📚 3 (x, ẋ) Phase Diagrams for Other Linear Equations; Stability | 其他线性方程的(x, ẋ)相图与稳定性
Phase diagrams provide a powerful qualitative tool for understanding the behaviour of solutions to autonomous first-order differential equations without solving them explicitly. By plotting the derivative ẋ = dx/dt against the state variable x, we can immediately identify equilibrium points and determine their stability. This article extends the basic linear case ẋ = kx to more general linear equations of the form ẋ = ax + b, exploring how algebraic forms control the phase line and how stability criteria emerge from the slope of the right‑hand side function. We will construct (x, ẋ) diagrams for these other linear equations, classify fixed points, and link the geometric picture to rigorous stability conditions used in IB Mathematics analysis.
相图为定性理解一阶自治微分方程的解的行为提供了有力工具，无需显式求解。通过将导数 ẋ = dx/dt 相对于状态变量 x 作图，我们可以立即识别平衡点并判断其稳定性。本文将基本的线性情形 ẋ = kx 拓展到更一般的线性方程 ẋ = ax + b，探讨代数形式如何控制相线以及稳定性判据如何从右端函数的斜率得出。我们将为这些其他线性方程构建 (x, ẋ) 图，对不动点进行分类，并将几何图像与 IB 数学分析中严格的稳定性条件联系起来。
1. Autonomous Equations and the (x, ẋ) Plane | 自治方程与 (x, ẋ) 平面
An autonomous first-order ordinary differential equation takes the form ẋ = f(x), where the rate of change depends only on the current value of x. The (x, ẋ) phase diagram is a Cartesian plot with x on the horizontal axis and ẋ on the vertical axis. The graph of f(x) in this plane intersects the x‑axis at equilibrium points, where f(x) = 0. For any x above the axis, ẋ > 0, so x increases; below the axis, ẋ < 0 and x decreases. This simple sign analysis allows us to draw arrows on the x‑axis (the phase line) indicating the direction of motion.
自治一阶常微分方程的形式为 ẋ = f(x)，变化率只依赖于当前的 x 值。(x, ẋ) 相图是以 x 为横轴、ẋ 为纵轴的笛卡尔图。函数 f(x) 在该平面中的图线与 x 轴相交于平衡点，即 f(x)=0。对于 x 轴上方的任意点，ẋ > 0，因此 x 增加；在轴下方，ẋ < 0 且 x 减小。这种简单的符号分析使我们能够在 x 轴（相线）上画出指示运动方向的箭头。
When the function f(x) is linear, the phase diagram consists of a straight line. By varying the slope and intercept, we obtain a family of lines that correspond to all possible linear autonomous equations. Although the algebra is simple, changing the parameters produces qualitatively different phase portraits, which are fundamental to understanding linear stability theory.
当函数 f(x) 为线性时，相图由一条直线构成。通过改变斜率和截距，我们得到一族对应于所有可能的线性自治方程的直线。虽然代数形式简单，但改变参数会产生定性不同的相图，这对理解线性稳定性理论至关重要。
2. The Prototype: ẋ = kx and Its Phase Diagram | 原型：ẋ = kx 及其相图
Start with the simplest linear equation, ẋ = kx, where k is a real constant. The point x = 0 is the unique equilibrium because k × 0 = 0. The (x, ẋ) diagram is a straight line passing through the origin with slope k. When k > 0, the line lies in the first and third quadrants: for x > 0, ẋ > 0, so x moves further to the right; for x < 0, ẋ < 0, so x moves further to the left. The equilibrium at x = 0 is therefore unstable because any small perturbation pushes the state away from zero.
