The notion of "tipping point" keeps coming up in many areas. One of the most frequent is in relation to climate change and global warming. The idea is one which I have been involved with my entire scientific career. I did my postdoctoral work under the mentorship of the late Aharon Katzir-Katchalsky at the Weizmann Institute in Israel from 1963 to 1965. The big new thing was linear non-equilibrium thermodynamics being applied to the modeling of transport of materials through biological membranes. It took a little while for those of us who got in on the ground floor with these new ideas to realize that the real world is anything but linear.

So often in science, especially theoretical science, the breakthroughs follow from breakthroughs in mathematics and it was such a set of breakthroughs that got us into closer touch with the real, non-linear world. The Tipping point idea arises from these breakthroughs and has many applications:

Tipping point (climatology), in which the system is the global climateThis short list is interesting because it illustrates the role of non-linear thinking in our current models for change. I am a systems scientist and I like to look at how the list of areas interact as one big system. This gets very exciting. Read on below and see what I mean.

Tipping point (physics), in which the system is the position of a physical object

Tipping point (sociology), is the event of a previously rare phenomenon becoming rapidly and dramatically more common

Planetary boundaries, in which living within the boundaries' stable state retains planetary habitability on Earth

In catastrophe theory, the value of the parameter in which the set of equilibria abruptly change

Angle of repose, the maximum angle of a stable slope of granular materials

In economics, the point at which a dominant technology or player defines the standard for an industry-resulting in "winner-take-all" economies of scale and scope

Early on, people like Rene Thom and Christopher Zeeman explored the phenomenon of bifurcations in the way systems behave.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour.Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs), and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them.The description above is a glimpse of the world they, among many others, opened up.

The name of the area of research they explored is Catastrophe theory.

In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.Complicated as this may all appear, it has some rather simple ways of being visualized.Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.

Catastrophe theory, which originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s, considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function).

Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.

The idea is that a system's evolution in time can be represented as occuring as a path on a surface. In other words a line on a sheet of paper if you like. A linear system has its trajectory drawn on a flat sheet. When the system is nonlinear the sheet of paper is actually bent and can assume some interesting shapes. If the paper is bent enough the line depicting the system's trajectory can curl under the original surface and when this happens it "falls off" and a jump occurs.

Such models have many realizations and books are out there about them. I want to fast forward a bit because there is a reason to be skeptical about these models because they are really too simple. What they do not deal with in any convenient manner is the links and interactions between the various aspects of the system. The change in one part of the system can change the defining equations for other parts and vice versa.

Look back at the list above. It includes climate, physical events, sociology, economics, and is only a part of what our world system is. Certainly the political systems we are so vitally interested in here are wrapped up in it all in very complex ways.Here is what I am driving at: we are now facing a variety of instabilities all at once. Again a partial list:

Climate, ocean acdity, warming, pollution, etc

elections

wars

bees and birds, insecticides

agricultural crises

water, polution and drought

economic turmoil

communications, internet, technology

many, many more

No one of these can be studied adequately isolated from the others yet we try so desperately to do so.

Just in climate science alone we keep seeing new things as change progresses that potentially change everything.

Maybe we humans are an "intelligent" species. Yet we are also arrogant if we believe the progress I have briefly described here is enough to adequately deal with what is coming and seems to be coming faster and faster. Is there an answer? My only answer is to become as flexible as possible and maybe some adaptation will be possible.

** 7:29 PM PT**: Thank you or adding this to the rec list. I hope it gets us thinking.

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