📚 Constructing (x, ẋ) Phase Diagrams in IB Mathematics | IB数学：构建(x, ẋ)相图的方法
A phase diagram is a powerful visual tool for analyzing differential equations. It replaces the search for explicit solutions with a picture of all possible states and behaviours. This article explains how to construct an (x, ẋ) phase diagram step by step, with worked examples that commonly appear in IB Mathematics.
相图是分析微分方程的有力可视化工具。它用一幅展示所有可能状态和行为的图像，替代了对显式解的追求。本文将逐步解释如何构建 (x, ẋ) 相图，并配有 IB 数学中常见的例题。
The horizontal axis represents the variable x, and the vertical axis represents its first derivative ẋ. Every point in this plane corresponds to a particular position and velocity. By drawing enough arrows and curves, we can understand the long-term behaviour of the system without solving it explicitly.
横轴表示变量 x，纵轴表示其导数 ẋ。平面上的每个点对应一个特定的位置与速度。通过画出足够多的箭头和曲线，我们无需显式求解就能理解系统的长期行为。
1. What Is an (x, ẋ) Phase Diagram? | 什么是 (x, ẋ) 相图？
An (x, ẋ) phase diagram uses the horizontal axis for the variable x and the vertical axis for its first derivative ẋ = dx/dt. Every point (x, ẋ) represents a possible instantaneous state of a dynamical system. As time passes, the state moves along a curve called a trajectory.
(x, ẋ) 相图以横轴表示变量 x，纵轴表示其导数 ẋ = dx/dt。每一个点 (x, ẋ) 都代表动力学系统的一个可能瞬时状态。随着时间推移，状态沿一条称为轨迹的曲线移动。
For a first-order ODE ẋ = f(x), the phase diagram is a one-dimensional line. For a second-order ODE, the state space is two-dimensional and the phase diagram is genuinely a plane. This two-dimensional view reveals fixed points, oscillations, and stability in a single glance.
对于一阶微分方程 ẋ = f(x)，相图是一维直线。对于二阶微分方程，状态空间是二维的，相图实际上是平面。这种二维视图可以一眼揭示不动点、振荡与稳定性。
2. Converting a Second-Order ODE to a System | 将二阶微分方程转化为系统
Consider a general second-order ODE of the form ẍ = f(x, ẋ). To construct its phase diagram, we first introduce a new variable v = ẋ. Then the second-order equation becomes a system of two first-order equations.
考虑一般形式的二阶微分方程 ẍ = f(x, ẋ)。为了构建其相图，我们首先引入新变量 v = ẋ。于是二阶方程变为两个一阶方程组成的系统。
ẋ = v, v̇ = f(x, v)
The derivative of x is simply v, and the derivative of v is the original acceleration. Each point in the (x, v) plane corresponds to a particular position and velocity, and the system tells us how that point moves.
x 的导数就是 v，而 v 的导数就是原来的加速度。(x, v) 平面上的每个点对应一个特定的位置和速度，系统告诉我们该点如何运动。
This transformation is useful because systems of first-order ODEs can be studied geometrically. The variable v acts as a bridge between the original equation and its phase portrait. In IB problems, you may also be given the system directly; the same sketching method applies.
这种变换之所以有用，是因为一阶方程组可以用几何方法研究。变量 v 充当了原方程与其相图之间的桥梁。在 IB 题目中，有时也直接给出方程组；同样的作图方法仍然适用。
3. Finding Equilibrium Points | 求平衡点
Equilibrium points, also called fixed points, are states where the system does not change. Mathematically, we require ẋ = 0 and v̇ = 0 simultaneously.
平衡点，也称为不动点，是系统不发生变化的状态。数学上，需要同时满足 ẋ = 0 和 v̇ = 0。
ẋ = v = 0, v̇ = f(x, 0) = 0
Because v = 0, every equilibrium point lies on the x-axis. Its x-coordinate is a root of f(x, 0) = 0. For example, for the harmonic oscillator ẍ = −ω²x, we have f(x,0) = −ω²x, so the only equilibrium is at the origin (0,0).
