<Ordinary Differential Equations> (by Vladimir I. Arnold ) Notes

<Ordinary Differential Equations> (by Vladimir I. Arnold ) Note

为什么研究相空间：the description of the states of the process as the points of a suitable phasespace often turns out to be extremely useful.
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The motion of the entire system is described by the motion of a point overa curve in the phase space. The velocity of the motion of the phase point overthis curve is defined by the point itself. Thus at each point of the phase spacea vector is given - it is called the phase velocity vector. The set of all phasevelocity vectors forms the phase velocity vector field in the phase space. Thisvector field defines the differential equation of the process
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方法论：The fundamental problem of the theory of differential equations is to determine or study the motion of the system using the phase velocity vectorfield. This involves, for example, questions about the form of phase curves
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In general form this problem does not yield to the methods of modernmathematics and is apparently unsolvable in a certain sense。Computers make it possible to find approximatelythe solutions of differential equations on a finite interval of time, but do notanswer the qualitative questions about the global behavior of phase curves.In what follows, along with methods for explicitly solving special differentialequations, we shall also present some methods for studying them qualitatively.The concept of a phase space reduces the study of evolutionary processesto geometric problems about curves defined by vector fields. We shall beginour study of differential equations with the following geometric problem.
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We remark that the form of the differential equation of the process, andalso the very fact of determinacy, finite-dimensionality, and differentiability ofa given process can be established only by experiment, and consequently onlywith limited accuracy. In what follows we shall not emphasize this circumstance every time, and we shall talk about real processes as if they coincidedexactly with our idealized mathematical models.
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积分法求解微分方程的局限：In the general case the problem of finding integral curves does not reduceto the operation of integration: even for very simply defined direction fields inthe plane the equations of the integral curves cannot be represented by finitecombinations of elementary functions and integrals
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Such an equation, whoseright-hand side is independent oft, is called autonomous. The rate of evolutionof an autonomous system,is determined entirely by the state of the system: the laws of nature are timeindependent.
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differential equation of normal reproduction
dx/dt= kx, k > 0.
From the form of the direction fieldit is clear that x increases with t, but is not clear whether infinite values of xwill be reached in finite time (whether an integral curve will have a verticalasymptote) or whether the curve will remain finite for all t.这就是为什么要研究相空间
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The same differential equation with negative k describes radioactive decay
这个方程的背后的意义，a natural law according to which "every" function is approximately linearlocally
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6. Exan1ple: The Equation of Normal Reproduction
dx/dt = kx,    k> 0
自然生长率 r(x) ≡dx/dt * 1/x
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7. Example: The Explosion Equation
dx/dt = kx²
 r(x) =kx
（1）
dx/dt = kx类比于，物种在1D（2D区域的边界）区域上分布，所以dx/dt与区域长度成线性关系
dx/dt = kx²类比于， 物种在2D(2D区域的面积）区域上分布，所以dx/dt与区域面积成线性关系
（2）
r(x) = k，自然生长率是常数
r(x) =kx，自然生长率与密度成线性正比
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Logistic Curve: dx/dt = (1- x)x
Harvest Quotas: dx/dt = (1 - x )x – c,    c is called the quota
Harvesting with a Relative Quota: dx/dt = ( 1- x )x- px
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Predator-Prey System
Lotka- Volterra model:
dx/dt =kx - axy,
dy/dt =-ly + bxy
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1. dx/dt含有 kx项，所以X会随自己的数量增加而增加。所以X是prey。
dy/dt 含有-ly项，所以Y会随着自己数量的增加而减少，所以Y是Predator。
X对自身是正反馈，Y对自身是负反馈。
2. 可以用自然生长率来描述该方程组：
r(x) = k – ay
r(y) = -l + bx
3. 为什么不是dx/dt =kx – ay？也就是说，为什么导致X减少的因素里与X的数量有关？
假设X均匀分布在区域S内。因为X与Y是捕食关系，所以不能交织地生活在一起；所以有理由假设S内没有Y，即Y在S外部；所以只有位于S边界上的X才有被捕食的可能。假设S内部X的数量为x，则S边界上X的数量可以近似地表示为sqrt(x)。 这可以解释 “为什么导致X减少的因素里与X的数量有关”。
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a property of the model (9) a 【robust property】 if it (or a closely similar property) also holds for every system ( 9ε) for sufficiently small ε.
