Glossary

 Alternative stable states Different (multiple) states (equilibria) of a system under the same external conditions. The state to which a system converges is path-dependent. Attractor The dynamic regime to which a system converges after some time. Examples of attractors: point, cyclic (periodic), quasiperiodic, chaotic Basin of attraction Set of initial conditions that lead to a particular state (equilibrium). Bifurcation A critical threshold in conditions at which the qualitative behavior of a system changes. Bistability The case where two alternative stable states exist. Catastrophic bifurcation Bifurcation where the current state of a system disappears and the system is forced to move to an alternative state. Catastrophic shift An abrupt shift in the state of a system induced by a small perturbation that pushes the system across the border of the basin of attraction. Critical transition Abrupt shift in the behavior of a system when certain parameters reach a threshold. Most pronounced example is a catastrophic bifurcation. Critical slowing down The phenomenon that the return time of a disturbance back to equilibrium increases close to a bifurcation. Eigenvalue (dominant) Maximum factor that expresses how much linearized deviations from equilibrium diverge in time. It approximates the recovery rate back to equilibrium after a perturbation. Equilibrium The condition at which competing processes are balanced. At a stable equilibrium, a system returns to it upon a small perturbation. At an unstable equilibrium, a system moves away from it upon a small perturbation. Fold bifurcation The threshold (in a parameter) at which a stable and an unstable equilibrium collide. It marks the disappearance of both equilibria. Hysteresis As conditions are changing in a bistable system, the system remains on the same state until a catastrophic bifurcation is reached at which it shifts to the alternative state. If conditions are changed in the opposite direction, the system jumps back to the original state only until it meets another catastrophic bifurcation. The distance (in parameter space) between the two catastrophic bifurcations defines the size of the hysteresis. The bigger the size, the more difficult for a catastrophic shift to be reversed. Leading indicators Divergence of the statistical properties in the pattern of fluctuations of a system close to a critical transition. Positive feedback A process through which something has a positive effect on itself. Regime shift A sharp change from one regime (state) to a contrasting one. A regime is a dynamic ‘state’ of a system: it can be a stable point or a cycle. Resilience (ecological) The magnitude of disturbance a system can tolerate before it shifts into a different state. Self-organized patterns Patterns in space that emerge from the interaction between many units. Transition Discontinuous (first-order): Abrupt change in the qualitative behavior of a system. Continuous (second-order): Smooth change in the qualitative behavior of a system. Noise-induced: Change in the qualitative behavior of a system in the presence of high noise intensity. Threshold A point where the system is very sensitive to changing conditions. Tipping point A point where the system may flip to another state.