因为 v = 0，所有平衡点都在 x 轴上。其横坐标是 f(x, 0) = 0 的根。例如，对于谐振子 ẍ = −ω²x，有 f(x,0) = −ω²x，因此唯一的平衡点在原点 (0,0)。
Some systems have multiple equilibria. For instance, ẍ = x − x³ gives f(x,0) = x − x³ = x(1−x)(1+x), so there are three fixed points. A good phase portrait must show the behaviour near every fixed point.
有些系统有多个平衡点。例如 ẍ = x − x³ 给出 f(x,0) = x − x³ = x(1−x)(1+x)，因此有三个不动点。好的相图必须展示每个不动点附近的行为。
4. Nullclines and the Direction Field | 零线与方向场
The nullclines are curves where one of the derivatives vanishes. The x-nullcline is given by ẋ = 0, which is simply the line v = 0 (the horizontal axis). On this line, trajectories cross vertically because the horizontal velocity is zero.
零线是其中一个导数为零的曲线。x-零线由 ẋ = 0 给出，即直线 v = 0（横轴）。在这条线上，由于水平速度为零，轨迹垂直穿过。
The v-nullcline is given by v̇ = f(x, v) = 0. On this curve, trajectories cross horizontally. The intersection points of the two nullclines are exactly the equilibrium points. Between the nullclines, we can draw short arrows showing the direction of movement; this collection of arrows is the direction field.
v-零线由 v̇ = f(x, v) = 0 给出。在这条曲线上，轨迹水平穿过。两条零线的交点正好是平衡点。在零线之间，我们可以画出短箭头表示运动方向；这些箭头的集合就是方向场。
To draw the arrows, choose a few sample points. At (x, v), the tangent vector is (v, f(x,v)). If v > 0, the trajectory points right; if v < 0, it points left. The sign of f(x,v) tells us whether the trajectory moves up or down. Combining these signs on each region of the plane gives the flow.
为了画箭头，选择若干采样点。在 (x, v) 处，切向量为 (v, f(x,v))。若 v > 0，轨迹指向右；若 v < 0，指向左。f(x,v) 的符号告诉我们轨迹向上还是向下。将平面每个区域中的符号结合起来，就得到了流向。
5. Linearization and Stability Analysis | 线性化与稳定性分析
To classify an equilibrium point, we linearize the system around it. For a system ẋ = F(x,v), v̇ = G(x,v), the Jacobian matrix at a fixed point (x*, v*) is formed by the partial derivatives of F and G.
为了对平衡点进行分类，我们在其附近将系统线性化。对于系统 ẋ = F(x,v)，v̇ = G(x,v)，在不动点 (x*, v*) 处的雅可比矩阵由 F 和 G 的偏导数构成。
J = [ ∂F/∂x ∂F/∂v ; ∂G/∂x ∂G/∂v ]
The eigenvalues of J determine the local shape of trajectories. Real eigenvalues of the same sign give a node (stable or unstable); opposite signs give a saddle; complex eigenvalues give a spiral or centre. In IB problems, you rarely need the full Jacobian; you can often sketch the phase portrait using known forms.
J 的特征值决定了轨迹的局部形态。同号实特征值给出结点（稳定或不稳定）；异号给出鞍点；复特征值给出螺旋点或中心点。在 IB 考题中，通常不需要完整计算雅可比矩阵；利用常见形式即可画出相图。
The table below summarises the possible local structures and their typical shapes.
下表总结了可能的局部结构及其典型形状。
| Eigenvalues | Type | Shape | 特征值 | 类型 | 形状 |
|---|---|---|---|---|---|
| Real, negative | Stable node | Arrows toward point | 实根，均为负 | 稳定结点 | 箭头指向该点 |
| Real, positive | Unstable node | Arrows away from point | 实根，均为正 |
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