Dissipative structure theory is a generalized thermodynamic theory put forward by Brussels School headed by Belgian chemical physicist I.llyaPrigogine after 20 years' research, which has been widely used in many fields.

Dissipative structure theory is based on the in-depth analysis of two pairs of contradictions in nature and the irreversibility of time. Two pairs of contradictions refer to the contradiction between Newtonian mechanics and thermodynamics, and the contradiction between the "degeneration theory" of thermodynamics and Darwin's "evolution theory". In Newton's classical mechanics, time is reversible, and things develop and evolve according to certain established laws. From now on, we can infer the future and the past from the present. The image described by Newtonian mechanics is a static physical image. Although thermodynamics describes a constantly evolving physical image, and time is irreversible, he believes that the evolution of things is always in a balanced direction, that is, in a unified, single and simple direction, that is, from order to disorder, which is essentially a kind of degradation. Darwin's "evolution theory" discusses another kind of evolution completely different from thermodynamics. Its evolution is a kind of "evolution", which is complex and unbalanced. Of course, time is irreversible. For these incompatible evolutionary images in the same world, the research results of Prigogine and others show that the key is that the nature of the system itself and the state of the system are different. There are isolated systems, closed systems and open systems. An isolated system refers to a system that has neither material exchange nor energy exchange with the outside world; A closed system refers to a system with only energy exchange but no material exchange with the outside world; An open system has both material exchange and energy exchange with the outside world. The different properties of the system determine the different expressions of the second law of thermodynamics. For systems with different properties, the expression of system entropy change is as follows:

Isolation system:

ds≥0 ( 1- 19)

Closed system:

Nonlinear geotechnical mechanics foundation

Open system:

ds=des+dis ( 1-2 1)

In the equation (1-19) ~ (1-21), ds is the entropy change of the system; Q is heat; T is the temperature. Des is an entropy change caused by the material and energy exchange between the system and the outside world, which is called entropy exchange. Des can be positive, negative or zero. Dis is the entropy generated by various reversible processes in the system, which is called entropy generation. Because entropy can only be generated but not destroyed, entropy generation is non-negative. Equation (1-2 1) shows that in an open system, if DES < 0 and | DES | > DIS at the same time, there are:

Ds=des+dis2) dimension is the servo principle.

Principle of maximum information entropy: Any system always has a certain degree of freedom, which leads to various elements in the system being in different states. The quantitative measure of state diversity (complexity and chaos) is called entropy, and the entropy of the system will strive for (or present) the greatest degree of freedom to maximize entropy. Therefore, the total information of the system has a maximum at the phase transition point. From the principle of maximum information entropy, we can infer the specific position of the critical point in the process of system from disorder to order.

The main method to solve the evolution equation of order parameters in synergetics is analytical method, that is, the exact or approximate analytical expression of order parameters and the analytical discriminant of instability are obtained by mathematical analytical method. Bifurcation theory in mathematics is often used in the analysis of instability, and catastrophe theory can also be applied in the special case of potential. Synergetics also often uses numerical methods, especially when studying transient processes and chaotic phenomena.

Synergetics has many similarities with dissipative structure theory and general system theory, and there are both connections and differences between them. General system theory puts forward the relationship among order, purpose and system stability, but does not answer the specific mechanism of this stability. Dissipative structure theory solves this problem from another side, pointing out that non-equilibrium state can be the source of order. Although synergetics also comes from the study of the ordered structure of non-equilibrium systems, it breaks away from the limitations of classical thermodynamics and further clarifies the specific mechanism of system stability and purpose.

Three. Chaos dynamics [12 ~ 17]

Chaos first appeared in the paper "Chaos in the Third Cycle" published by Li and Yorke 1975. Chaos is a common phenomenon in nonlinear systems, but so far, there is no good operational definition of chaos. However, no matter what the definition of chaos is, there is a common feature: in a certain parameter space, deterministic nonlinear systems show sensitive dependence of long-term behavior on initial values. In chaotic motion, two orbits with very close initial values will be exponentially separated with the development of time. In other words, it is impossible to accurately predict the long-term behavior of the orbit.

