Constructing a Phase Diagram for (x, ẋ) | 构建 (x, ẋ) 相图

📚 Constructing a Phase Diagram for (x, ẋ) | 构建 (x, ẋ) 相图

Phase diagrams provide a powerful visual tool for understanding the behaviour of dynamical systems without solving differential equations analytically. When the state of a system is described by a position x and its time derivative ẋ (velocity), the phase plane with coordinates (x, ẋ) reveals how trajectories evolve over time, highlighting equilibrium points, periodic orbits, and stability. Constructing such a diagram by hand is a fundamental skill in IB Mathematics, particularly in the study of second-order differential equations.

相图为理解动力系统的行为提供了强大的可视化工具，无需解析求解微分方程。当系统的状态由位置 x 及其时间导数 ẋ（速度）描述时，以 (x, ẋ) 为坐标的相平面可以展示轨迹如何随时间演变，突出平衡点、周期轨道和稳定性。手工构建这样的相图是IB数学中一项基本技能，尤其在二阶微分方程的学习中尤为重要。

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1. From Second-Order ODE to First-Order System | 从二阶常微分方程到一阶系统

Many mechanical and electrical systems are modelled by a second-order differential equation of the form d²x/dt² = f(x, dx/dt). To construct a phase diagram, we introduce a new variable y = dx/dt (so y = ẋ), transforming the single second-order equation into a pair of first-order equations: dx/dt = y, dy/dt = f(x, y). The phase plane is then the (x, y) plane, where the horizontal axis represents position x and the vertical axis represents velocity ẋ.

许多机械和电气系统由形如 d²x/dt² = f(x, dx/dt) 的二阶微分方程建模。为了构建相图，我们引入新变量 y = dx/dt（即 y = ẋ），将单个二阶方程转化为一阶方程组：dx/dt = y，dy/dt = f(x, y)。于是相平面就是 (x, y) 平面，其横轴代表位置 x，纵轴代表速度 ẋ。

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2. Equilibrium Points and Stability | 平衡点与稳定性

Equilibrium points occur where both derivatives are zero simultaneously: y = 0 and f(x, y) = 0. These points correspond to constant solutions (no motion). Their stability can be classified by linearising the system near the equilibrium and evaluating the eigenvalues of the Jacobian matrix. Common types include stable nodes, unstable nodes, saddle points, stable spirals, unstable spirals, and centres.

平衡点出现在两个导数同时为零的位置：即 y = 0 且 f(x, y) = 0。这些点对应常数解（无运动）。它们的稳定性可通过在平衡点附近线性化系统并计算雅可比矩阵的特征值来分类。常见类型包括稳定结点、不稳定结点、鞍点、稳定焦点、不稳定焦点和中心点。

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3. Nullclines: The Backbone of the Phase Portrait | 零等斜线：相图的主干

Nullclines are curves along which one of the derivatives is zero. The x-nullcline is given by dx/dt = 0, so y = 0 (the horizontal axis). The y-nullcline is defined by dy/dt = 0, so f(x, y) = 0. At any intersection of the x-nullcline and a y-nullcline, we have an equilibrium point. Plotting these curves first divides the plane into regions where the signs of the derivatives are constant.

零等斜线是其中某个导数为零的曲线。x-零等斜线由 dx/dt = 0 给出，即 y = 0（水平轴）。y-零等斜线由 dy/dt = 0 定义，即 f(x, y) = 0。任何 x-零等斜线与 y-零等斜线的交点都是平衡点。首先画出这些曲线能将平面划分为导数符号保持恒定的区域。

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4. Sign Analysis of the Vector Field | 向量场的符号分析

Once the nullclines are plotted, we determine the signs of dx/dt and dy/dt in each region. Since dx/dt = y, the horizontal velocity component is positive above the x-axis (trajectories move right) and negative below (trajectories move left). The sign of dy/dt depends on f(x, y). By testing a representative point in each region, we can draw small direction arrows that guide the trajectory sketching.

