The General Phase Plane | 一般相平面

📚 The General Phase Plane | 一般相平面

The phase plane is one of the most powerful geometric tools for analysing systems of ordinary differential equations. Instead of solving explicitly for x(t) and y(t), we study how solutions evolve in the (x, y) plane, revealing the qualitative behaviour of the whole system at a glance. This article explains the key ideas behind the general phase plane, including direction fields, nullclines, critical points, linearisation, and how to classify equilibrium points. You will learn to sketch phase portraits and interpret stability properties – skills highly valued in IB Higher Level and further mathematics.

相平面是分析常微分方程组最强有力的几何工具之一。我们不显式求解 x(t) 和 y(t)，而是观察解在 (x, y) 平面中的演化，从而一眼看穿整个系统的定性行为。本文阐述一般相平面背后的核心思想，包括方向场、零等倾线、临界点、线性化以及平衡点的分类方法。你将学会绘制相图并解读稳定性特征——这些技能在 IB 高水平和进阶数学中备受重视。

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1. Phase Plane: A Visual Tool | 相平面：一种可视化工具

When we work with a system of two first‑order autonomous differential equations, such as dx/dt = f(x, y) and dy/dt = g(x, y), each pair (x, y) represents the state of the system at time t. The phase plane is simply the xy‑plane where every point has a velocity vector (f, g) attached to it. Plotting a selection of these vectors creates a direction field, which already suggests how trajectories flow without solving the system algebraically. This geometric viewpoint transforms a calculus problem into a dynamical portrait, allowing us to assess long‑term tendencies, closed orbits, and stability in a unified way.

当我们研究两个一阶自治微分方程构成的系统——例如 dx/dt = f(x, y)、dy/dt = g(x, y)——每一组 (x, y) 都代表系统在时刻 t 的状态。相平面就是附有速度向量 (f, g) 的 xy 平面。选取部分向量绘制出的方向场，不做代数求解就能揭示轨线的走向。这种几何视角把微积分问题转换成动力学肖像，使我们能够统一评估长期趋势、闭轨以及稳定性。

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2. From Higher‑Order ODEs to First‑Order Systems | 从高阶ODE到一阶系统

Many physical problems are initially expressed as a second‑order differential equation, like my” + by’ + ky = 0. To enter the phase plane, we convert this into a pair of first‑order equations. A standard trick is to set x = y and introduce a new variable z = y’. The original equation then becomes dx/dt = z and dz/dt = –(b/m)z – (k/m)x. The phase plane now uses axes x (displacement) and z (velocity). Any higher‑order linear ODE with constant coefficients can be reduced in a similar fashion, always producing a system of the form X’ = AX for a 2×2 matrix A. This step is essential because the phase‑plane methods apply directly to first‑order autonomous systems.

许多物理问题起初表现为二阶微分方程，例如 my” + by’ + ky = 0。为了进入相平面，我们把它转化为一对方程。经典技巧是设 x = y，再引入新变量 z = y’。原方程随即变为 dx/dt = z 和 dz/dt = –(b/m)z – (k/m)x。此时相平面的坐标轴分别为位移 x 和速度 z。任何高阶常系数线性 ODE 都可类似降阶，最终得到一个形如 X’ = AX 的系统，其中 A 是 2×2 矩阵。这一步至关重要，因为相平面方法直接适用于一阶自治系统。

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3. Direction Fields and Trajectories | 方向场与轨迹

Given a system dx/dt = f(x,y), dy/dt = g(x,y), we can compute the slope dy/dx at any regular point as g/f (provided f ≠ 0). By drawing short line segments with that slope on a grid, we obtain the direction field. A trajectory is a parametrised curve (x(t), y(t)) that obeys the system; its tangent at each point matches the direction field. Importantly, distinct trajectories never cross except at critical points where both f and g vanish. Even without numerical solvers, sketching a few arrows often reveals whether solutions spiral inwards, head towards a node, or form closed loops. This qualitative insight is the first step in constructing a full phase portrait.

给定系统 dx/dt = f(x,y)、dy/dt = g(x,y)，我们可以计算任意正则点处的斜率 dy/dx = g/f（假设 f ≠ 0）。在网格上画出具有该斜率的短线段，即得方向场。轨迹是满足系统的参数化曲线 (x(t), y(t))；其每一点的切线都与方向场吻合。重要的是，不同轨迹仅在 f 和 g 同时为零的临界点处相交。即使没有数值求解器，草绘若干箭头也常能揭示解是向内螺旋、趋向结点，还是形成闭合环路。这种定性洞察是构建完整相图的第一步。

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4. Nullclines: Where the Flow Becomes Vertical or Horizontal | 零等倾线：流向垂直或水平处

Nullclines are curves in the phase plane where one of the derivatives is zero. The x‑nullcline is defined by f(x,y) = 0; along it the flow is purely vertical (dx/dt = 0). The y‑nullcline is defined by g(x,y) = 0; along it the flow is purely horizontal (dy/dt = 0). Intersections of the two nullclines are exactly the critical points of the system. Shading regions between nullclines according to the signs of f and g helps partition the plane into zones with consistent arrow directions. This structured approach simplifies the assembly of a rough phase portrait and is particularly useful for nonlinear systems where explicit solutions are unavailable.

