📚 7. Classification of Linear Equilibrium Points | 线性平衡点的分类
In the study of differential equations and dynamical systems, an equilibrium point tells us where a system can rest. For a two-dimensional linear system of the form ẋ = Ax, the origin is the only equilibrium point when the matrix A is invertible. The behaviour of solution curves near this point can be classified into a small number of geometric types – nodes, saddles, spirals, centres and degenerate cases – all determined purely by the eigenvalues of A. This classification is not only central to IB Mathematics analysis and approaches but also forms the foundation for understanding non-linear systems through linearisation.
在研究微分方程与动力系统时，平衡点表示系统可以静止的位置。对于二维线性系统 ẋ = Ax 而言，当矩阵 A 可逆时，原点是唯一的平衡点。解曲线在该点附近的行为可以被归纳为少数几种几何类型——结点、鞍点、焦点、中心以及退化情形——这些完全由矩阵 A 的特征值决定。这一分类不仅是 IB 数学分析与方法的核心内容，也是通过线性化理解非线性系统的基础。
1. What Is an Equilibrium Point? | 什么是平衡点？
An equilibrium point (or critical point) for a system dx/dt = f(x, y), dy/dt = g(x, y) is a point where both derivatives vanish simultaneously, so the system stays constant. In the linear homogeneous case ẋ = Ax, the condition Ax = 0 gives x = 0 as the unique equilibrium when det A ≠ 0. Near this point we study how trajectories approach, leave, or circle around it.
对于系统 dx/dt = f(x, y), dy/dt = g(x, y)，平衡点（临界点）是指两个导数同时为零的点，这样系统就保持不变。在线性齐次情形 ẋ = Ax 中，条件 Ax = 0 在 det A ≠ 0 时给出唯一的平衡点 x = 0。我们就在该点附近研究轨线如何趋近、远离或环绕它。
- Equilibrium at the origin is isolated when det A ≠ 0. | 当 det A ≠ 0 时，原点处的平衡点是孤立的。
- If det A = 0, there is a whole line of equilibria (degenerate). | 若 det A = 0，则存在一整条平衡点直线（退化情形）。
- Stability is determined by whether all eigenvalues have negative real parts. | 稳定性取决于所有特征值是否都具有负实部。
2. Linear Systems in the Plane | 平面线性系统
A general two-dimensional linear system is written as ẋ = a x + b y, ẏ = c x + d y, where A = [[a, b], [c, d]]. The qualitative behaviour of all solution curves near the origin is completely captured by the eigenvalues λ₁, λ₂ of A. Thus, classifying equilibrium points reduces to analysing the possible combinations of real, repeated, or complex conjugate eigenvalues.
一般二维线性系统可写为 ẋ = a x + b y, ẏ = c x + d y，其中 A = [[a, b], [c, d]]。所有解曲线在原点附近的定性行为完全由矩阵 A 的特征值 λ₁, λ₂ 决定。因此，线性平衡点的分类就归结为分析实特征值、重根特征值以及共轭复特征值可能组合的情形。
The trace T = a + d and determinant D = ad − bc give the characteristic polynomial: λ² − Tλ + D = 0. The discriminant Δ = T² − 4D determines whether eigenvalues are real and distinct, repeated, or complex conjugate.
迹 T = a + d 与行列式 D = ad − bc 给出特征多项式：λ² − Tλ + D = 0。判别式 Δ = T² − 4D 决定了特征值是相异实根、重根，还是共轭复根。
3. Eigenvalues: The Key to Classification | 特征值：分类的关键
Every type of phase portrait near the origin arises from the nature of λ₁ and λ₂. Real eigenvalues give exponential growth or decay along distinct directions; complex eigenvalues produce oscillations. The sign of the real part dictates whether trajectories move inwards or outwards with time.