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The oscillations describable by stable cycles are called 【self-oscillations】, in contrast to the forced oscillations caused by periodic external action。
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first-order homogeneous linear equation： dy/dx = f(x)y
first-order inhomogeneous linear equation:  dy/dx = f(x)y + g(x)
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Definition. A 【δ-shaped sequence】 is a sequence hN of nonnegative smooth functions equal to 0 outside neighborhoods that tend to 0 as N -->∞ and each possessing an integral equal to 1.
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∫ δ(x-ζ) g (x) dx = g(ζ),    δ(x -ζ) is the "δ-function concentrated at the point ζ."。 用这种方式来描述“ζ时刻的脉冲”，进而描述扰动
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Definition. The solution of the equation
dy/dx = f(x) y + δ(x-ζ)
with initial condition y(0) = 0, is called the 【influence function脉冲函数】 of the perturbation at the instant ~ on the solution at the instant x (or the Green's function)and is denoted y = G($_ζ$)(x ).
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§ 4. Phase Flows
- deterministic process的形式化产生了one-parameter transformation group(也称作phase flow)
（p61）This follows from the fact that, by the definition of determinacy, each state uniquely  determines both the future and the past of the process.
- A 【transformation】 of a set is a one-to-one mapping of the set onto itself.
- A collection of transformations of a set is called a 【transformation group】 if it contains the inverse of each of its transformations and the product of any two of its transformations.
群的4个性质：1.封闭性；2结合律：f(gh) = (fg)h；3.单位元；4.逆元。
The concept of a transformation group is one of the most fundamental in all of mathematics and at the same time one of the simplest: the human mind naturally thinks in terms of  invariants of transformation groups (this is connected with both the visual apparatus and our power of abstraction).
A mapping φ : G ---> H of the group G into the group H is called a 【homomorphism】 if it takes products into products and inverses into inverses: 
φ(fg)=φ(f)φ(g), φ(g($^{-1}$))=φ($^{-1}$)(g)
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A group is called 【commutative (or Abelian)交换群】 if the product is independent of the order of the factors: fg = gf for any two elements of the group. The operation in an commutative  group is usually denoted +
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Definition. A 【one-parameter group of transformations(也称作phase flow)】 of a set is an action on the set by the group of all real numbers.
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A one-parameter transformation group is the mathematical equivalent of the physical concept of a "two-sided deterministic process."
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The orbits of a phase flow are called its 【phase curves (or trajectories)】.
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The points that are phase curves are called 【fixed points】 of the flow。（用这种方式来定义不动点！）
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A 【diffeomorphism】 is a mapping that is smooth, along with its inverse。
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Example 1. M = R, g($^t$) is（可以是）translation by 2t (i.e., g($^t$)x = x + 2t). Properties 1) and 2) are obvious.
Example 1. M = R, g($^t$) is（可以是） multiplication by e($^{kt}$).
Example 2. M = R² , g($^t$) is（可以是） rotation about 0 by the angle t.
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hyperbolic rotation， xy = C，area preserved
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If the phase flow describes the course of a process with arbitrary initial conditions, then the differential equation  determines the local law of evolution of the process; the theory of differential equations is supposed to reconstruct the past and predict the future knowing this law of evolution.
The statement of a law of nature in the form of a differential equation reduces any problem about the evolution of a process (physical, chemical, ecological, etc.) to a geometric problem of the behavior of the phase curves of the given vector field.
所以我们可以放心大胆地用相图来研究微分方程了。
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Definition. The phase flow of the differential equation x = v(x) is the one-parameter diffeomorphism group for which v(.) is the phase velocity vector field.
