Definition of the tipping point

Question

I have seen this mentioned in papers (e.g. here). I have a vague idea that a tipping point of a dynamical system is the state where small fluctuations can cause major changes in behavior. Is there a rigorous (or at least clearer) definition of what a tipping point means in math?

Answer 1

There is no consensus on defining tipping points, but in many contexts that concern dynamical systems, it is usually some kind of bifurcation. A bifurcation is a sudden change of the dynamics of a dynamical system when a parameter is infinitesimally changed.

There are some kinds of bifurcations that I would not expect to be referred to as tipping points, namely those which do not lead to a first-order discontinuity in phase space. By first-order discontinuity, I mean that reasonable observables of the dynamics are discontinuous with respect to a parameter (a jump in the bifurcation diagram), as opposed to their derivative being discontinuous (a kink in the bifurcation diagram). This equivalent to the union of attractors undergoing a discontinuous change. For example, period doublings are only second-order discontinuous and thus usually not considered tipping points.

Often tipping points are considered to happen in a noise-free version of the actual dynamics or a comparable simplification. A typical example for this is when the tipping point makes the system bistable, and the other attractor becomes relevant as it can be reached through noise.