📚 IB Mathematics: Phase Portraits and Stability Analysis of Linear Systems | IB数学：线性方程组相图与稳定性分析
Phase portraits provide a geometric way to understand the behaviour of solutions to systems of linear differential equations. Instead of solving the system algebraically, we visualise how trajectories move in the phase plane, revealing stability, oscillation, and long-term trends at a glance.
相图提供了一种几何方法，帮助我们理解线性微分方程组解的行为。我们不必代数求解整个系统，而是通过观察相平面中轨线的运动，一眼看出稳定性、振荡以及长期趋势。
1. What Is a Linear System and a Phase Portrait? | 什么是线性系统与相图？
A two-dimensional linear system can be written as:
二维线性系统可以写成：
dx/dt = ax + by, dy/dt = cx + dy
where a, b, c, d are constants. In matrix form, this is d𝐱/dt = A𝐱 with 𝐱 = (x, y)ᵀ s (x(t), y(t)) in the xy-plane, with arrows indicating the direction of increasing time t.
其中 a、b、c、d 为常数。矩阵形式为 d𝐱/dt = A𝐱，这里 𝐱 = (x, y)ᵀ。相图是在 xy 平面中绘制若干代表性轨线 (x(t), y(t))，并用箭头指示时间 t 增加的方向。
2. The Equilibrium Point and Nullclines | 平衡点与零斜线
An equilibrium (fixed point) occurs when both derivatives vanish: dx/dt = 0 and dy/dt = 0 nes are the lines where dx/dt = 0 and dy/dt = 0; their intersection gives the equilibrium.
平衡点（不动点）出现在两个导数都为零时：dx/dt = 0 且 dy/dt = 0。对于 det(A) ≠ 0 的线性系统，唯一平衡点为原点 (0, 0)。零斜线是满足 dx/dt = 0 与 dy/dt = 0 的直线，它们的交点给出平衡点。
Nullclines help sketch the flow: crossing the x-nullcline, trajectories move vertically; crossing the y-nullcline, trajectories move horizontally.
零斜线有助于绘制流的方向：穿越 x-零斜线时，轨线垂直运动；穿越 y-零斜线时，轨线水平运动。
3. Eigenvalues and Eigenvectors: The Key Tool | 特征值与特征向量：关键工具
The stability of the origin is determined by the eigenvalues λ of the coefficient matrix A. We compute them from the characteristic equation:
原点的稳定性由系数矩阵 A 的特征值 λ 决定。我们通过特征方程计算：
det(A − λI) = λ² − (trace A)λ + det(A) = 0
he discriminant Δ = T² − 4D determines whether eigenvalues are real, repeated, or complex.
迹为 T = a + d，行列式为 D = ad − bc。判别式 Δ = T² − 4D 决定特征值是实数、重根还是复数。
4. Classification by Eigenvalues | 按特征值分类
Using λ₁ and λ₂, we classify the equilibrium:
利用 λ₁ 和 λ₂，我们可以对平衡点分类：
- Real distinct eigenvalues with opposite signs → saddle point (unstable).
- Real distinct negative eigenvalues → stable node.
- Real distinct positive eigenvalues → unstable node.
- Complex eigenvalues with negative real part → stable spiral.
- Complex eigenvalues with positive real part → unstable spiral.
- Pure imaginary eigenvalues → centre (stable but not asymptotically).
- 异号实特征值 → 鞍点（不稳定）。
- 两个负实特征值 → 稳定结点。
- 两个正实特征值 → 不稳定结点。
- 实部为负的复特征值 → 稳定螺线。
- 实部为正的复特征值 → 不稳定螺线。
- 纯虚特征值 → 中心（稳定但非渐近稳定）。
5. Saddle Points: Stable and Unstable Manifolds | 鞍点：稳定流形与不稳定流形
For a saddle, λ₁ < 0 < λ₂. The eigenvectors give straight-line solutions. Along the eigenvector for λ₁, solutions approach the origin; along the eigenvector for λ₂, solutions move away. All other trajectories follow hyperbolic curves that first approach then leave the origin.
对于鞍点，λ₁ < 0 < λ₂。特征向量给出直线解。沿着 λ₁ 的特征向量方向，解趋向原点；沿着 λ₂ 的特征向量方向，解远离原点。所有其他轨线沿双曲型曲线运动，先靠近原点再离开。
Saddle points are always unstable because there is at least one direction along which trajectories escape.
鞍点总是不稳定的，因为至少存在一个方向，轨线沿该方向逃逸。
6. Nodes: Stable and Unstable | 结点：稳定与不稳定
If both eigenvalues are real with the same sign, the origin is a node. When λ₁ < λ₂ < 0, all trajectories enter the origin tangent to the eigenvector corresponding to the eigenvalue with smaller absolute value. The phase portrait shows curves that bend toward the slow direction.
如果两个特征值均为实数且同号，则原点为结点。当 λ₁ < λ₂ < 0 时，所有轨线沿绝对值较小的特征值所对应的特征向量方向进入原点。相图显示轨线弯曲朝向慢方向。
lled a proper (star) node if A is diagonalisable, or an improper (degenerate) node if not.
如果两个特征值均为正，则流动方向相反：所有轨线离开原点，形成不稳定结点。若 λ₁ = λ₂，当 A 可对角化时称其为星形结点，否则称为退化结点。
7. Spirals and Centres | 螺线与中心
genvalues are complex, λ = α ± iβ, the solutions involve sine and cosine terms. The term eαt unstable spiral). If α = 0, trajectories form closed ellipses, and the origin is a centre.