从最简单的线性方程 ẋ = kx 开始，其中 k 为实常数。点 x = 0 是唯一的平衡点，因为 k × 0 = 0。(x, ẋ) 图是一条过原点且斜率为 k 的直线。当 k > 0 时，直线位于第一和第三象限：对 x > 0，ẋ > 0，故 x 向右移动；对 x < 0，ẋ < 0，故 x 向左移动。因此 x = 0 处的平衡是不稳定的，因为任何微小摄动都会使状态远离原点。
When k < 0, the line passes through the second and fourth quadrants. For x > 0, ẋ is negative, pulling x back toward zero; for x < 0, ẋ is positive, also pushing x toward zero. The equilibrium is stable, indeed asymptotically stable, because all trajectories approach zero as time increases. The case k = 0 makes every x an equilibrium point, representing neutral stability, but this is a degenerate scenario.
当 k < 0 时，直线穿过第二和第四象限。对 x > 0，ẋ 为负，将 x 拉回零点；对 x < 0，ẋ 为正，也将 x 推向零点。平衡点是稳定的，实际上是渐近稳定的，因为随着时间增加所有轨迹都趋近于零。k = 0 的情形使所有 x 都是平衡点，代表中性稳定，但属于退化情况。
3. Stability Classification from the Sign of k | 由 k 的符号判断稳定性分类
The stability of the equilibrium x* = 0 in ẋ = kx is completely determined by the sign of k. Geometrically, if the graph of f(x) = kx crosses the x‑axis with a positive slope, the equilibrium is unstable; if it crosses with a negative slope, the equilibrium is stable. This geometric rule will generalise to any differentiable function f(x): the derivative f ‘(x*) governs local stability. For the linear prototype, f ‘(0) = k, so sign of k directly gives the stability type.
方程 ẋ = kx 中平衡点 x* = 0 的稳定性完全由 k 的符号决定。从几何上看，如果 f(x)=kx 的图线以正斜率穿过 x 轴，平衡点不稳定；如果以负斜率穿过，则稳定。这一几何规则将推广到任何可微函数 f(x)：导数 f ‘(x*) 决定局部稳定性。对线性原型，f ‘(0)=k，因此 k 的符号直接给出稳定性类型。
In IB terminology, an equilibrium is called stable if solutions starting sufficiently close remain close for all future time; it is asymptotically stable if they also tend to the equilibrium as t → ∞. For ẋ = kx with k < 0, x = 0 is asymptotically stable, often simply referred to as a 'sink'. With k > 0, x = 0 is an unstable ‘source’.
在 IB 术语中，如果从足够近处出发的解在未来所有时间都保持接近，平衡点称为稳定的；如果当 t → ∞ 时它们还趋向该平衡点，则称为渐近稳定。对于 ẋ = kx 且 k < 0，x = 0 是渐近稳定的，常简称为“汇”。当 k > 0 时，x = 0 是不稳定的“源”。
4. Other Linear Equations: The General Form ẋ = ax + b | 其他线性方程：一般形式 ẋ = ax + b
Consider the affine linear equation ẋ = ax + b, where a and b are constants and a may be zero. This is the most general linear first‑order autonomous ODE. The (x, ẋ) diagram is a straight line with slope a and ẋ‑intercept b. Equilibria occur where ax + b = 0, giving the fixed point x* = –b/a, provided a ≠ 0. If a = 0, the equation reduces to ẋ = b; there is no equilibrium unless b = 0, in which case all x are equilibria.
考虑仿射线性方程 ẋ = ax + b，其中 a 和 b 为常数且 a 可以为零。这是最一般的线性一阶自治常微分方程。(x, ẋ) 图是一条斜率为 a、ẋ 截距为 b 的直线。平衡点发生在 ax + b = 0 处，得到不动点 x* = –b/a，假设 a ≠ 0。如果 a = 0，方程化为 ẋ = b；除非 b = 0（此时所有 x 都是平衡点），否则没有平衡点。
Shifting from ẋ = kx to ẋ = ax + b moves the equilibrium from the origin to x* = –b/a without changing the qualitative stability determined by a. The slope a still dictates whether the equilibrium is stable (a < 0) or unstable (a > 0). Therefore, the entire family of lines with the same slope a but different intercepts b shares identical stability properties at their respective fixed points.
