First, let's observe a differential equation:
If we divide the formula (1) by the formula (2) as we solved the Lotka-Volterra model in the previous section, we will find that the sum of the variables on both sides of the equation cannot be separated, let alone the analytical solution of the equation.
But if polar coordinate transformation is adopted, the original equation is transformed into:
At this point, the analytical solution can be obtained by dividing the two formulas.
This method is called "variable change". The specific operation of substitution is easy to understand, but what is the mathematical basis of substitution? What is the theoretical system of substitution? This requires the introduction of "group" and a large number of related concepts.
"One-to-one mapping? Mapping): Under a certain mapping, different elements in a set (sometimes called a "domain") have different images in the set, whereas each element in the set has an original image in the set.
"Transformation": A one-to-one mapping from a set to the set itself. For example, it is not a transformation, because the set elements cannot find the original image; After the change, it was changed.
The "product" of transformation: transformation and the product of transformation are defined as carrying out transformation first and then carrying out transformation, that is.
The "inverse" of transformation: the inverse satisfaction of transformation.
Transformation group: a group consisting of a set of transformations.
"Group (group; Or abstract group: if (1) the product of any two elements in a transformation group is still in the transformation group (additive closure), and (2) the inverse of any element in the transformation group is still in the transformation group (inverse operation closure), then the transformation group is called abstract group.
"Exchange group; ; Or Abel group: a group satisfied by any two elements in the group.
Action: Each element in the group corresponds to a transformation about the set. At the same time, the product of any two elements in the group also corresponds to the transformation, and the inverse of any element corresponds to the transformation. This is the so-called group's influence on the set. This sounds very abstract. Let's make a simple analogy: a transformation is a machine, a group is a group of machines, and a collection is a pile of products to be processed. One element represents a product, and one action means that the machine is turned on and applied to the product for processing. Obviously, a group can act on the whole set or only one element.
"Orbit": The set formed by the group acting on an element in the set is called the locus of points.
Smooth manifold (or differential manifold): A differentiable topological manifold, such as Euclidean space. It is the same concept as "set", but smaller and stricter than "set".
Differential homeomorphism: the mapping from smooth manifold to smooth manifold. If they are all smooth, they are called differential homeomorphisms.
"Single parameter group": refers to a group consisting of a series of transformations. Where is the parameter, called "time". According to the definition of group, a single parameter group satisfies: for any real number. Yes For example, it stands for translation unit.
"Phase flow": The physical meaning of a single parameter group can also be understood as follows: a process is in an initial state, and then it is transformed every time the time step of 1 is increased. So at this moment, the phase point is located. At this time, the phase point is located at. In this sense, a single parameter group is also called phase flow, just like a particle flowing with water in phase space. The trajectory of a point in the phase space under the action of phase flow is the phase curve.
"phase velocity vector": for "one-parameter differential homeomorphism? Difference) "(therefore differentiable), the phase velocity vector of a phase flow at a point in a smooth manifold is defined as the velocity when it is about to leave that point. The phase velocity vectors of all points in a smooth manifold constitute the phase velocity field.
"Solution": For a single parameter differential homeomorphism group, the mapping can be regarded as the solution of the differential equation related to the phase flow.
To sum up, the above concepts can be roughly expressed in the following figure:
With the above concepts, let's go back to the initial question-what is the mathematical theoretical basis of substitution? The purpose of replacing differential equations is to find a suitable differential homeomorphism, thus simplifying the phase velocity vector field.
"Vector diagram": There is a vector in the set, which is the phase velocity vector when the phase point leaves the point. After the mapping acts on a vector, the vector becomes another vector in the set. Is the phase velocity vector when the phase point leaves the point. It is called an image under the action of mapping (Figure 2).
Tangent space: the space where all phase velocity vectors located at a certain point in the set are located. For example, if it is a sphere, it is a plane (Figure 3).
With these, it is natural to ask: What is the relationship between harmony and disharmony? How to ask?
It can be obtained by Taylor expansion, so it is the derivative of the mapping (Figure 4). When a set is a multidimensional space, it is a matrix satisfying the following conditions:
"Tangent vector": The phase velocity vector is defined when the coordinate system is clear, but when the coordinate system is not clear, the concept of phase velocity vector needs to be extended to a more general case, that is, tangent vector. Tangent vector refers to the tangent vector of the midpoint of a set, and it refers to a displacement (time axis, phase space) that satisfies the starting point.
Image of vector field: Just like the singular and plural forms of English words, the image of vector field is similar to the plural form of vector image. Another vector field generated by the differential homeomorphism of the vector field in the set is called the image of the vector field, which is recorded as (Figure 5).
Now suppose that the set is a one-dimensional space, and the starting point of a vector in the set is the length of this vector. Now using the transformation, the length of the vector is 2 (which also satisfies the formula). For the convenience of description, let's remember that the vector field in the set is "(basis vector field)". In this way, after the transformation, we can directly write:, thus knowing that the transformed vector field is before the transformation (the whole base vector field is reduced by half, which is equivalent to pulling all vectors in the vector field twice).
For any differential homeomorphism, its effect on the vector field is as follows:
When a set belongs to a dimensional space, any vector field can be expressed as with coordinates, where is the base vector field with coordinates.
At the same time, under the action of differential homeomorphism, the integral curve corresponding to the original phase velocity vector field in the set is also mapped to the set, which is the substitution of differential equations.
Mapping can be applied not only to collections, but also to the mapping itself. That is, one mapping can become another mapping under the action of one mapping.
"Phase flow image": Under the action of differential homeomorphism, the phase flow image satisfies (that is, Figure 6).
? At this point, we call it phase flow equivalent (or similar or * * * yoke). Differential homeomorphism is called "equivalence" (or "conjugate differential homeomorphism").
Let's review the meaning of phase flow again.
The position of Malthusian theory-"the first law of ecology" is mentioned in two books: Theory-based Ecology-Darwin's Method and Principles and Applications of Theoretical Ecology. In ecology, Malthusian theory has continuous form and discrete form. Compare the two formulas (see Notes on Theoretical Ecology (II) for details). Now let's look at their mathematical basis.
Define the phase flow satisfaction (it satisfies the closure of addition and the closure of inverse operation). Obviously, the phase curve produced by the phase flow acting on any phase point is the discrete Malthus model.
Phase velocity vector of phase flow.
Therefore, the differential equation "related" to the phase flow is obtained.
Then, order is a Malthusian model in continuous form.
Solve the equation in turn and get its solution. So the phase flow is the solution of the equation satisfying the initial conditions, that is.
Therefore, phase flow is a discrete form of ordinary differential equation, and ordinary differential equation is a continuous form of phase flow.
However, please note that not all ordinary differential equations have corresponding phase flows.
For example, the solution satisfying the initial condition is, let, where is a single parameter transformation. But time is not smooth, so it is not a differential homeomorphism, and naturally it cannot be a phase flow.
Substitution is one of the main methods to solve differential equations, and the essence of substitution is the transformation of phase velocity vector field by differential homeomorphism.
From set to set, it is called mapping; From the set to the set itself, it is called transformation; Smooth mapping is called differential homeomorphism; The "complex number" of transformation is called transformation group; The transformation group satisfying the closure of addition and the closure of inverse operation can be simply called group; A group of one-parameter differential homeomorphisms is called phase flow (figure 1).
The influence of phase flow on phase point can be regarded as a discrete evolution process. Many phase flows have corresponding differential equations.
Reading Notes: Ordinary Differential Equation (II)- Lotca-volterra Model
Reading Notes: Ordinary Differential Equation (IV)-Homogeneous Equation