原点附近每种类型的相图都源自 λ₁ 与 λ₂ 的性质。实特征值产生沿不同方向的指数增长或衰减；复特征值产生振荡。实部的符号决定了轨线随时间向内还是向外运动。
- Real, distinct, same sign → Node | 相异实根且同号 → 结点 (Node)
- Real, distinct, opposite sign → Saddle | 相异实根且异号 → 鞍点 (Saddle)
- Repeated, two independent eigenvectors → Star/Proper node | 重根且有两个独立特征向量 → 星形结点/正常结点
- Repeated, one eigenvector → Improper/degenerate node | 重根且只有一个特征向量 → 退化结点
- Complex with non-zero real part → Spiral/Focus | 复根且实部非零 → 焦点/螺旋 (Spiral)
- Purely imaginary → Centre | 纯虚根 → 中心 (Centre)
4. Real Distinct Eigenvalues | 相异实特征值
When λ₁ and λ₂ are real and distinct, we can find two linearly independent eigenvectors. Trajectories approach or leave the origin along these eigen-directions. If both eigenvalues are positive, all trajectories move away from the origin – an unstable node. If both are negative, they move towards the origin – a stable node. The eigenvector corresponding to the eigenvalue with larger absolute value dominates the direction of flow near the equilibrium.
当 λ₁ 与 λ₂ 为相异实根时，我们可以找到两个线性无关的特征向量。轨线沿着这两个特征方向趋近或离开原点。若两个特征值均为正，所有轨线远离原点——这是 不稳定结点。若均为负，轨线趋近原点——这是 稳定结点。绝对值较大的特征值对应的特征向量在平衡点附近主导流动方向。
λ₁ > λ₂ > 0 → Unstable node (source)
λ₁ < λ₂ < 0 → Stable node (sink)
5. Saddle Points | 鞍点
If λ₁ > 0 and λ₂ < 0 (or vice versa), the equilibrium is a saddle point. There are two special trajectors: the stable manifold along the eigenvector with negative eigenvalue (solutions approach the origin) and the unstable manifold along the eigenvector with positive eigenvalue (solutions move away). All other trajectories follow hyperbolic curves that avoid the origin. A saddle is always unstable.
若 λ₁ > 0 且 λ₂ < 0（或反之），则平衡点为 鞍点。存在两条特殊轨线：沿负特征值特征方向的稳定流形（解趋近原点）和沿正特征值特征方向的不稳定流形（解远离原点）。其余轨线呈双曲线形状，绕过原点。鞍点总是 不稳定的。
Recognising saddles is crucial because even a single positive eigenvalue makes the whole system unstable, no matter how negative the other eigenvalue may be.
识别鞍点至关重要，因为即使另一个特征值再负，只要存在一个正特征值，整个系统就是不稳定的。
6. Repeated Real Eigenvalues | 重实特征值
When Δ = T² − 4D = 0, we have a repeated eigenvalue λ = T/2. The classification then depends on the number of linearly independent eigenvectors. If there are two independent eigenvectors, A is a scalar multiple of the identity and trajectories form a star node (proper node) – all straight lines through the origin. If there is only one eigenvector, we obtain an improper (degenerate) node where trajectories curve towards the single eigen-direction.
当 Δ = T² − 4D = 0 时，特征值为重根 λ = T/2。此时分类取决于线性无关特征向量的个数。若存在两个独立特征向，矩阵 A 是单位阵的标量倍，轨线形成 星形结点（正常结点） ——所有轨线均为通过原点的射线。若仅有一个特征向，我们就得到 退化结点，轨线最终弯向唯一特征方向。
- λ > 0: Unstable star or improper node. | 不稳定星形结点或退化结点。
- λ < 0: Stable star or improper node. | 稳定星形结点或退化结点。
7. Complex Conjugate Eigenvalues | 共轭复特征值
When T² − 4D < 0, eigenvalues are complex conjugates λ = α ± iβ. The origin now becomes a spiral (focus) if α ≠ 0, or a centre if α = 0. In a spiral, trajectories wind around the origin infinitely many times while moving inward (α < 0, stable spiral) or outward (α > 0, unstable spiral).
当 T² − 4D < 0 时，特征值为共轭复数 λ = α ± iβ。此时原点在 α ≠ 0 时为 焦点（螺旋）；在 α = 0 时为 中心。对于焦点，轨线绕原点旋转无穷多次，同时向内（α < 0，稳定焦点）或向外（α > 0，不稳定焦点）运动。
A centre occurs when all trajectories are closed periodic orbits surrounding the origin. Centres are neutrally stable – nearby trajectories neither spiral in nor out, but linear analysis alone cannot guarantee a centre for a non-linear system (we need constants of motion). For strictly linear systems, however, pure imaginary eigenvalues always give a centre.