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P65.  The reason why the two fields just given have no phase flows lies in the noncompactness of the phase space.  a smooth vector field on a compact manifold always defines a phase flow.（其实，也就是要排除奇点）
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wiki： In mathematics, a slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation.
Given an ordinary differential equation y'=f(x,y), the slope field for that differential equation is the vector field that takes a point (x,y) to a unit vector with slope f(x,y). The vectors in a slope field are usually drawn without arrowheads, indicating that they can be followed in either direction.
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Definition. The 【linear operator f($_{*x}$)】 is called the derivative of the mapping f  at the point x
The derivative of the mapping fat the point xis a linear operator f($_{*x}$)
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Tangent vectors move forward under the mappings g : M ---> N   (  v -->g($_{*x}$)v )
Functions move backward under the mappings g : M ---> N  (   g($^*$) f   ←-- f)
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Definition. The image of a vector field under a diffeomorphism onto is the vector field whose value at each point is the image of the vector of the original field at the pre-image of the given point. The image of the field v under the diffeomorphism g is denoted g($_*$) v
In other words, the image g($_*$)v of the field v in M under a diffeomorphism g of a domain M onto N is the field ω inN defined by the formula ω(y) = (g($_*$)x)v(x), where x = g($^{-1}$) y
外微分就是在计算g($_*$)
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Can every smooth direction field in a domain of the plane be extended
to a smooth vector field?
Answer. No, if the domain is not simply connected
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5. The Action of a Diffemnorphism on a Phase Flow
Let {g($^t$) : M ->M} be a one-parameter diffeomorphism group, and let f: M -> N be another onto diffeomorphism.
Definition. The image of the flow {g($^t$) } under the action of the diffeomorphism f is the flow {h($^t$): N-> N}, where h($^t$) = f g($^t$) f($^{-1}$)
M-----g($^t$)---->M
|f …...................|f
N-----h($^t$)---->N
If we regard the diffeomorphism f as a "change of variables," then the transformation h($^t$) is simply the transformation g($^t$) "written in new coordinates."
Remark. The flows {g($^t$)} and {h($^t$)} are sometimes called 【equivalent (or similar or  conjugate)】, and the diffeomorphism f is called an 【equivalence (or a conjugating diffeomorphism)】.
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Theorem. The diffeomorphism f takes the field v into the field w; conversely, if a diffeomorphism takes v into w, then it takes {g($^t$)} into { h($^t$)}.
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Definition. A diffeomorphism g : M --> M is called a 【symmetry of the vector field】 v on M if it maps the field into itself: g($_*$) v = v. We also say that the field v is 【invariant】 with respect to the symmetry g.
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Problem 1. Suppose a diffeomorphism takes the phase curves of a vector field into
one another. Is it a symmetry of the field?
Answer. Not necessarily
Problem 2. Supppose a diffeomorphism maps the integral curves of a direction
field into one another. Is it a symmetry of the direction field?
Answer. Yes.（direction field和vector field的不同之处）
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A field is said to be invariant with respect to a group of diffeomorphisms
if it is invariant with respect to each transformation of the group. In this case
we say that the field admits this symmetry group.
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The use of similarity considerations originated with Galileo, who explained the limitations in size of land animals with it. The weight grows in proportion to the cube of the linear dimension and bone strength in proportion to the square. 
 Numerous applications of these considerations in various areas of natural science bear such names as: similarity theory, dimension theory, scaling, self-modelling, and others.(很有用的样子，但这一节没看懂。我还是先去撸丁同仁的那本ODE吧)
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所有这些解法都有两个根本缺点：1.类似dx/dt = x²-t这样的方程解不了----无法表示为“初等函数和代数函数（以及它们的积分）的有限的组合”。2.有些精确解的表达式非常复杂，还不如近似解有实用价值。（但精确解还是有价值的，比如可以用作证明，在数学物理里可以建模）
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