For conservative systems, solutions satisfying different initial conditions will not tend to the same point set at the same time; For dissipative systems, solutions satisfying different initial conditions may tend to the same point set, which is called attractor. The attractor of chaotic motion usually has non-integer dimensions, so it is also called strange attractor.

Fourth, fractal theory [5 ~ 1 1]

Fractal was put forward by American scientist Mandelbrot in 1977. He called coastlines, snowflakes, chaos and other seemingly chaotic but fine-structured figures fractal. The fine structure here mainly refers to the self-similar structure, that is, it has no unified feature scale, but the images on all scales are the epitome of the whole image and are similar to each other. The main description of fractal is fractal dimension, that is, its capacity dimension is not an integer but a fraction. Size is an important characteristic quantity of geometric objects, and it is the number of independent coordinates needed to determine the position of a point in geometric objects. In Euclidean space, people are accustomed to regard space as three-dimensional, plane as two-dimensional, and straight line or curve as one-dimensional.

The basic idea of fractal geometry is that objective things have self-similar hierarchical structure, and local and whole things have statistical similarity in shape, function, information, time and space, which is called self-similarity. This self-similar hierarchical structure, appropriately enlarge or reduce the geometric size, the overall structure remains unchanged.

In fact, most fractal phenomena in nature can't strictly meet the conditions of self-similarity, such as undulating mountain contours, winding rivers, fracture images displayed after material fracture, etc. Their self-similarity is approximate and self-similarity in statistical sense. The detailed theory of statistical fractal can be found in reference [9].

Verb (abbreviation of verb) mutation theory [1 ~ 7,12 ~18,23]

Catastrophe theory is a mathematical discipline developed in 1970s, which was formally founded by French mathematician Thom in 1972. Mutation mainly means that in the process of things' development and change, they often change from one state to another by leaps and bounds, or after a period of slow continuous change, under certain external conditions, they will produce a discontinuous change. This sudden change phenomenon is very common in geotechnical engineering such as earthquake, rockburst, landslide and collapse.

Catastrophe theory is mainly based on topology and structural stability theory, and puts forward a new principle to distinguish catastrophe from jump, that is, under strict control conditions, if the intermediate transition state experienced in qualitative change is stable, it is a gradual process. Thom pointed out that there are seven types of mutations under the control of four factors: three-dimensional space and one-dimensional time: fold mutation, cusp mutation, dovetail mutation, butterfly mutation, hyperbolic umbilical cord, elliptical umbilical cord and parabolic umbilical cord. Among the seven catastrophe models, the second one, the cusp catastrophe model, is the most commonly used, as shown in figure 1-5. As can be seen from figure 1-5, the coordinates of three-dimensional space are control parameters a and b and state variable x respectively. The image of the bifurcation set is a semi-cubic parabola on the plane controlled by the control parameters (a, b), that is, the projection of the upper and lower leaves of the balance surface on the plane of the system control parameters (a, b), that is, the topological mapping from the balance surface to the control parameter plane. Therefore, the bifurcation point set (a, b) divides the control parameter plane into two regions, one in the bifurcation triangle region and the other outside the bifurcation triangle region.

According to Figure 1-5, the main features of the catastrophe model can be summarized as follows:

1), the system may have two or more different states, that is, for some control parameter ranges, the potential of the system may have two or more minima;

2) Unreachable, that is, there is an unstable equilibrium position in the middle of the folding of the equilibrium surface, and the system cannot be in this equilibrium position (that is, unreachable). From the point of view of differential equation solution, unattainable corresponds to unstable solution;

3) Sudden jump, a small change of control parameters will cause a big change of state variables, which will cause the system to jump from one local minimum critical point to another. When a sudden jump occurs, the potential energy (or state) will shift from a gradually disappearing local minimum to another critical point of the global or local minimum, which is completed in a sudden way, that is, the change of potential energy is discontinuous;

