EME 504
Foundations in Sustainability Systems

5.2 Mathematical Definitions in Tipping Point Analysis

The Lamberson and Page (2012) and Lenton (2013) papers assigned for this module are quite abstract! Here, I will try to break down the key concepts for you. You may want to look over this page or print it while you are reading the paper. I anticipate that you may need to re-read some of paper to really understand the material presented. Even so, you may have many unanswered questions after doing that. Make sure that you take advantage of the class forum, since some of your classmates may have valuable insights on this area!

In the previous page, we introduced the simplest mathematical definition of a Discontinuity. Lamberson and Page (2012) define Tipping Points as discontinuities between current and future states of a system. Lenton (2013) defines tipping points as points in which small perturbations generate abrupt, often irreversible changes in the future of the system. Note that the fact that perturbation does not need result in immediate change means that tipping points are difficult to predict.

Lenton (2013) presents a graphic way to understand tipping points in Figures 1 and 2 of the article. Basically, the variable of interest is moving between potential energy wells (e.g., equilibrium or metastable equilibrium in thermodynamic systems). These potential wells are termed Attractors. Systems with tips ought to have strong positive feedback (runaway) loops that will magnify any perturbations. These positive feedback loops create alternative metastable states, eventually driving transitions between attractor states (e.g., equilibrium states) in the system.

In the mathematical framework presented by Lamberson and Page (2012), large changes in the path of the variable of interest (or the tipping variable (xt)) occur as a consequence of small changes in another system variable (or the threshold variable (yt)) around a tipping point τ belonging to yt space.

The assigned reading strongly relies on dynamical systems constructs to address the following challenges in understanding tipping points:

  • The tipping variable (xt) in the system may depend on multiple other system variables.
  • The change generated by crossing a threshold point for any of the system variables may not occur immediately. To express this time refraction, use the function LΔ = xt+Δ (how long it takes for the consequences of critical changes in the system to be observable).
  • The tipping variable (xt) in the system may depend on multiple other system variables.
  • The change generated by crossing a threshold point for any of the system variables may not occur immediately. To express this time refraction, use the function LΔ = xt (how long it takes for the consequences of critical changes in the system to be observable).

If you may find the notation chosen by the authors unfamiliar, the table below may be useful to map the different concepts introduced.

Table 5.1: Definitions for analysis of tipping points modified from Lamberson and Page (2012)
Notation Definition Alternative Notation Notes
x Variable or state of interest.
Tipping variable.
x(t), xt The initial condition of the state is represented by x0 = x(t=0).
y Threshold variable. y(t), yt A direct tip occurs if x = y.
z Place-holding variable. z(t), zt Other variable(s) that may influence the value of the tipping variable x.
Continuous equation of motion. dx/dt = f (xt,yt) ẋ = 0 at relative and absolute minimae and maximae of the tipping variable x.
xt+1 Discrete equation of motion. xt+1 =F (xt,yt) ***
Ω (e.g., Ωx, Ωy, Ωz) State space. *** ***

From a simplified perspective, a local maximum of the tipping variable may represent a tipping point.

Keep in mind that a model may or may not have a tip:

  • A model with no available metastable states has no tipping points. Mathematically, this is a function with an absolute and no relative minimum.
  • Tipping points may be the result of data artifacts, such as apparent kinks due to lack of data.

Forum Discussion 5.1:

The paper by Lamberson and Page (2012) uses a computational approach to define tipping points. The Lenton (2013) paper emphasizes a graphic description of tipping points. For this discussion, you will translate the terminologies given by Lamberson and Page (2012) and by Lenton (2013) into a  consistent language (i.e. how definitions correspond between these two papers). Using this terminology provide a brief example for both direct and contextual tipping points. You may relate this example to your case study (although it is not neccessary).