绘制出零等斜线后，我们在每个区域确定 dx/dt 和 dy/dt 的符号。由于 dx/dt = y，水平速度分量在 x 轴上方为正（轨迹向右移动），下方为负（轨迹向左移动）。dy/dt 的符号由 f(x, y) 决定。通过在每个区域内测试一个代表性点，我们可以绘出小的方向箭头，指导轨迹的勾画。

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5. Sketching Phase Trajectories | 绘制相轨迹

Starting from a chosen initial condition (x₀, y₀), we can follow the local vector field to trace out a smooth trajectory. Trajectories cannot intersect except at equilibrium points, where the vector field vanishes. By combining the nullclines and direction arrows, we can sketch representative orbits, including closed loops (periodic solutions), spirals, and paths that approach or leave equilibria.

从选定的初始条件 (x₀, y₀) 出发，我们可以跟随局部向量场描绘出一条平滑的轨迹。除向量场消失的平衡点外，轨迹不能相交。结合零等斜线和方向箭头，我们可以勾画出代表性轨道，包括闭合环路（周期解）、螺线以及趋近或离开平衡点的路径。

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6. Example 1: Simple Harmonic Oscillator | 示例1：简谐振子

Consider the undamped oscillator d²x/dt² + ω²x = 0. The equivalent system is dx/dt = y, dy/dt = –ω²x. The x-nullcline is y = 0; the y-nullcline is x = 0. The only equilibrium is (0,0). In the four quadrants, dx/dt and dy/dt have signs that produce circular trajectories. For instance, in the first quadrant (x > 0, y > 0), dx/dt > 0 but dy/dt < 0, so the trajectory curves rightwards and downwards, forming part of a closed orbit.

考虑无阻尼振子 d²x/dt² + ω²x = 0。等价系统为 dx/dt = y，dy/dt = –ω²x。x-零等斜线为 y = 0；y-零等斜线为 x = 0。唯一的平衡点是 (0,0)。在四个象限中，dx/dt 和 dy/dt 的符号产生圆形轨迹。例如，在第一象限 (x > 0, y > 0) 中，dx/dt > 0 而 dy/dt < 0，因此轨迹向右下方弯曲，形成闭合轨道的一部分。

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7. Example 2: Damped Harmonic Oscillator | 示例2：阻尼谐振子

Adding a linear damping term gives d²x/dt² + 2ζω dx/dt + ω²x = 0. The system becomes dx/dt = y, dy/dt = –2ζω y – ω²x. The x-nullcline is still y = 0. The y-nullcline is the line y = –(ω/(2ζ)) x. For 0 < ζ < 1 (underdamped), trajectories spiral into the origin (a stable spiral). For ζ > 1 (overdamped), trajectories approach the origin along a stable node, without oscillation. The nullclines and direction arrows reveal these qualitative differences clearly.

加入线性阻尼项得到 d²x/dt² + 2ζω dx/dt + ω²x = 0。系统变为 dx/dt = y，dy/dt = –2ζω y – ω²x。x-零等斜线仍为 y = 0。y-零等斜线是直线 y = –(ω/(2ζ)) x。对于 0 < ζ < 1（欠阻尼），轨迹螺旋式趋近原点（稳定焦点）。对于 ζ > 1（过阻尼），轨迹沿稳定结点无振荡地趋近原点。零等斜线和方向箭头清楚地揭示了这些定性差异。

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8. Example 3: Nonlinear Pendulum | 示例3：非线性单摆

The undamped pendulum equation is d²θ/dt² = –(g/l) sin θ. Let x = θ, y = dθ/dt. Then dx/dt = y, dy/dt = –(g/l) sin x. The x-nullcline is y = 0; y-nullclines are x = nπ (n integer). Equilibria occur at (0,0), (±π,0), (±2π,0), etc. Linearising shows that (0,0) is a centre and (±π,0) are saddle points. The phase portrait consists of closed orbits around the origin, separatrices connecting saddle points, and running trajectories outside the separatrices representing whirling motions.

无阻尼单摆方程为 d²θ/dt² = –(g/l) sin θ。令 x = θ，y = dθ/dt。则 dx/dt = y，dy/dt = –(g/l) sin x。x-零等斜线为 y = 0；y-零等斜线为 x = nπ（n为整数）。平衡点出现在 (0,0)、(±π,0)、(±2π,0) 等处。线性化显示 (0,0) 为中心点，而 (±π,0) 为鞍点。相图由围绕原点的闭合轨道、连接鞍点的分界线以及分界线外代表旋转运动的流动轨迹组成。

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9. Energy Method for Conservative Systems | 保守系统的能量法

For a conservative system where d²x/dt² = –dV/dx, a first integral is the total energy E(x, y) = ½y² + V(x), which is constant along trajectories. This allows us to sketch phase curves by plotting level sets of E. For the pendulum, E = ½y² – (g/l) cos x. The level curves give the separatrix at E = g/l and bounded oscillations for |E| < g/l, matching the nullcline analysis.

