Potential: elementary catastrophe theory studies potential system. In a strict mechanical sense, potential is the relatively conservative position energy of force field. In a thermodynamic system, the thermodynamic potential is free energy, and the direction of system evolution is determined by it. Think of "potential" as a system capable of taking a certain trend. Potential is determined by the relative relationship and interaction of system components and the relative relationship between system and environment, so system potential can describe the behavior of the system through system behavior variables (state variables) and external control parameters.
In this way, behavior space and control space can be formed under the conditions of various possible sets of external control parameters and internal behavior variables. We call the n-dimensional behavior space (also called state space) composed of n behavior variables Rn, and the m-dimensional space composed of m control variables control space and Rm. The comprehensive space is represented as rn+m, and catastrophe theory is to put the change, development and evolution of things in this RN+M space to study their behavior catastrophe. Generally speaking, RN+M is a high-dimensional space, and the control and behavior space formed by the behavior of the studied object constitutes a high-dimensional state surface, which is mathematically called hypersurface. Given the change range of control parameters, that is, in a given control space, it is mathematically called projecting the behavior of the system onto the control space to see how the behavior parameters of the system change. Another important methodological feature of catastrophe theory is to project the high-dimensional surface RN+M space onto the control space Rm, and to study how the properties of things' potential change when the control parameter C changes continuously. When m is not large, the complexity of the problem can be greatly reduced. For example, when m = 3, especially m = 2, it is very convenient and easy to study the potential function, and it has obvious geometric significance.
Steady-state point: catastrophe theory calls the point that satisfies the condition that the potential derivative of the sliding function is zero "steady-state point". Stations have different classifications under different conditions. For example, when n = 1, there are three types of stagnation points: maximum, minimum and inflection point; When n = 2, different potential functions have more types of stationary points. The deeper difference lies in the degradation and non-degradation of stagnation point. The degenerate stationary point is called singularity, because the system has a lot of strange behaviors near this point [7 1]. Discontinuous results caused by continuous changes appear at singularities.
The local nature of potential: the nature of isolated point is of little significance, and we are interested in the change of system behavior near this point. Through research, catastrophe theory finds the local characteristics of the system near the points represented by a series of definitions and theorems: ① local singularity of unsteady points; (2) Non-degenerate stagnation point has no singularity locally; ③ Degenerate stagnation location (Morse and non-Morse parts), in which the non-Morse part contains singularity.
Attractor Attractor is a limit state of system trend. Generally speaking, the system will gradually tend to a unique limit state. However, there may be multiple limit points. According to the specific situation, the limit state can be a closed trajectory or a more complex figure, such as a curved surface or a larger flow shape. The connected set of these limit points is called the "attractor" of the system. According to Thom's definition, given such an attractor A, the set of trajectories tending to A in the dynamic field constitutes a region of space Rk, which is called the basin of attractor A. If the system has multiple disjoint attractors, these attractors will be in a competitive state, and attractor A may be destroyed and decomposed into multiple attractors [Poincare called this phenomenon "bifurcation"]. This situation is similar to the "phase space" formed by all the behaviors of a small ball rolling in scraggy's golf course, and its behavior is bound to be "attracted" by the competition of various low-lying areas divided by high ground. At this time, the behavior of the ball is different in different places, and it will become unstable in some places, so the structural space of its behavior also appears local structural instability.
6. 1. 1.2 basic theory
(1) Topology: It is a related discipline similar to the study of topography. Topology is a branch of geometry, but this geometry is different from the usual plane geometry and solid geometry. Usually, the research object of plane geometry or solid geometry is the positional relationship between points, lines and surfaces and their measurement properties. Topology has nothing to do with the measurement nature and quantitative relationship of the length, size, area and volume of the research object. It studies the property that geometric objects remain unchanged after continuous transformation, and it only considers the positional relationship between objects. The essence of topology: ① topological equivalence; ② Perforated and non-perforated; ③ There are boundary constraints. This paper only introduces a well-understood topological property-topological equivalence.