4) Divergence. In the area near the critical point, the small change (disturbance) of the initial value of the control parameters may lead to the huge difference of the final state, which indicates that the small disturbance of the parameters will cause the change of the physical process or the state nature of the system;

5) Lag, when any physical system can't repeat a certain change process strictly in reverse, there will be lag. For example, cusp mutation does not occur in the bifurcation set, but on the bifurcation set line, and the position of jumping from the bottom page to the top page is different from that of falling from the top page to the bottom page;

6) Multipath, the state variable is in a certain state in the balance surface, which can be realized by controlling different paths of parameter change.

Figure 1-5 cusp catastrophe model

Because catastrophe theory plays an important role in the nonlinear theoretical analysis of geotechnical system, it will be introduced in detail below.

1. gradient system, mutation and its conditions

In mechanical systems, Newton's second law of particles with mass m can be expressed as

Nonlinear geotechnical mechanics foundation

Where represents the damping term; F stands for the external force term, if the external force has potential V, that is

Nonlinear geotechnical mechanics foundation

Newton's equation (1-24) can be expressed as

Nonlinear geotechnical mechanics foundation

If the acceleration of the system is small, the equation (1-25) can be approximately written as follows

Nonlinear geotechnical mechanics foundation

It represents the balance between damping force and external force. Equation (1-26) is called gradient system.

Let the equilibrium point of the gradient system be x* and satisfy:

Nonlinear geotechnical mechanics foundation

Therefore, the equilibrium point x* is the critical point or stagnation point of potential, and its stability is determined by the sign of this point: if (the minimum point of potential v), the equilibrium point x* is stable, which is called the attractor of gradient system; If (the maximum point of potential V), the equilibrium point x* is unstable, which is called the repulsion of gradient system. To sum up:

Nonlinear geotechnical mechanics foundation

The bifurcation point of attractor and repulsion satisfies:

Nonlinear geotechnical mechanics foundation

It is the turning point of potential, and it is often a place with unstable structure.

2. Pan-extension and codimension

For the convenience of discussion, we usually take a polynomial whose potential V(x, μ) is x, and let x* and μ satisfy the condition (1-29) be (x*, μ) = (0,0), where μ is the control parameter.

When μ=0, the potential function V(x) has the following form:

v(x)= a3x 3+a4 x4+a5x 5+……(μ= 0)( 1-30)

If a3≠0, high-order terms such as x4 and x5 can be ignored near x=0, and a3= 1/3 can be transformed, then the formula (1-30) can be approximately written as follows.

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Extension (1-3 1), Thom suggested the standard form of this extension as follows.

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When μ=0, the formula (1-32) degenerates into the formula (1-3 1). Because the potential v expressed by equation (1-32) only contains one control parameter μ, the codimension of the system is called 1.

By analogy, the general generalizations of V(x)=(μ= 0) are as follows

Nonlinear geotechnical mechanics foundation

Nonlinear geotechnical mechanics foundation

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The complementary dimensions of their corresponding gradient systems are 2, 3 and 4 respectively.

The mutations in the gradient system composed of formula (1-32), formula (1-33), formula (1-34) and formula (1-35) are called folded, pointed, dovetail and butterfly mutations respectively. Below we focus on folding mutation and cusp mutation.

3. Folding mutation

The gradient system of folding mutation is obtained by formula (1-32):

Nonlinear geotechnical mechanics foundation

The image of its potential function V(x, μ) is shown in figure 1-6, and the black dots in the figure represent the position of the system. It can be seen from the equation (1-36) that when μ < 0, the system has two equilibrium points:

Nonlinear geotechnical mechanics foundation

Figure 1-6 folded catastrophe potential function image

As can be seen from figure 1-6(a), it is the maximum point of potential v and the only one; It is the minimum point of potential v and an attractor. When μ > 0, the equation (1-36) has no equilibrium point.

