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Global change has been predominantly studied from the prism of 'how much' rather than 'how fast' change occurs. Associated to this, there has been a focus on environmental drivers crossing a critical value and causing so‐called regime shifts. This presupposes that the rate at which environmental conditions change is slow enough to allow the ecological entity to remain close to a stable attractor (e.g. an equilibrium). However, environmental change is occurring at unprecedented rates. Equivalently to the classical regime shifts, theory shows that a critical threshold in rates of change can exist, which can cause rate‐induced tipping (R‐tipping). However, the potential implications of R‐tipping in ecology remain understudied. We aim to facilitate the application of R‐tipping theory in ecology with the objective of identifying which properties (e.g. level of organisation) increase susceptibility to rates of change. First, we clarify the fundamental difference between tipping caused by the magnitude as opposed to the rate of change crossing a threshold. Then we present examples of R‐tipping from the ecological literature and seek the ecological properties related to higher sensitivity to rates of change. Specifically, we consider the role of the level of ecological organisation, spatial processes, eco‐evolutionary dynamics and pair–wise interactions in mediating or buffering rate‐induced transitions. Finally, we discuss how targeted experiments can investigate the mechanisms associated to increasing rates of change. Ultimately, we seek to highlight the need to better understand how rates of environmental change may induce ecological responses and to facilitate the systematic study of rates of environmental change in the context of current global change. [ABSTRACT FROM AUTHOR]

Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an "internal" drive. When the oscillators are mismatched, phase synchronization occurs, while in a master slave configuration, generalized synchrony results. These effects are demonstrated in a system of coupled chaotic Rossler oscillators.

Recently, the phase-flip bifurcation has been described as a fundamental transition in time-delay coupled, phase-synchronized nonlinear dynamical systems. The bifurcation is characterized by a change of the synchronized dynamics from being in-phase to antiphase, or vice versa; the phase-difference between the oscillators undergoes a jump of pi as a function of the coupling strength or the time delay. This phase-flip is accompanied by discontinuous changes in the frequency of the synchronized oscillators, and in the largest negative Lyapunov exponent or its derivative. Here we illustrate the phenomenology of the bifurcation for several classes of nonlinear oscillators, in the regimes of both periodic and chaotic dynamics. We present extensive numerical simulations and compute the oscillation frequencies and the Lyapunov spectra as a function of the coupling strength. In particular, our simulations provide clear evidence of the phase-flip bifurcation in excitable laser and Fitzhugh-Nagumo neuronal models, and in diffusively coupled predator-prey models with either limit cycle or chaotic dynamics. Our analysis demonstrates marked jumps of the time-delayed and instantaneous fluxes between the two interacting oscillators across the bifurcation; this has strong implications for the performance of the system as well as for practical applications. We further construct an electronic circuit consisting of two coupled Chua oscillators and provide the first formal experimental demonstration of the bifurcation. In totality, our study demonstrates that the phase-flip phenomenon is of broad relevance and importance for a wide range of physical and natural systems.

We study the dynamics of oscillators that are mutually coupled via dissimilar (or "conjugate") variables and find that this form of coupling leads to a regime of amplitude death. Analytic estimates are obtained for coupled Landau-Stuart oscillators, and this is supplemented by numerics for this system as well as for coupled Lorenz oscillators. Time delay does not appear to be necessary to cause amplitude death when conjugate variables are employed in coupling identical systems. Coupled chaotic oscillators also show multistability prior to amplitude death, and the basins of the coexisting attractors appear to be riddled. This behavior is quantified: an appropriately defined uncertainty exponent in the coupled Lorenz system is shown to be zero.