Discontinuous changes in which things suddenly jump from one state form to another fundamentally different state form, including the instantaneous process of sudden changes, are called catastrophes. Catastrophe theory takes discontinuity as the research object, and uses mathematical tools such as topology, singularity theory and structural stability to study the transition of a system (process) from one stable state to another. Catastrophe theory uses a set of parameters to describe the state of the system. When the system is in a steady state, a function representing the state of the system takes a certain value (such as energy minimization or entropy maximization). When the parameters change in a certain range and the function value has multiple extreme values, the system is bound to be in an unstable state. If the parameters change slightly, the unstable system can enter another stable state, that is, the state suddenly changes at this moment.
A point in the parameter space can correspond to multiple steady-state solutions (asymptotically stable and unstable) of the system. Only when there are multiple steady-state solutions, it is possible for the system to transition between asymptotically stable steady-state solutions, and then sudden changes will occur. The existence of multiple steady-state solutions comes from nonlinearity, so mutation will only occur in complex systems with nonlinearity.
Catastrophe makes the state space of the system differentiable, and the traditional calculus method is powerless to describe and explain the catastrophe. Catastrophe theory is a mathematical model of catastrophe phenomenon and a theory about singularity. One of its important features is to study the discontinuous mutation phenomenon in the natural and social fields, explain the real things and predict the future mutation of things, so as to control the occurrence of mutation.
Catastrophe theory not only studies the dynamic stability of the system, but also studies the structural stability of the system, pointing out that the stability of the system should meet the stability of the dynamic system and the structural stability at the same time, so the stability of the system can be controlled by changing the system structure.
(1) Mutation Theory and Mutation
Every natural process has its stable state and unstable state. The characteristic of steady state is that it can still maintain its original state under the action of various small fluctuations. Once it is disturbed very little, it quickly changes its original state, which is an unstable state. Changing from unstable state to stable state is the movement trend of natural process; However, the change from stable state to unstable state requires external work.
The stable state and unstable state in catastrophe theory are relative to certain control conditions. Under the control of external force, the stable state and unstable state of the system can be freely converted. If the initial state of the system is stable, the state of the system will also change under the continuous control force or tectonic force. When the control factors reach a certain limit (critical value), the state of the system also reaches the critical state of stability and instability. At this time, although the control factors will not change, the state of the system will still jump away from the critical surface or change to another new stable state quickly, which is catastrophe. When the control factor reaches the critical value, the mutation does not occur, but occurs in the waiting period after the critical value. The types of mutations can include jumping discontinuous changes and gradual continuous changes. The difference between the two is that the former happens so fast that it is difficult to find the process of mutation, while the latter lasts for a short time.
(2) The development of catastrophe theory and the description of catastrophe characteristics.
Catastrophe theory is a new branch of mathematics to study discontinuous phenomena, which is developed on the basis of system structural stability theory, topology and singularity theory. Although using differential equation model to describe natural phenomena has achieved great success in many disciplines, differential equation is only suitable for describing continuous changing phenomena, but not for describing discontinuous phenomena, that is, abrupt phenomena. Catastrophe theory takes the position of topological surface in three-dimensional space as the model of catastrophe process.
Many interesting phenomena in nature involve discontinuity. This discontinuity can be reflected in time, such as the breaking of waves, the division of cells or the collapse of bridges; It can also be reflected in space, such as the boundary of an object or the interface between two biological tissues. However, most of the skills available to applied mathematicians are designed and applied for continuous quantitative research. These methods are mainly based on calculus, but they have been greatly refined and expanded since Newton and Leibniz, which has greatly improved people's understanding of nature.
We know that everything that is considered to be complex has the characteristics of mutation. Mutation has two meanings here: one is the emergence of complex things as a whole or at a certain level; The second kind refers to the sudden change of various decomposable parts within a complex thing or within a certain level. As long as the nature of our research object involves the same change between different material levels, the mutation characteristics will appear immediately. No matter from the higher level or the lower level of the system, the results of those newly generated attributes are abrupt changes.
The sudden appearance of traits seems to have no preparation stage, and it is difficult to find that some changes are directly related to their local living environment at that time. Catastrophe theory studies the problems reflected by this phenomenon. Mutation shows that only mutation is the real qualitative change, which has two remarkable characteristics-completely driven by internal factors and jumping of change. Through the study of catastrophe theory, we can find that no matter whether the controlling factors change or remain unchanged after reaching the critical point, they can not affect the catastrophe process, and the catastrophe process is only determined by internal factors.
(3) Two hypotheses of catastrophe theory
Catastrophe theory explains catastrophe under the following two assumptions:
1) It is assumed that the state of the system at any moment can be completely determined by given n variables (x 1, x2, ..., xn). N is finite, but it can be large. (x 1, x2, …, xn) is called the state variable or internal variable of the system. At the same time, it is assumed that the system is controlled by m independent variables (u 1, u2, …, um), that is, the values of these variables determine the values of Xi (I = 1, 2, …, n). (ul, u2, …, um) is called the control variable or external variable of the system.