System mutation should satisfy the following formula, namely

Nonlinear geotechnical mechanics foundation

The solution is (x*, μ) = (0,0). The variation of the solution of gradient system (1-36) with μ is shown in figure 1-7. As can be seen from Figure 1-7, as μ turns from negative to positive, the attractor (stable solution) originally at μ < 0 disappears when μ=0, and then follows t→+∞, x→-∞.

In fact, as can be seen from Figure 1-6, when the control parameter μ changes from μ < 0 to μ > 0, the particles that were originally at the potential minimum value gradually rise, and when μ=0, the particles are already at the potential inflection point, which is bound to suddenly decline and enter the state of t→+∞, x→-∞.

In figure 1-7, two equilibrium points of μ < 0 are projected on the μ axis and folded together.

Fig. 1-7 diagram of the change of solution of folded catastrophe system with control parameters

4. cusp mutation

The gradient system of cusp catastrophe is obtained by formula (1-33).

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In order to better analyze the formula (1-39), we first analyze its two special cases, and then analyze its general case.

(1)μ 1= constant (μ 1=0)

At this point, the formula (1-39) becomes

Nonlinear geotechnical mechanics foundation

The potential functions of the corresponding system are

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Its image is shown in Figure 1-8, and the black dots in the figure represent the position of the system.

Nonlinear geotechnical mechanics foundation

The equilibrium point of the system can be determined by equation (1-40), i.e.

Nonlinear geotechnical mechanics foundation

When μ 2 < 0, it is the maximum point of potential V, which is a repulsive force; It is the minimum point of potential v, and it is two attractors with equal potential. When μ 2 > 0, only one equilibrium point is the minimum point of potential V, which is an attractor.

When a mutation occurs, the following formula should be satisfied, namely

Nonlinear geotechnical mechanics foundation

The solution is (x*, μ 2) = (0,0).

The solution of the system (1-40) changes with the control parameter μ2 as shown in Figure 1-9. As can be seen from Figure 1-9, as μ2 turns from negative to positive, the two attractors (stable solutions) that were originally at μ 2 < 0 merge at μ2=0, which means that the particle that was originally at the minimum potential gradually increases with the minimum potential, and when it reaches μ2=0, it is already at the new minimum potential.

The relationship between the solution of figure 1-9 system (1-40) and the control parameter μ2.

(2)μ2= constant (μ2=-3)

System (1-39) is converted to:

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At this time, the potential functions of the system are:

Nonlinear geotechnical mechanics foundation

Its image is shown in figure 1- 10, and the black dots in the figure represent the position of the system.

The equilibrium point of the system can be determined by equation (1-44), i.e.

x3-3x+μ 1=0 ( 1-46)

The number of balance points depends on:

Nonlinear geotechnical mechanics foundation

When | μ 1 | > 2, d > 0, and the potential v is only minimal (attractor, figure 1- 10(a) and (e))；); ); When | μ 1 | < 2, d < 0, and the potential v has a maximum (repulsion, figure 1- 10(c)) and two minima (attractor, figure1-10). When | μ 1 | = 2 and D=0, mutation will occur (Figure1-kloc-0/0 (b) and (d)).

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According to the formula (1-45), the mutation point shall meet the following conditions:

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The solution is (x*,) = (1, 2).

The solution of the system (1-44) varies with the control parameter μ 1, as shown in figure1-1. From the figure 1- 1 1, it can be seen that if the original particle is to the left of q', there is only one minimum potential. After entering q', there is one maximum potential and two minimum potentials before reaching p, but the particle is still in the original minimum position. When it reaches p, the original minimum position of the particle becomes an inflection point. As long as μ 1 is slightly larger than 2, the particle will suddenly jump to another minimum position Q. On the contrary, with the parameter μ 1 changing from large to small, the state will change from Q to P along the QP ′ line, and the particle will suddenly jump to Q ′. This is the mutation phenomenon represented by the system (1-44). Moreover, the mutation of μ 1 from small to large and from large to small occurs in different positions, which is called hysteresis.

The relationship between the solution of figure1-1system (1-44) and the control parameter μ 1.