从 ẋ = kx 变为 ẋ = ax + b 会将平衡点从原点平移到 x* = –b/a，而不改变由 a 决定的定性稳定性。斜率 a 依然决定平衡点是稳定（a < 0）还是不稳定（a > 0）。因此，具有相同斜率 a 但截距 b 不同的整个直线族在各自的不动点处具有相同的稳定性性质。
5. Constructing Phase Lines for ẋ = ax + b | 构建 ẋ = ax + b 的相线
To draw the phase line, locate the equilibrium x* = –b/a on the x‑axis. Then evaluate the sign of ẋ to the left and right of x*. Because f(x) is linear, the sign changes at most once at x*. For a > 0, the straight line rises from left to right, so ẋ is negative when x < x* and positive when x > x*. Arrows on the phase line point toward the right for x > x* and toward the left for x < x*, indicating that trajectories move away from x*—an unstable equilibrium.
要绘制相线，先在 x 轴上标出平衡点 x* = –b/a。然后判断 x* 左右两侧 ẋ 的符号。由于 f(x) 是线性的，符号最多在 x* 处改变一次。对于 a > 0，直线从左向右上升，因此当 x < x* 时 ẋ 为负，当 x > x* 时 ẋ 为正。相线上的箭头在 x > x* 时指向右，在 x < x* 时指向左，表明轨迹远离 x*——这是一个不稳定的平衡点。
For a < 0, the line falls from left to right. Then ẋ > 0 when x < x* and ẋ < 0 when x > x*. Consequently, arrows point toward x* from both sides, making the equilibrium asymptotically stable. If a = 0 and b ≠ 0, the entire x‑axis has only one arrow direction—rightwards if b > 0, leftwards if b < 0—and no equilibrium exists. This corresponds to constant‑rate growth or decay with no steady state.
对于 a < 0，直线从左向右下降。此时当 x < x* 时 ẋ > 0，当 x > x* 时 ẋ < 0。因此箭头从两侧指向 x*，使平衡点渐近稳定。若 a = 0 而 b ≠ 0，整个 x 轴只有单一箭头方向——若 b > 0 则向右，若 b < 0 则向左——且不存在平衡点。这对应于没有稳态的恒定速率增长或衰减。
6. Visualising Stability in the (x, ẋ) Diagram | 在 (x, ẋ) 图中可视化稳定性
Plot the line f(x) = ax + b on the (x, ẋ) plane. The equilibrium is the x‑intercept. The stability is immediately visible: when the line crosses the x‑axis from the upper‑left to the lower‑right (negative slope), the equilibrium is stable; when it crosses from lower‑left to upper‑right (positive slope), it is unstable. This geometric crossing rule is a visual shortcut that works for all differentiable functions, not just linear ones. The steeper the negative slope, the stronger the local restoring tendency.
在 (x, ẋ) 平面上画出直线 f(x) = ax + b。平衡点即 x 轴截距。稳定性一目了然：当直线以从左上到右下的方式穿过 x 轴（负斜率）时，平衡点是稳定的；当穿过方式为从左下到右上（正斜率）时，则不稳定。这一几何穿越规则是一个视觉捷径，对所有可微函数（不仅是线性函数）都成立。负斜率的绝对值越大，局部回归趋势越强。
For IB students, it is essential to connect this visual inspection with the analytic criterion: compute f ‘(x*) = a. If a < 0, the equilibrium is stable; if a > 0, it is unstable. Because f(x) is linear, f ‘(x*) is constant and equal to a everywhere; the stability conclusion is global, meaning all trajectories behave qualitatively the same way regardless of starting point.