当特征值为复数时，λ = α ± iβ，解包含正弦和余弦项。eαt 控制振幅。若 α < 0，轨线向内螺旋至原点（稳定螺线）。若 α > 0，轨线向外螺旋（不稳定螺线）。若 α = 0，轨线形成闭合椭圆，原点为中心。
For a centre, trajectories are periodic but not asymptotically stable: nearby solutions stay nearby but never converge to the origin.
对于中心，轨线是周期的，但不是渐近稳定的：邻近的解保持邻近，但永远不会收敛到原点。
8. Stability Definitions in IB | IB中的稳定性定义
thematics, stability is usually described in terms of the behaviour of solutions as t → ∞:
在IB数学中，稳定性通常根据 t → ∞ 时解的行为来描述：
- Asymptotically stable: all solutions tend to the equilibrium as t → ∞.
- Stable (neutral): solutions stay bounded but do not necessarily approach the equilibrium.
- Unstable: at least one solution moves away from the equilibrium.
- 渐近稳定：所有解在 t → ∞ 时趋向平衡点。
- 稳定（中性）：解保持有界，但不一定趋向平衡点。
- 不稳定：至少有一个解远离平衡点。
Linear systems have global stability: stability of the origin is determined entirely by the eigenvalues, independent of initial conditions.
线性系统具有全局稳定性：原点的稳定性完全由特征值决定，与初始条件无关。
9. The Trace-Determinant Plane | 迹-行列式平面
The parabola D = T²/4 separates real eigenvalues (below) from complex eigenvalues (above).
我们可以在 (T, D) 平面中总结所有可能的行为，其中 T = trace(A)，D = det(A)。抛物线 D = T²/4 将实特征值（下方）与复特征值（上方）分开。
| Condition | Type | Stability |
| D < 0 | Saddle | Unstable |
| D > 0, T² > 4D, T < 0 | Stable node | Asymptotically stable |
| D > 0, T² > 4D, T > 0 | Unstable node | Unstable |
| D > 0, T² < 4D, T < 0 | Stable spiral | Asymptotically stable |
| D > 0, T² < 4D, T > 0 | Unstable spiral | Unstable |
| D > 0, T = 0 | Centre | Stable |
2 matrix, you can immediately identify the type of equilibrium without solving the system.
这个图是一个强大的复习工具：给定任意 2×2 矩阵，无需解系统即可立即判断平衡点类型。
10. Worked Example | 例题解析
Consider the system: dx/dt = −2x + y, dy/dt = x − 2y.
考虑系统：dx/dt = −2x + y，dy/dt = x − 2y。
scriminant Δ = T² − 4D = 16 − 12 = 4. Eigenvalues are λ = (−4 ± 2)/2, so λ₁ = −1, λ₂ = −3.
矩阵为 A = [[−2, 1], [1, −2]]。迹 T = −4，行列式 D = 4 − 1 = 3。判别式 Δ = T² − 4D = 16 − 12 = 4。特征值为 λ = (−4 ± 2)/2，即 λ₁ = −1，λ₂ = −3。
d to the origin as t → ∞, entering along the eigenvector for λ₁ = −1 (the slow direction).
两个特征值均为负且不相等，因此原点为稳定结点。所有轨线在 t → ∞ 时趋向原点，沿 λ₁ = −1 对应的特征向量（慢方向）进入。
11. Drawing Phase Portraits by Hand | 手绘相图技巧
Follow these steps to sketch a reliable phase portrait:
按照以下步骤绘制可靠的相图：
- Find the eigenvalues and eigenvectors of A.
- Draw the straight-line solutions along the eigenvector directions.
- Draw the nullclines to indicate the horizontal and vertical flow directions.
- Determine the type of equilibrium and sketch nearby trajectories accordingly.
- Add arrows showing the direction of increasing t.
- 求 A 的特征值和特征向量。
- 沿特征向量方向绘制直线解。
- 绘制零斜线以指示水平和垂直流动方向。
- 确定平衡点类型并据此绘制邻近轨线。
- 添加箭头表示 t 增加的方向。
Symmetry and careful plotting of a few solution curves using a differential equation solver can verify your sketch.
利用对称性以及用微分方程求解器绘制几条解曲线，可以验证你的草图。
12. Connecting to Real-World Models | 联系现实世界模型
Linear systems model many IB contexts: predator-prey populations (linearised near equilibrium), coupled spring-mass systems, and electrical circuits. Stability analysis tells us whether a small disturbance will decay, persist, or grow.
线性系统建模了许多IB场景：捕食者-猎物种群（在平衡点附近线性化）、耦合弹簧-质量系统、以及电路。稳定性分析告诉我们微小扰动是会衰减、持续还是会增长。
For example, in a linearised predator-prey model, a centre corresponds to neutrally stable oscillations, while a spiral corresponds to damped or growing oscillations. Understanding phase portraits helps you interpret these systems qualitatively and connect mathematics to scientific contexts.
例如，在线性化的捕食者-猎物模型中，中心对应于中性稳定的振荡，而螺线对应于阻尼或增长振荡。理解相图有助于你定性地解释这些系统，并将数学与科学情境联系起来。
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