零等倾线是相平面中某个导数为零的曲线。x‑零等倾线由 f(x,y) = 0 定义；沿此线流动是纯竖直的（dx/dt = 0）。y‑零等倾线由 g(x,y) = 0 定义；沿此线流动是纯水平的（dy/dt = 0）。两条零等倾线的交点正是系统的临界点。根据 f 和 g 的正负号对零等倾线之间的区域进行着色，可将平面划分为箭头方向一致的区域。这种结构化的方法简化了粗略相图的构建，对无法获得显式解的非线性系统尤为有用。

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5. Critical Points and Their Importance | 临界点及其重要性

A critical point (also called an equilibrium or fixed point) occurs wherever f(x₀, y₀) = 0 and g(x₀, y₀) = 0 simultaneously. If a system starts exactly at a critical point, it stays there forever (dx/dt = dy/dt = 0). The nature of the surrounding trajectories determines whether the equilibrium is stable, unstable, or a saddle. Small perturbations near a stable node or spiral decay back to the point, whereas near a saddle they are repelled along at least one direction. Identifying and classifying all critical points of a system is thus the central task of phase‑plane analysis, as the whole global portrait is organised around these landmarks.

临界点（亦称平衡点或不动点）出现在 f(x₀, y₀) = 0 与 g(x₀, y₀) = 0 同时成立的位置。若系统精确始于临界点，它将永远停留于此（dx/dt = dy/dt = 0）。周围轨迹的性质决定了该平衡点是稳定、不稳定还是鞍点。靠近稳定结点或螺旋的微小扰动会衰减并回归该点，而在鞍点附近至少沿某一方向会被排斥。因此，找出并分类系统的所有临界点是相平面分析的中心任务，整个全局肖像正是围绕这些标志性点组织起来的。

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6. Linearisation: Approximating Near Equilibrium | 线性化：平衡点附近的近似

For a nonlinear system, we can extract local behaviour by linearising around a critical point. If the equilibrium is at (a,b), we write u = x – a, v = y – b and expand f,g in a Taylor series, retaining only the linear terms. The resulting linear system is

du/dt = (∂f/∂x)|₀ u + (∂f/∂y)|₀ v
dv/dt = (∂g/∂x)|₀ u + (∂g/∂y)|₀ v

where the partial derivatives are evaluated at (a,b). This is simply u’ = J u, with J the Jacobian matrix. The eigenvalues of J determine the type and stability of the critical point. The Hartman–Grobman theorem guarantees that, provided no eigenvalue has zero real part, the phase portrait of the linearised system is topologically conjugate to that of the original nonlinear system in a neighbourhood of the equilibrium. This powerful result allows us to classify almost all generic equilibria without solving the full nonlinear equations.

对于非线性系统，我们可以通过在临界点附近线性化来提取局部行为。设平衡点为 (a,b)，令 u = x – a、v = y – b，并将 f,g 作泰勒展开，仅保留线性项。得到的线性系统为

du/dt = (∂f/∂x)|₀ u + (∂f/∂y)|₀ v
dv/dt = (∂g/∂x)|₀ u + (∂g/∂y)|₀ v

其中偏导数均在 (a,b) 处取值。这其实就是 u’ = J u，J 为雅可比矩阵。J 的特征值决定了临界点的类型与稳定性。Hartman–Grobman 定理保证，只要没有特征值的实部为零，线性化系统的相图在平衡点附近与原始非线性系统拓扑共轭。这一有力结论使我们无需求解完整的非线性方程就能对几乎所有通有平衡点进行分类。

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7. Classification of Linear Systems | 线性系统的分类

For the two‑dimensional linear system X’ = AX, the eigenvalues λ₁, λ₂ of A dictate the phase portrait. The main classes are:

• Real, distinct, same sign: node (stable if negative, unstable if positive).
• Real, distinct, opposite signs: saddle (always unstable).
• Complex conjugate with non‑zero real part: spiral (stable if real part < 0, unstable if > 0).
• Purely imaginary: centre (stable but not asymptotically stable, closed orbits).
• Repeated eigenvalue, complete eigenvectors: star node.
• Repeated eigenvalue, defective: improper (degenerate) node.

Using the trace T = λ₁ + λ₂ and determinant D = λ₁ λ₂, one can quickly place any matrix in the (T, D) plane: a parabola D = T²/4 separates nodes from spirals, the axis T = 0 separates stable from unstable, and D < 0 gives saddles. This compact graphical summary makes classification almost mechanical, which is ideal for exam conditions.