当所有轨线是围绕原点的闭合周期轨道时，出现中心。中心是 中性稳定的 ——邻近轨线既不向内螺旋也不向外螺旋；但仅凭线性分析无法保证非线性系统中的中心（需要运动常数）。然而，对于严格线性系统，纯虚特征值总是给出中心。
α ± iβ with α ≠ 0 → Spiral; α = 0 → Centre
8. The Trace-Determinant Diagram | 迹-行列式图
All possible linear phase portraits can be mapped onto the (T, D) plane. The parabola D = T²/4 divides real eigenvalues (inside the wedge D < T²/4) from complex eigenvalues (above the parabola, D > T²/4). Above the T-axis (D > 0), saddles lie below the diagonal D = 0? Actually saddles occur when D < 0, regardless of T. This powerful diagram lets you classify an equilibrium by computing just T and D.
所有可能的线性相图都可以映射到 (T, D) 平面上。抛物线 D = T²/4 将实特征值（楔内 D < T²/4）与复特征值（抛物线上方，D > T²/4）分隔开来。在 D > 0 区域，鞍点出现在 D < 0 时（与 T 无关）。这个强大图示使你只需计算 T 和 D 就能对平衡点进行分类。
| Region on (T,D) plane | Eigenvalue type | Phase portrait |
|---|---|---|
| D < 0 | Real, opposite signs | Saddle (unstable) |
| 0 < D < T²/4, T < 0 | Real, both negative | Stable node |
| 0 < D < T²/4, T > 0 | Real, both positive | Unstable node |
| D > T²/4, T < 0 | Complex with negative real part | Stable spiral |
| D > T²/4, T > 0 | Complex with positive real part | Unstable spiral |
| D > 0, T = 0 | Purely imaginary | Centre (neutrally stable) |
When D = 0, at least one eigenvalue is zero – the system is degenerate and the phase portrait consists of lines of equilibria.
当 D = 0 时，至少有一个特征值为零——此时系统是退化的，相图由平衡点直线组成。
9. Stability of Equilibrium Points | 平衡点的稳定性
Stability is described in terms of what happens as t → ∞. A point is asymptotically stable if every trajectory starting sufficiently close tends to the equilibrium. This happens precisely when both eigenvalues have strictly negative real parts (Re(λ) < 0). It is unstable if at least one eigenvalue has positive real part. It is neutrally stable if eigenvalues are purely imaginary (Centre).
稳定性是根据 t → ∞ 时的行为来描述的。若所有从足够近处出发的轨线都趋近平衡点，则该点是 渐近稳定的。当且仅当所有特征值的实部严格为负（Re(λ) < 0）时，才会出现这种情况。若至少一个特征值实部为正，则是不稳定的。若特征值为纯虚数（中心），则为中性稳定。
- Stable node and stable spiral: globally asymptotically stable for linear system. | 稳定结点和稳定焦点：对线性系统是全局渐近稳定的。
- Centre: trajectories neither approach nor leave; small perturbations yield nearby cycles. | 中心：轨线既不趋近也不离开；微小扰动得到邻近的周期轨道。
- Stability of degenerate cases requires analysis of higher-order terms. | 退化情况的稳定性需要高阶项分析。
10. Sketching Phase Portraits Step by Step | 逐步绘制相图
To sketch a full phase portrait for a linear system, follow these steps:
绘制线性系统完整相图，遵循以下步骤：
Step 1: Compute eigenvalues λ₁, λ₂ of matrix A. Identify their type and stability.
步骤1: 计算矩阵 A 的特征值 λ₁, λ₂。确定它们的类型与稳定性。
Step 2: If eigenvalues are real, find eigenvectors v₁, v₂ and draw the eigendirections. For stable nodes, indicate motion towards origin along both axes; for unstable nodes, away.
步骤2: 若为实特征值，求特征向量 v₁, v₂ 并画出特征方向。对于稳定结点，标出沿两轴趋近原点的运动；对于不稳定结点，则远离。
Step 3: For a saddle, draw the stable and unstable manifolds. Add hyperbolic curves in other quadrants.
步骤3: 对鞍点，画出稳定和不稳定流形。在其余象限中添加双曲线轨线。
Step 4: For spirals, indicate the sense of rotation (clockwise or counter clockwise) by testing a sample point, e.g. (1,0), to see the direction of velocity vector.