Topology is also called "rubber plate geometry" figuratively. If you draw a fish on the rubber board. Then as long as the rubber sheet is stretched or compressed, it can be continuously transformed from one image to another. This operation can be understood as topological transformation or differential homeomorphism. Two geometric objects are considered topologically equivalent or homeomorphic if they can be continuously deformed to the other without any tearing or adhesion at different points. Differential homeomorphism refers to one-to-one continuously differentiable transform. Two geometric objects are diffeomorphic if they are diffeomorphic and the deformation will not cause wrinkles or flatten wrinkles. As long as two geometric objects are topologically equivalent, their qualitative properties remain unchanged after topological transformation, that is, the structure is stable, or the structure is stable. For example, for a dynamic system, if it is possible to control parameters to change continuously, the number of singularities in its phase space and the properties of attractors and repulsions remain unchanged. Although the trajectory distribution around the singularity has changed, we think its structure is moldy, that is, the topology is equivalent before and after the change. Catastrophe theory is to study the topological structure of branch point sets in a more general sense.
(2) Singularity theory: British mathematician Sanders [18] pointed out: "Singularity theory of catastrophe theorists". The so-called singularity is for regular points. Generally speaking, there are a large number of regular points, while singularities are individual. Because the singularity is unique, it plays an important role in mathematics. From a mathematical point of view, catastrophe phenomenon is also called "unstable singularity".
Catastrophe theory mainly considers the transition of a system or process from one stable state to another. The so-called steady state refers to the continuous appearance of a certain state of a system or process. The state of the system can be described by a set of parameters. When the system is in a steady state, the function of the system state takes a unique extreme value, such as minimizing energy and maximizing entropy. When the parameters can change within a certain range, if the function has multiple extreme values, then the system must be in an unstable state.
Considering whether a system is stable from a mathematical point of view, it is often necessary to require the extreme value of a function, that is, the point where the derivative of the function is zero, which is the simplest singularity or critical point.
Let the critical point of function Fuv(x) be the solution of differential equation. When u and v are given.
Water inrush from coal mine floor
One or several critical points x can be obtained. Therefore, the critical point can be regarded as a single-valued or multi-valued function of the parameters u, v, and denoted as X = L (U, v). Obviously, such a function can geometrically determine a three-dimensional Euclidean space, that is, the nearest (u, v, x) reddest surface, that is, the geometry of the critical point, which is called the critical surface. The point where the function takes the extreme value is called the stable point, and the critical point is not necessarily the stable point, so the critical point may make the system stable or unstable. Therefore, studying the stability and instability of the system is the smallest change of the function Fab(x), which is called the potential function.
The general description of a dynamic system is as follows. Let the system dynamics equation described by n state variables (also called internal variables) x 1, …xn and m control parameters (also called external variables or control variables) u 1, …um be written as follows.
Water inrush from coal mine floor
The equation becomes an autonomous dynamic equation (the right side of the equation does not contain time variables), and the right side of the equation can be expressed as the gradient of the potential function v (x {j}, u {a}), namely
Water inrush from coal mine floor
This equation is called gradient dynamic system, and its steady-state solution is obtained by the following formula. The steady-state solution {{xj0}} appears as a singularity in the phase space. Catastrophe theory is a theory that uses potential function V to study how the singularity changes with control parameters and the topological invariant relationship between V and {{xj}} and {{ua}}.
(3) Structural stability theory:
Ahesen matrix and congruent rank number: the stability of singularity can be determined by the second derivative of potential function, and the δ minimum point of potential function is attractor and the maximum point is repulsive. The gradient of the potential function determines the singularity, and the properties of the singularity are determined by its second-order partial derivative matrix, namely
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matrix
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It's called the Heisenberg matrix.
