Qualitative Considerations
So it looks like were left with nothing?
However, lets 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 (19372010), 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 (18541912), «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.
Lets 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.
Beyond the boundaries
Beyond the boundaries of physics as an exact science, there are attempts to generalize the formation and self-organization of structures in open systems that are far from thermodynamic equilibrium. As conceived by its creator, Professor Hermann Haken, synergetics is an interdisciplinary direction that is called upon to play the role of a kind of metascience, noticing and studying the general nature of those patterns and dependencies that private sciences considered «their own».
«Synergetics (from other Greek prefixes «syn» with the meaning of compatibility and «ergon» activity) «together action».
«It should be emphasized that synergetics is by no means one of the frontier sciences such as physical chemistry or mathematical biology, emerging at the intersection of the two sciences.»
This is a fairly new area of science there are still no clear boundaries, or even clear definitions of areas. The area of research in synergetics is not clearly defined and can hardly be limited, since its interests extend to all branches of natural science. A common feature is the consideration of the dynamics of any irreversible processes and the emergence of fundamental innovations.
Since there are no clear boundaries and definitions, there is an inevitable dispute in scientific and pseudo-scientific circles. Anyone is ready to declare himself a zealot of true metaknowledge, and to call everyone else pseudo-scientists, scholastics and charlatans.
I must say that this is a completely normal phenomenon for science. True Science differs from faith or ideology in that it itself calls into question all its achievements. Especially on the frontier of knowledge, that is, exactly where new knowledge about the self-organization of matter is being formed today.
Maybe our approach will be accused of being pseudo-scientific. Real academics are simply obliged to do this.
Truth, scientists say, is just the most appropriate concept to explain facts. I think this principle is good for business as well. Lets test our theory with practice.
So, what do we get from the sciences at the start of the path? No strict laws, no numerical mathematical apparatus. The more complex and larger the system, the less numerical accuracy, the more qualitative considerations. Is this enough to put such knowledge into practice and apply it successfully?
Here well see. What difference does it make whether it is physics or something else, as long as the conclusions work! That is, they would help us in business.
Chaos that creates
Progress
The evolution of matter today is already widely considered as part of a global synergetic process. And the emergence of life on the planet, and the origin of the mind, the evolution of conscious life as part of the epic self-organization of matter.
The origin of life itself is still a big mystery. And were not going to solve it. But, since we live in this world, now we are concerned about another question how to live? For us, it is only important that with the transition from inanimate nature to living nature, the essential feature of systems changes. Inanimate systems are passive systems. And self-organization arises only when the system is pumped with energy from outside. The water in the kettle starts to move (convection), and then boils just because we turned on the stove. Whirlwinds in the atmosphere cyclones and tornadoes are the result of its heating by the energy of the sun. If the energy supply is cut off, the systems will rush to an equilibrium state, any heterogeneity will sooner or later be resolved.