2) It is assumed that the dynamic model of the system can be derived from the smooth potential. Smooth potential leads to the discontinuous form of dynamics through the disappearance of steady state.
Based on the above assumptions, catastrophe theory holds that the number of possible discontinuous structures with different properties does not depend on the number of state variables (which may be large), but on the number of control variables (which are generally small). Especially if the number of control variables does not exceed 4, there are only seven different types of mutations, and none of them involves more than two state variables.
(4) Singularity theory, equilibrium surface and bifurcation point set
Marx once said: "Any science can be truly perfect only by making full use of mathematics". The mathematical foundation of catastrophe theory is quite rich, and some contents are still profound, involving group theory, manifold and singularity theory of mapping in modern mathematics, especially topological methods. Let's briefly introduce some mathematical foundations of catastrophe theory.
British mathematician P.T. Sanders pointed out: "As a part of mathematics, catastrophe theory is a theory about singularity". What is a singularity? The so-called singularity is relative to the regular point Generally speaking, there are a large number of regular points, and singularities are individual. It is precisely because of the singularity that it occupies a prominent position in mathematics.
The main point of catastrophe theory is to study the transition of a system or process from one stable state to another. What is steady state? The so-called stability means that a system or process continues to appear in a certain state. External interference may make the system deviate from a certain state, resulting in instability, and when the interference is eliminated, it will return to the original state and continue to be stable. Therefore, stability not only means that things are unchanged, but also means that things have certain anti-interference ability, or when interference makes things deviate from the stable state, things can rely on some role to return to the stable state. The upright state of the tumbler is a stable state, and it can be returned to the upright state no matter which direction the disturbance causes it to deviate.
We know that the state of the system can be described by a set of parameters. When the system is in a stable state, a function that marks the state of the system takes a unique extreme value (such as energy minimization and entropy maximization). ). When the parameters change in a certain range and the function has multiple extreme values, then the system must be in an unstable state. Therefore, to examine whether a system is stable from a mathematical point of view, it often needs the extreme value of a function. To find the extreme value, we must first find the point where the derivative of the function is zero, and the point where the derivative value is zero is the simplest singularity, or critical point. If the function is Fuv(x), where u and v are parameters, then finding the critical point of the function Fuv(x) is to find the solution of the differential equation. When the values of u and v are given, one or several critical points x can be found. Therefore, the critical point X can be regarded as a single-valued or multi-valued function of parameters U and V, and this function is denoted as x = l (u, V). A surface in v), that is, the set of critical points, is called a critical surface, and the point where the function takes the extreme value is called a stable point. The critical point is not necessarily a stable point, so the critical point may or may not make the system stable.
If the state of the system can be described by a function with m parameters and n variables (x 1, x2, …, xn), the problem will be much more complicated. Because there is more than one variable, the extreme value of multivariate function should be solved by partial derivative, not just an equation, but a group of partial differential equations, that is, the critical point should satisfy the group of partial differential equations.
Dissipative structure, self-organization, catastrophe theory and earth science
Or a point abbreviated as sum delta ≡ det {H(v)}, where, as far as X is concerned, det stands for determinant, and h (v) is Hessen matrix of V, and Hesse matrix is the second-order partial derivative matrix as follows:
Dissipative structure, self-organization, catastrophe theory and earth science
However, at this time, the critical interface formed by the critical point is a high-dimensional surface in the N+M dimensional space, and the graph can't be drawn, which is almost unimaginable just by intuition. However, Thom has insightful views on the topological properties of high-dimensional surfaces, so this difficulty was solved by Thom and their standard formulas were found, which is the famous Thom classification theorem.
In catastrophe theory, we call those quantities that may mutate as state variables or internal variables, while the factors that cause mutation and change constantly are called control variables or external variables. For example, in the phase change model of water from liquid to gas (called gas-liquid phase change model), temperature t and pressure p are continuous variables that cause sudden change of water, so they are control variables, and the density of water is the state variable in the process of water boiling. The high-density state corresponds to the liquid state, and the low-density state represents the gas state.
It is assumed that the state of the system at any moment can be completely determined by given n state variables (x 1, x2, ... xn), where n is finite, but it can be large; At the same time, it is also assumed that the system is controlled by m independent variables (u 1, u2, …, um), that is, the values of these variables determine the value of xi, but they are not completely unique. We assume that m is relatively small, usually not greater than 5.
Let the dynamics of the system be derived from a smooth potential function, let the potential function V be given, and define its equilibrium surface M by an equation, where subscript x represents the gradient represented only by variable X, and this surface M consists of all critical points of V, that is, all equilibrium points (stable or unstable) of the system. It can be proved mathematically that m is a manifold, that is, a smooth surface with good behavior. Singularity set S is defined as a subset of M consisting of all degenerate critical points of V. These points satisfy the sum δ ≡ det {h (v)} = 0. The projection of s in control space c is called branch point set. The method of finding the branch point set can be obtained by eliminating all state variables from the definition equation of S. The branch point set is the set of all points in C that change the form of V.