If the determinant det vij≠0≠0 of Hessen matrix, the gradient is
Water inrush from coal mine floor
A definite singularity is called an isolated singularity (or Morse singularity). Hessen matrix is symmetric and can be decomposed into diagonal matrix by linear transformation (such as orthogonal transformation). The diagonal matrix elements ω 1, ω2, ... ω n are the eigenvalues of Hessen matrix. Singularity is related to control parameters, so the eigenvalue is also a function of control parameters. If ωI(I = 1, 2, ... l) is 0, and the control parameters u 1, u2, ... um take Morse specific values, then the Hess-Sen matrix is not a full rank matrix, that is, detvij = 0, then the singularity determined by δ V = 0 and detvij = 0 is a non-isolated singularity.
B Morse lemma: the potential function v (x {j}, u {a}) can be expanded into Taylor series near the singularity. Assuming that the δ singularity is at the origin of the phase space, the constant term in this expansion can be taken as zero. According to the definition of singularity, the first-order partial derivative
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It is also zero. So there was water inrush from coal mine floor.
If the above formula has transformed Hessen matrix into diagonal shape by linear transformation, and the rank of Hessen matrix is n, the singularity will be completely determined by the eigenvalue ωi of Hessen matrix, and the higher-order term has no influence. The potential function at this time is called Morse potential, and its structure is stable.
If the rank of Hessen matrix is n-l, there is ω 1 = ω 2 = … = ω l = 0. The second-order partial derivative cannot determine the singularity of state variables x 1, x2, …, xl, so the third-order partial derivative of potential function should be considered. In this way, the Hessen matrix with cofunction L can divide the potential function into Morse part and non-Morse part, namely
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Where is the Morse part corresponding to the isolated singularity? The non-Morse part corresponding to the non-isolated singularity of VNM. Obviously, VNM is a cubic term about x 1, x2, …xl in Taylor series expansion. If the third-order partial derivative of the potential function is zero, the quartic term should be considered as zero, and so on.
Structural instability is limited to the state variables x 1, x2, …xl, and other state variables XL+ 1, XL+2, …xn have nothing to do with the properties of the potential function and can be ignored. This result shows that the potential function can be divided into Morse part and non-Morse part. At the same time, state variables are also divided into two parts, substantive variables related to structural stability and immaterial variables unrelated to structural stability. This conclusion is called Morse Lemma, and the second part can be omitted when analyzing mutation types. The tree of possible mutation type depends not on the number n of state variables, but on the number l of substantive variables, that is, on the congruence rank number l of Hessen matrix.
Codimensionality of C-universal expansion: VNM in the above potential function is a generation term of mutation, and its structure is unstable. By adding some items to make it a structural stability function in the above sense, this addition process becomes an extension.
For example, take VNM = x4 as an example to illustrate the concept of universal extension. X4 → X4+AX2 is still unstable in structure, because adding other items will produce different types from X4+AX2. If x4 is extended to
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Then the structure is stable, because adding a quintic or higher term will not affect its type, and there is no lower term to add. It is a structurally stable extension of x4, which is called a complete extension.
In fact, it is not necessary to add all the low-order terms, and a stable extension of the structure can be obtained. Through coordinate translation, cubic term and constant term can be eliminated, and the following results are obtained.
Water inrush from coal mine floor
V(x) is also a complete extension of x4, which is topologically equivalent, that is, all types encountered in the function family also appear in the function family V(x), but V(x) only uses two extension parameters, u and v, and completely extends the extension parameters to universal extension.
The number of expansion parameters required for universal expansion is equal to the complementary size. Complementary dimension is defined as the difference between the dimension of geometric object and the dimension of space, which indicates the number of equations needed to describe geometric object. For example, a two-dimensional surface (geometric object) in three-dimensional space needs an equation to describe its complementary dimension1; One-dimensional curve in three-dimensional space has a residual dimension of 2 and needs two equations; The zero-dimensional point of three-dimensional space needs three equations, and the remaining dimensions are also three.
Codimension has two important properties: divisibility means that no matter what the dimension of a geometric object is, its space can be divided into two different parts only when its codimension is 1; Invariance means that when the state variable of a potential function is divided into substantial and immaterial parts, the codimension remains unchanged after omitting the immaterial part.