对于二维线性系统 X’ = AX，A 的特征值 λ₁、λ₂ 决定了相图。主要类型如下：

• 相异实根，同号：结点（负值稳定，正值不稳定）。
• 相异实根，异号：鞍点（总是不稳定）。
• 共轭复根，实部非零：螺旋（实部 < 0 稳定，> 0 不稳定）。
• 纯虚根：中心（稳定但非渐近稳定，产生闭轨）。
• 重根，完备特征向量：星形结点。
• 重根，亏损：退化结点。

利用迹 T = λ₁ + λ₂ 与行列式 D = λ₁ λ₂，可快速将任何矩阵置于 (T, D) 平面：抛物线 D = T²/4 把结点与螺旋分开，轴 T = 0 分隔稳定与不稳定，D < 0 给出鞍点。这幅紧凑的图解让分类近乎机械，非常适合考试使用。

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8. Stability and Eigenvalues | 稳定性与特征值

Stability is read directly from the real parts of the eigenvalues. If all eigenvalues have negative real part, trajectories approach the equilibrium as t → ∞ – asymptotic stability. If at least one eigenvalue has positive real part, the equilibrium is unstable. The borderline case of pure imaginary eigenvalues yields a centre, which is Lyapunov stable but small errors in modelling or tiny nonlinearities can alter it. For nonlinear systems, linearisation faithfully predicts the stability of hyperbolic equilibria (no eigenvalues on the imaginary axis). When the linearised system gives a centre, the nonlinear system may actually possess a stable or unstable spiral, so a deeper investigation (e.g. using a Lyapunov function) is required. In IB contexts, centres in linear systems are accepted as neutrally stable.

稳定性可直接从特征值的实部读出。若所有特征值实部为负，轨迹当 t → ∞ 时趋近平衡点——渐近稳定。若至少一个特征值实部为正，则平衡点不稳定。纯虚特征值的临界情形产生中心，它是 Lyapunov 意义下稳定的，但建模中的微小误差或非线性扰动可能改变它。对于非线性系统，线性化忠实预测双曲平衡点（没有位于虚轴的特征值）的稳定性。当线性化系统给出中心时，实际非线性系统可能具有稳定或不稳定的螺旋，因此需要更深入的研究（例如使用 Lyapunov 函数）。在 IB 范围内，线性系统的中心被视为中性稳定。

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9. Drawing a Phase Portrait: A Step‑by‑Step Guide | 绘制相图：分步指南

Here is a reliable procedure for sketching a phase portrait for an autonomous system dx/dt = f(x,y), dy/dt = g(x,y):
Step 1: Find all critical points by solving f = 0, g = 0 simultaneously.
Step 2: Draw the nullclines f = 0 (vertical arrows) and g = 0 (horizontal arrows) and mark their intersections.
Step 3: Determine the signs of f and g in each region to infer arrow directions; sketch a representative grid of direction vectors.
Step 4: For each critical point, compute the Jacobian and its eigenvalues, then classify the point (node, saddle, spiral, etc.).
Step 5: Sketch local trajectories respecting the linearised behaviour near equilibria.
Step 6: Connect the local portraits into a global picture, ensuring that trajectories do not cross and that separatrices (e.g. saddle ins/outs) are drawn carefully.
Step 7: Indicate arrows on trajectories to show the direction of increasing t.
With practice, this method becomes fast and yields accurate exam‑ready solutions.

以下是绘制自治系统 dx/dt = f(x,y)、dy/dt = g(x,y) 相图的可靠步骤：
第一步：联立求解 f = 0、g = 0，找出所有临界点。
第二步：画出零等倾线 f = 0（竖直箭头）和 g = 0（水平箭头），并标记它们的交点。
第三步：确定每个区域内 f 和 g 的正负号以推断箭头方向；草绘代表性的方向向量网格。
第四步：对每个临界点，计算雅可比矩阵及其特征值，然后分类（结点、鞍点、螺旋等）。
第五步：根据平衡点附近的线性化行为绘制局部轨迹。
第六步：将局部肖像连接成全局图像，确保轨迹不相交，并仔细画出分界线（如鞍点的入出线）。
第七步：在轨迹上标注箭头以指示 t 增大的方向。
经过练习，这一方法将变得快速，并能产生准确的考试级解答。

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10. Summary and Exam Tips | 总结与考试技巧

The general phase plane transforms differential equations into a geometric story. Remember the core flow: system → nullclines → critical points → linearisation → classification → phase portrait. Familiarity with the eigenvalue–trace–determinant chart saves time. In an exam, always label axes, mark critical points clearly, and draw arrows on several trajectories. If a linearisation yields a centre, state that the linearised system has a centre but note that the true nonlinear system might differ. The phase plane connects directly to modelling in physics, biology and economics, so appreciating its qualitative power is as important as algorithmic skill. Practise with standard linear systems and then move to simple predator–prey or competition models to build confidence.

一般相平面把微分方程转化为几何故事。牢记核心流程：系统 → 零等倾线 → 临界点 → 线性化 → 分类 → 相图。熟悉特征值–迹–行列式图表可节省时间。考试时，务必标注坐标轴，清晰标出临界点，并在多条轨迹上绘制箭头。若线性化给出中心，说明线性化系统具有中心，但需指出真实非线性系统可能不同。相平面与物理、生物和经济学中的建模直接相关，因此领悟其定性力量与算法技能同样重要。可从标准线性系统开始练习，再过渡到简单的捕食者–猎物或竞争模型，以建立信心。

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