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So it looks like we’re left with nothing?

However, let’s listen to the greats. It seems that not everything is so hopeless!

The mathematics of describing nonlinear effects is highly non-trivial. But, as Academician V. I. Arnold (1937—2010), one of the greatest mathematicians of the 20th century, said:

«These objective laws of the functioning of nonlinear systems cannot be ignored. Only the simplest qualitative conclusions have been formulated above. The theory also provides quantitative models, but qualitative conclusions seem to be more important and at the same time more reliable: they depend little on the details of the functioning of the system, the structure of which and the numerical parameters may not be well known.»

Henri Poincaré (1854—1912), «the last of the great universal mathematicians,» also said that only a limited amount of qualitative information is needed to understand qualitative changes in the behavior of systems.

So there are no formulas. They are useless. But there is good news! It turns out that it is important not to calculate the exact trajectory of changes, but to be ready for the phenomenon – for the critical point and for the qualitative transition that will follow. Actually, this is what we do in the morning when we boil water for tea. We do not calculate or measure anything, we just wait for the moment of a qualitative transition – we wait for the water to boil. And this is enough for us to understand that the moment has come, you can make tea.

Let’s return to our elastic band, to our manual bifurcation. When we squeezed it and got a deflection, we can play with it further, for example, try to put pressure on the bulge.

Fig. 6. Longitudinal and transverse action on an elastic object

Our «antistress» with a certain effort will begin to flip in the opposite direction. If we draw a set of solutions to the equation in the parameter space: Deflection / Longitudinal pressure / Transverse pressure, then we will find a funny surface in it, similar to the assembly of a fabric. This surface is in a section of mathematics called Catastrophe Theory and is called Cusp catastrophe.

On this decision surface, we will see the buckling path under longitudinal pressure, which we have already seen in Fig. 5. To do this, it is enough to cut our Assembly with a vertical plane, for which the transverse pressure is equal to zero.

Fig. 7. Surface of the state of an elastic object. Buckling under longitudinal compression

The area of instability is represented by a triangular «tongue», indicated by a dotted line in the middle of the Cusp, where the system can get and stay in this state for some time, until any infinitely small impact throws it into one of the stability zones – a deflection in one direction or another.

We can also trace the trajectory of the state of the object under the influence of transverse pressure.

Fig. 8. Transverse route on the state surface. Memory effect

By itself, the understanding of a mathematical catastrophe, as a kind of map, a qualitative picture of the space of possible states, already allows us to understand a lot about the behavior of an object, to be prepared for surprises, and moreover, to use these properties. Despite the fact that our pictures of catastrophic behavior do not promise any quantitative accuracy, the operating point of the system, having fallen into the zone of instability, for example, does not know when and where it will leave it.

The transverse route – the transfer of such systems from one state to the opposite (the so-called hysteresis phenomenon) is used in many places, for example, in binary memory cells. And in order to use this memory, it turns out that it is necessary to control only one control parameter, which switches the cell.

Is a high-quality picture enough to expect and manage high-quality transitions in our systems, including business ones? We will see this below.