Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Multistability refers to the coexistence of different stable states in nonlinear dynamical systems. This phenomenon has been observed in laboratory experiments and in nature. In this introduction, we briefly introduce the classes of dynamical systems in which this phenomenon has been found and discuss the extension to new system classes. Furthermore, we introduce the concept of critical transitions and discuss approaches to distinguish them according to their characteristics. Finally, we present some specific applications in physics, neuroscience, biology, ecology, and climate science.

Multistability and tipping: From mathematics and physics to climate and brain-Minireview and preface to the focus issue.

The Atlantic meridional overturning circulation (AMOC) represents the zonally integrated stream function of meridional volume transport in the Atlantic Basin. The AMOC plays an important role in transporting heat meridionally in the climate system. Observations suggest a heat transport by the AMOC of 1.3 PW at 26°N-a latitude which is close to where the Atlantic northward heat transport is thought to reach its maximum. This shapes the climate of the North Atlantic region as we know it today. In recent years there has been significant progress both in our ability to observe the AMOC in nature and to simulate it in numerical models. Most previous modeling investigations of the AMOC and its impact on climate have relied on models with horizontal resolution that does not resolve ocean mesoscale eddies and the dynamics of the Gulf Stream/North Atlantic Current system. As a result of recent increases in computing power, models are now being run that are able to represent mesoscale ocean dynamics and the circulation features that rely on them. The aim of this review is to describe new insights into the AMOC provided by high-resolution models. Furthermore, we will describe how high-resolution model simulations can help resolve outstanding challenges in our understanding of the AMOC.

The Atlantic Meridional Overturning Circulation in High-Resolution Models

The Atlantic Meridional Overturning Circulation (AMOC), a major ocean current system transporting warm surface waters toward the northern Atlantic, has been suggested to exhibit two distinct modes of operation. A collapse from the currently attained strong to the weak mode would have severe impacts on the global climate system and further multi-stable Earth system components. Observations and recently suggested fingerprints of AMOC variability indicate a gradual weakening during the last decades, but estimates of the critical transition point remain uncertain. Here, a robust and general early-warning indicator for forthcoming critical transitions is introduced. Significant early-warning signals are found in eight independent AMOC indices, based on observational sea-surface temperature and salinity data from across the Atlantic Ocean basin. These results reveal spatially consistent empirical evidence that, in the course of the last century, the AMOC may have evolved from relatively stable conditions to a point close to a critical transition. The Atlantic Meridional Overturning Circulation (AMOC) is currently strong, but transition to its weak mode could see significant changes in the climate system. This work presents an observation-based early-warning system for such transitions and shows that the AMOC may be approaching a transition.

Observation-based early-warning signals for a collapse of the Atlantic Meridional Overturning Circulation

The concept of tipping points and critical transitions helps inform our understanding of the catastrophic effects that global change may have on ecosystems, Earth system components, and the whole E...

Evasion of tipping in complex systems through spatial pattern formation.

We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down), we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift between two asymptotically autonomous systems with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system.

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Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

The Atlantic meridional overturning circulation (AMOC) transports substantial amounts of heat into the North Atlantic sector, and hence is of very high importance in regional climate projections. T...

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0051

Basin bifurcations, oscillatory instability and rate-induced thresholds for Atlantic meridional overturning circulation in a global oceanic box model

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how con...

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2021.0059

Phase tipping: how cyclic ecosystems respond to contemporary climate

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to ``parameter shifts'' between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous R\"ossler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.

Weak tracking in nonautonomous chaotic systems.

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-toequilibrium connections that we determine using Lin’s method for an augmented system. Considering weak tracking for a nonautonomous Rössler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor.

Hassan Alkhayuon

Papers

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

Basin bifurcations, oscillatory instability and rate-induced thresholds for Atlantic meridional overturning circulation in a global oceanic box model

Phase tipping: how cyclic ecosystems respond to contemporary climate

Weak tracking in nonautonomous chaotic systems.

Rate-induced transitions for parameter shift systems