步骤4: 对焦点，通过检验样本点如 (1,0) 的速度向量方向来表示旋转方向（顺时针或逆时针）。
Step 5: Label axes and draw a few representative trajectories to convey global behaviour. Always add arrowheads to show direction of increasing t.
步骤5: 标注坐标轴并画出若干代表性轨线以传达全局行为。始终添加箭头以显示 t 增加的方向。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Confusing T and D signs: A negative determinant D immediately signals a saddle, regardless of T. Students often misjudge stability by looking only at the trace.
混淆 T 和 D 的符号： 负行列式 D 直接表明鞍点，与 T 无关。学生常只看迹就对稳定性做出错误判断。
Misreading complex eigenvalues: For λ = α ± iβ, stability depends on α, not on β. Even if β is very large, a positive α still makes the spiral unstable.
误读复特征值： 对 λ = α ± iβ，稳定性取决于 α 而非 β。即使 β 很大，正的 α 仍使焦点不稳定。
Assuming all repeated eigenvalues give star nodes: Star nodes only occur when the matrix is a scalar multiple of identity. An improper node is far more common and requires care with eigenvectors.
误认为所有重特征值都给出星形结点： 星形结点仅当矩阵为单位阵的标量倍时出现。退化结点远为常见，需要注意特征向量。
Thinking a centre in a linear system guarantees a centre in an approximating non-linear system: This is false; non-linearities can turn a centre into a weak spiral. IB typically asks for linear classification only, but you should be aware of this subtlety.
认为线性系统中的中心能保证在近似非线性系统中也是中心： 这是不对的；非线性项可能把中心变成弱焦点。IB 通常只要求线性分类，但你应了解这一细微差异。
12. Worked Example and Quick Reference | 实例解析与速查
Example: Classify the equilibrium at (0,0) for the system ẋ = 3x + 2y, ẏ = x + 4y. Compute T = 7, D = (3)(4) − (2)(1) = 10. Characteristic equation λ² − 7λ + 10 = 0 ⇒ (λ−5)(λ−2)=0, so eigenvalues λ₁=5, λ₂=2. Both positive real → unstable node. Eigenvectors: for λ=5, (A−5I)v=0 gives v₁ = (1,1)ᵀ; for λ=2, v₂ = (−2,1)ᵀ. Phase portrait shows trajectories leaving the origin, with the steeper direction along v₁ (larger eigenvalue).
实例： 对系统 ẋ = 3x + 2y, ẏ = x + 4y，分类原点 (0,0) 处的平衡点。计算 T = 7, D = (3)(4) − (2)(1) = 10。特征方程 λ² − 7λ + 10 = 0 ⇒ (λ−5)(λ−2)=0，特征值 λ₁=5, λ₂=2。均为正实根 → 不稳定结点。特征向量：对 λ=5，(A−5I)v=0 得 v₁ = (1,1)ᵀ；对 λ=2，v₂ = (−2,1)ᵀ。相图展示轨线从原点离开，较陡的方向沿 v₁（较大特征值）。
| Eigenvalues | Type | Stability |
|---|---|---|
| λ₁ > λ₂ > 0 | Unstable node | Unstable |
| λ₁ < λ₂ < 0 | Stable node | Asymptotically stable |
| λ₁ > 0, λ₂ < 0 | Saddle | Unstable |
| λ = α ± iβ, α < 0 | Stable spiral | Asymptotically stable |
| λ = α ± iβ, α > 0 | Unstable spiral | Unstable |
| λ = ± iβ | Centre | Neutrally stable |
| Repeated λ > 0, 2 eigenvectors | Unstable star node | Unstable |
| Repeated λ < 0, 1 eigenvector | Stable improper node | Asymptotically stable |
By mastering these eight distinct portrait types and their links to T and D, you can confidently tackle any IB exam question on phase plane analysis. Always draw a quick trace-determinant sketch in the margin to verify your reasoning before writing the final answer.
掌握这八种不同的相图类型及其与 T 和 D 的关联，你就能自信地应对 IB 考试中任何关于相平面分析的题目。在写出最终答案前，总是在草稿边上快速画一张迹-行列式草图来验证推理。
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