be invalid (Williamson and Lenton, 2015). In particular, when the forcing of a system has a comparable period to the timescale of the system, the forcing cannot be modelled as a slow, constant control parameter or a fast, random process; however they can still be thought of as a perturbation away from the system steady state that one can measure the recovery time from, an observation we exploit in this manuscript.…”

“…This is usually done by calculating the lag 1 autocorrelation for a sliding data window of the return map time series at least as long as the system timescale but not so long that any increasing trend in system timescale skews the autocorrelation estimate. It is also desirable to have many points within this window as the standard error of the estimate scales as 1/ √ m, where m is the number of cycles (points) within the window (see Williamson and Lenton (2015) for a discussion). For time series consisting of a small number of cycles this can be a limiting factor.…”