What is catastrophe theory and its application? Catastrophe theory studies the phenomenon and law of transition from one stable configuration to another. It points out any movement state of nature or human society. Let's look at catastrophe theory and its application.
What is catastrophe theory and its application 1 what is catastrophe theory?
With the birth of catastrophe theory, the "sudden jump" of the internal state integrity of the system is called catastrophe, which is characterized by continuous process and discontinuous results. Catastrophe theory can be used to identify and predict the behavior of complex systems.
"mutation"
The word "catastrophe" originally means "catastrophe" in French, emphasizing the meaning of discontinuity or sudden change in the process of change. In nature and human social activities, besides gradual and continuous smooth changes, there are a lot of sudden changes and jumps, such as rock fracture, bridge collapse, earthquake, tsunami, cell division, biological variation, human shock, emotional fluctuation, war, market change, enterprise closure, economic crisis and so on.
Basic mutation
Seven main mutations: folding mutation, cusp mutation, dovetail mutation, butterfly mutation, hyperbolic mutation, elliptic mutation and paraparabolic mutation. The secondary application research of catastrophe theory includes bifurcation theory, non-equilibrium thermodynamics, singularity theory, synergetics and topological dynamics.
Theoretical origin
Catastrophe theory originated in the late 1960s and is now considered as a part of chaos theory. 1972, French mathematicians published a book, which expounded this theory independently and systematically. His book is called Structural Stability and Morphogenesis, and Thom hopes to predict the behavior of complex and disorderly system changes through this book.
For many years, the continuous, gradual and steady movement and change process of many things in nature can be solved satisfactorily by calculus. For example, the earth goes around the sun regularly and continuously, so that people can accurately predict the future motion state, which needs to be described by classical calculus. However, there are many mutations and jumps in natural and social phenomena, and the discontinuity caused by flying over makes the behavior space of the system non-differentiable, and calculus cannot be solved. For example, water suddenly boils, ice suddenly melts, volcanoes erupt, earthquakes suddenly occur, houses suddenly collapse, and patients suddenly die.
This process from gradual change and quantitative change to sudden change and qualitative change is a sudden change phenomenon, which cannot be described by calculus. In the past, scientists encountered various difficulties in studying this mutation, the main difficulty of which was the lack of suitable mathematical tools to provide a mathematical model to describe them. So, is it possible to establish a general mathematical theory about catastrophe phenomenon to describe all kinds of jumping and discontinuous processes? This forces mathematicians to further study the mathematical theory describing the leap process and discontinuous phenomenon of catastrophe theory. 1972, French mathematician rene Toum clearly expounded the catastrophe theory in his book Structural Stability and Morphogenesis, and announced its birth.
basic content
Catastrophe theory is mainly based on topology and structural stability theory, and puts forward a new principle to distinguish catastrophe from leap: under strict control conditions, if the intermediate transition state experienced in qualitative change is stable, it is a gradual process. For example, if a wall is demolished, if bricks are demolished from above, the whole process is a process of gradually stabilizing the structure. If the wall is dismantled to a certain extent from the foot, it will destroy the structural stability of the wall and the wall will collapse. This structural instability is the process of mutation and leap. Another example is social change, from feudal society to capitalist society. The French Revolution was achieved by violence, while the Meiji Restoration in Japan was achieved gradually through a series of reforms. For the stability and instability of this structure, catastrophe theory uses the existence of potential function to express stability and the elimination of potential function to express instability, and has its own set of operating methods. For example, if a ball is stable at the bottom of a depression and unstable at the top of a protrusion, then the ball will roll down from the top and transition to a new depression, and things will suddenly change. When the ball is at the bottom of the new depression, it starts to be stable again, so the existence and disappearance of the potential function depression is the basis for judging the process of stability and instability, gradual change and mutation. Thom's catastrophe theory is to describe the jump of system state with mathematical tools and give the parameter region of the system in a stable state. When the parameters change, the state of the system changes, and when the parameters pass through some specific positions, the state suddenly changes.
Catastrophe theory puts forward a series of mathematical models to explain the discontinuous change process of natural and social phenomena and describe why various phenomena suddenly jump from one form to another. Such as rock fracture, bridge fracture, cell division, embryo mutation, market destruction and social structure upheaval. According to catastrophe theory, a large number of discontinuous events in natural and social phenomena can be represented by some specific geometric shapes. Tom pointed out that there are seven types of mutations controlled by four factors in three-dimensional space and one-dimensional space: folding mutation, spire mutation, dovetail mutation, butterfly mutation, hyperbolic umbilical mutation, elliptical umbilical mutation and parabolic umbilical mutation.
For example, hold an elastic steel wire with your thumb and middle finger and bend it upward, and then press it hard to deform it. To a certain extent, the steel wire will suddenly bend down and lose its elasticity. This is a common mutation phenomenon in life, which has two stable States: upward bending and downward bending. The state is determined by two parameters, one is the force of finger clamping (horizontal direction) and the other is the pressure of steel wire (vertical direction), which can be described by cusp catastrophe. Tip mutation and butterfly mutation are reversible models between several qualitative states. There are still some processes in nature that are irreversible. For example, death is a mutation, and the living can become dead, but not dead. This process can be described by the highest odd model of time functions such as folding catastrophe and dovetail catastrophe. Therefore, catastrophe theory uses an vivid and accurate mathematical model to describe the process of quality mutual change.