(3) When μ 1 and μ2 are variables.

Derive the equation (1-33), and determine the mutation point from the sum, that is

Nonlinear geotechnical mechanics foundation

The first equation is also the equation of equilibrium point. Get by eliminating x

Nonlinear geotechnical mechanics foundation

The cusp catastrophe of the graph 1- 12 system (1-33)

On the control parameter plane (μ2, μ 1), the image of is as shown in figure1-kloc-0/2, which is two curves with sharp points at (μ2, μ1) = (0,0). Because of this, this mutation is called cusp mutation. These two curves divide the parametric plane (μ2, μ 1) into two regions, where d < 0 (between the two curves) and d > 0 (outside the two curves). In the region of D < 0, the potential has two minima (attractor) and a maximum (repulsion), while in the region of D > 0, the potential V has only one minima (attractor). On a straight line with D=0, the potential v has only a minimum value and an inflection point, where a sudden change will occur.

Six, synergy and Harken controlled principle

When analyzing the bifurcation and catastrophe of the system, the behavior of the system is influenced by many control parameters. With the change of control parameters, the system presents diversified movements and finally presents orderly characteristics. Haken believes that this is the result of cooperation between subsystems of nonlinear open systems, which is called synergy [24].

Haken also believes that in this open dissipative system, the evolution of the system can be determined by only a few control parameters (called order parameters) due to internal interaction. The evolution process of order parameters is best described by gradient system and corresponding potential, which is a universal law of open nonlinear dissipative systems. Haken called it the controlled principle, and mathematically called it the central manifold theorem. Here is an example.

For the following nonlinear systems

Nonlinear geotechnical mechanics foundation

Obviously, the system (1-5 1) has only one equilibrium point:

(x*，y*)=(0，0) ( 1-52)

Its Jacobian matrix is

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The eigenvalue of the matrix is

Nonlinear geotechnical mechanics foundation

Therefore, the equilibrium point (x*, y *) = (0 0,0) is a saddle point. The eigenvector of s 1=-β (satisfying JX=-βX) is (x=0, y is arbitrary). From the formula (1-5 1), let x=0 and get:

Nonlinear geotechnical mechanics foundation

y(t)=y(0)e-βt ( 1-56)

Therefore, (x=0, y is arbitrary) is a stable manifold M*, which tends to the equilibrium point in the form of e-β T, but the manifold corresponding to s2=0 near the equilibrium point is more complicated. By the formula (1-5 1)

Nonlinear geotechnical mechanics foundation

Near the equilibrium point, the power series solution is used to make

Nonlinear geotechnical mechanics foundation

Substitute it into the equation (1-57) and compare the coefficients to get:

Nonlinear geotechnical mechanics foundation

Nonlinear geotechnical mechanics foundation

This is the invariant manifold obtained by s2=0, that is, the central manifold Mc, as shown in figure 1- 13.

Graph/manifold of kloc-0/-13 system (1-5 1)

Substitute the formula (1-60) into the formula (1-5 1). The first formula is

Nonlinear geotechnical mechanics foundation

Near x=0, the above formula is approximately as follows

Nonlinear geotechnical mechanics foundation

Nonlinear geotechnical mechanics foundation

So the central manifold x(t) approaches the equilibrium point in the form of y(t).

From the above analysis, we can see that the points on the stable manifold (x=0) of the nonlinear system tend to the equilibrium point in the form of e-βt, while the points on the central manifold tend to the equilibrium point in the form of sum. By comparing them, we can know that the second formula of formula (1-5 1) is a fast variable equation, and the first formula is a slow variable equation. This is very small after a short time, and can be set in the second formula of the fast-changing equation (1-5 1).

Nonlinear geotechnical mechanics foundation

The first formula of the slow variable equation (1-5 1) can be written as follows

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The above formula is the governing equation (1-5 1) that determines the evolution of the system. Equation (1-65) is a gradient system, and its potential is

Nonlinear geotechnical mechanics foundation