📂Dynamical Systems

Definition of a Tipping Point in Dynamics

Definition ¹

The dynamical changes that occur in a normal system as parameters vary are called bifurcations, whereas the dynamical changes that occur in non-normal systems are called tipping points.

Explanation

Typical tipping

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There can be several types of tipping points, but as of 2026 three are widely recognized: N-tipping, B-tipping, and R-tipping. Each will be treated in more detail in its own post, but from the perspective of the quasi-static attractor they can be briefly summarized as follows³.

• N-tipping: The system parameters do not change, but the state is shifted by noise occurring in the neighborhood of the quasi-static attractor. It is more likely to occur when the noise is stronger.
• B-tipping: Caused by a bifurcation of the quasi-static attractor due to a sudden or qualitative change in the system. In the figure above this is illustrated by a saddle–node bifurcation occurring over time, where a stable fixed point disappears and the state rapidly transitions to another fixed point.
• R-tipping: From the viewpoint of topological equivalence of systems, the system itself has not changed, but in a normal system the rate of change of the state is also time-dependent. R-tipping occurs when the system’s change is so rapid that, despite varying continuously, the trajectory fails to keep up with the quasi-static attractor.

Less well-known tipping

Other types such as S-tipping, A-tipping, and P-tipping have also been proposed, so they are worth being aware of⁴.