对于满足 d²x/dt² = –dV/dx 的保守系统，一个首次积分是总能量 E(x, y) = ½y² + V(x)，它沿轨迹保持恒定。这使得我们可以通过绘制 E 的水平集来勾画相曲线。对于单摆，E = ½y² – (g/l) cos x。水平集给出分界线 E = g/l 以及 |E| < g/l 时的有界振荡，与零等斜线分析一致。

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10. Step-by-Step Construction Guide | 逐步构建指南

To construct a phase diagram for (x, ẋ) systematically: (i) Write the system as dx/dt = y, dy/dt = f(x,y). (ii) Find all equilibrium points by solving y = 0 and f(x,y) = 0. (iii) Plot the x-nullcline (the x-axis) and the y-nullcline (f(x,y) = 0). (iv) Divide the plane into regions using these curves. (v) Determine the signs of dx/dt and dy/dt in each region by testing a sample point. (vi) Draw short arrows to represent the local direction. (vii) Sketch smooth trajectories by following the arrows, ensuring they cross nullclines with zero vertical or horizontal slope as appropriate. (viii) Indicate stability near equilibria with arrows converging or diverging.

系统构建 (x, ẋ) 相图的步骤：(i) 将系统写为 dx/dt = y，dy/dt = f(x,y)。(ii) 求解 y = 0 和 f(x,y) = 0，找到所有平衡点。(iii) 绘制 x-零等斜线（x轴）和 y-零等斜线（f(x,y) = 0）。(iv) 用这些曲线划分平面成若干区域。(v) 在每个区域内测试样本点，确定 dx/dt 和 dy/dt 的符号。(vi) 画出短箭头表示局部方向。(vii) 跟随箭头勾画平滑轨迹，确保轨迹在穿过零等斜线时分别具有零垂直斜率或零水平斜率。(viii) 通过收敛或发散的箭头表示平衡点附近的稳定性。

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11. Common Pitfalls and How to Avoid Them | 常见错误及其避免方法

One frequent mistake is forgetting that the x-nullcline is always y = 0 in this formulation, which means trajectories must cross the x-axis with a purely vertical tangent. Another is misinterpreting the direction field: a vertical arrow (dy/dt >> 0, dx/dt ≈ 0) does not mean the trajectory is vertical – it is simply strongly influenced in the vertical direction. Also, avoid drawing trajectories that intersect except at equilibria. Always check the sign analysis carefully near curved nullclines.

一个常见错误是忘记在此形式中 x-零等斜线始终为 y = 0，这意味着轨迹穿过 x 轴时切线必须是纯垂直的。另一个错误是误解方向场：垂直箭头（dy/dt >> 0，dx/dt ≈ 0）并不意味着轨迹是垂直的——它只是表明在垂直方向上受到强烈影响。同样，避免画出除平衡点外相交的轨迹。始终仔细检查弯曲零等斜线附近的符号分析。

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12. Connection to IB Exam Questions | 与IB考题的联系

IB Mathematics Analysis and Approaches HL and Applications and Interpretation HL may include questions that ask students to sketch a phase portrait for a given second-order ODE, identify the type and stability of equilibrium points, and describe the long-term behaviour of a system. Mastery of nullclines, direction fields, and energy methods is essential. Practice with linear and nonlinear examples, such as the pendulum or mass-spring systems, will build the confidence needed to handle these exam problems efficiently.

IB数学分析与方法HL以及应用与解释HL的考题可能要求学生为给定的二阶常微分方程画出相图，识别平衡点的类型与稳定性，并描述系统的长期行为。掌握零等斜线、方向场和能量法至关重要。通过线性和非线性例子（如单摆或弹簧振子系统）的练习，将建立起高效处理这些考试题目所需的信心。

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