Professor Ziman, a British mathematician, called catastrophe theory "an intellectual revolution in mathematics-the most important discovery after calculus". He also set up a research group to carefully study and expand its application. In just a few years, there have been more than 400 papers, which can be said to be the heyday. Because of this achievement, Tom won the Fields Prize, the highest prize in international mathematics.
Theoretical steps
Catastrophe theory is widely used in the fields of change management and organizational development. One form of change is smooth, continuous and gradual. A series of business process improvement ideas mostly follow this change model, such as Kaizen, TotalQualityManagement and SixSigma. In terms of catastrophe theory, it is a preset change on the basis of the existing stable interface.
Another form of change is catastrophic, sudden and radical, completely deviating from the state before the change. The result of this change is often caused by drastic changes such as business process reengineering. This type of change is "discontinuous". In terms of catastrophe theory, it is a catastrophe that completely defines another stable state.
Therefore, "real" changes are more similar to drastic changes such as business process reengineering. In addition, of course, there are simple changes, and what kind of changes to adopt depends on the needs of specific problems. This is a challenge for change experts. They must be able to decide when radical changes are needed and when gradual changes should be implemented. It is not easy to make the right choice, because radical changes will inevitably lead to a period of "chaos and disorder" in the organization, and then a new stable state can be discovered and defined. This requires changing the thawing/freezing method in management. In some cases, the organization will be forced to make radical changes. Moreover, in reality, there may not be such a clear path of "where to come from and where to go", which leads to continuous and gradual changes in the organization. In this case, the assumed route of change is meaningless.
Theoretical advantage
1. Catastrophe theory is helpful to understand the true face of change management and the ideological viewpoint of chaos theory. It reveals why real change is a dangerous activity.
2. Catastrophe theory interrupts the idea that organizations can show various forms based on diversified value spectrum, and there are only a few truly stable organizational forms.
3. Catastrophe theory also reveals why change cannot be "managed" but can only be "influenced".
4. The theory deals with the ideological form of "form" (Gestalt theory) and changes. It creates a new perspective to understand the organization.
limit
1. From the perspective of understanding organizational behavior, the significance of Thom research work is more reflected in qualitative analysis than quantitative analysis.
2. Predicting even the simplest system behavior is still challenging.
3. Considering the time limit of research, everything is not a "mutation", but the cumulative effect of various factors captured by researchers at a certain moment.
4.Thom's research failed to involve complex systems with multiple (more than 5) important variables, and it may be impossible to predict the behavior of complex systems (or organizations).
What is catastrophe theory and its application? 2. Research on catastrophe theory.
Catastrophe theory studies the phenomenon and law of transition from one stable configuration to another. It is pointed out that any movement state of nature or human society can be divided into stable state and unstable state. Under the action of small accidental disturbance factors, it can still maintain its original stable state; However, once disturbed, people who leave their original state quickly are unstable, and stable state and unstable state are intertwined. The transition of nonlinear system from one stable state (equilibrium state) to another stable state occurs in the form of mutation. Catastrophe theory, as a powerful mathematical tool to study the orderly evolution of systems, can better explain and predict sudden phenomena in nature and society, and has broad application prospects in mathematics, physics, chemistry, biology, engineering technology, social science and other fields.
Catastrophe theory is to describe the qualitative change process caused by the sudden interruption of continuous action with the mathematical model of images. This theory is related to chaos theory. Although they are two completely independent theories, it is generally believed that catastrophe theory is a part of chaos theory.
Although catastrophe theory is a mathematical theory, its core idea is helpful for people to understand system changes and system interruptions. If the system is at rest (that is, it has not changed), it will tend to obtain an ideal stable state, or at least be in a certain state range. If the system is affected by external changing forces, the system will initially try to absorb external pressure through reaction. If possible, the system will return to its original ideal state. If the changing force is too strong to be fully absorbed, catastrophic changes will occur and the system will enter another new stable state or another state interval. In this process, it is impossible for the system to return to the original stable state in a continuous way.
Give an example to explain this theory more vividly. Let people imagine that there is a glass bottle on the table, which is in a stable state without any change. This is a stable equilibrium. Now imagine gently pushing the bottleneck with your fingers, not too hard. At this time, the change occurred and the glass bottle shook. It absorbs changes in a continuous way, which is an unstable balance. If you stop pushing, the glass bottle will return to the ideal stable state. However, if you continue to push hard, when your thrust reaches a certain level, the glass bottle will fall down and enter a new stable equilibrium state. The state of the glass bottle suddenly changed at this moment, and a discontinuous change happened like this: during the falling process of the glass bottle, it is impossible to have a stable intermediate state until it completely falls on the table.
Thorne's catastrophe theory means that the change of the system is realized through continuous and discontinuous change modes. This process is related to chaos theory, because the glass bottle has only two States-standing or lying down. These two states are also possible results of the basin. See chaos theory. However, some states can never be achieved because they are inherently unstable.