对 IB 学生而言，必须将这一视觉观察与解析判据联系起来：计算 f ‘(x*) = a。若 a < 0，平衡点稳定；若 a > 0，它不稳定。由于 f(x) 是线性的，f ‘(x*) 处处恒为 a；稳定性结论是全局的，也就是说，无论出发点在何处，所有轨迹在定性上行为相同。
7. Algebraic Derivation of Stability: The Exact Solution | 稳定性的代数推导：精确解
Although phase diagrams are qualitative, solving the linear ODE confirms the conclusions. The general solution to ẋ = ax + b (a ≠ 0) is x(t) = –b/a + C e^(at), where C is an arbitrary constant. The equilibrium is x* = –b/a. If a < 0, the exponential term e^(at) decays to zero as t → ∞, so x(t) → x* for any C, proving asymptotic stability. If a > 0, the exponential grows without bound (unless C = 0), so the equilibrium is unstable: any small perturbation C ≠ 0 leads to deviation from x* that magnifies over time.
尽管相图是定性的，求解线性常微分方程可验证这些结论。方程 ẋ = ax + b (a ≠ 0) 的通解为 x(t) = –b/a + C e^(at)，其中 C 为任意常数。平衡点为 x* = –b/a。如果 a < 0，指数项 e^(at) 随 t → ∞ 衰减至零，因此对任意 C 有 x(t) → x*，证实了渐近稳定性。如果 a > 0，指数函数无界增长（除非 C = 0），故平衡点不稳定：任何微小扰动 C ≠ 0 都会导致随时间放大的偏离。
When a = 0, the solution is x(t) = x₀ + bt, representing linear growth or decay. There is no approach to a fixed point (unless b = 0, the case of a constant solution). This aligns with the phase diagram showing a horizontal line at ẋ = b, with no x‑intercept and a uniform arrow direction.
当 a = 0 时，解为 x(t) = x₀ + bt，表示线性增长或衰减。不存在趋近于不动点的情况（除非 b = 0，即常数解的情形）。这与相图显示的一条位于 ẋ = b 的水平线一致，既无 x 截距，箭头方向也全一致。
8. Parameter Analysis: How a and b Influence the Phase Portrait | 参数分析：a 和 b 如何影响相图
The parameter a, being the slope, controls the type of stability. As long as a ≠ 0, the sign of a alone determines whether the equilibrium is a source or a sink. The magnitude |a| affects the strength of attraction or repulsion: larger |a| corresponds to a steeper line and hence faster exponential growth or decay in the exact solution. In the phase diagram, a steeper slope means that a given displacement from x* produces a larger ẋ, implying more rapid movement along the phase line.
参数 a 作为斜率，控制稳定性类型。只要 a ≠ 0，a 的符号单独决定平衡点是源还是汇。|a| 的大小影响吸引或排斥的强度：较大的 |a| 对应更陡的直线，从而在精确解中意味着更快的指数增长或衰减。在相图中，更陡的斜率意味着给定偏离 x* 的位移会产生更大的 ẋ，表明沿相线的运动更快。
The parameter b translates the equilibrium horizontally. Changing b shifts the x‑intercept but leaves the slope a unchanged; therefore stability properties are invariant under translation. This is why we can view the equation ẋ = ax + b as a shifted version of the fundamental homogeneous equation ẋ = ax, centred now at x* = –b/a. For instance, ẋ = 2x – 6 has a = 2 > 0, so the equilibrium x* = 3 is unstable, exactly as x = 0 is for ẋ = 2x.
参数 b 水平地平移平衡点。改变 b 会移动 x 截距但保持斜率 a 不变；因此稳定性性质在平移下是不变的。这就解释了为什么我们可以将方程 ẋ = ax + b 视为基本齐次方程 ẋ = ax 的平移形式，只是现在中心位于 x* = –b/a。例如，ẋ = 2x – 6 有 a = 2 > 0，故平衡点 x* = 3 不稳定，和 ẋ = 2x 中 x = 0 的情况完全相同。
9. Worked Examples with (x, ẋ) Diagrams | 带 (x, ẋ) 图的解题示例
Example 1: Consider ẋ = –3x + 9. Rewrite as ẋ = –3(x – 3). The equilibrium is x* = 3, and the slope is a = –3 < 0. The (x, ẋ) line crosses the x‑axis at x = 3 with negative slope. For x = 0, ẋ = 9 > 0; for x = 5, ẋ = –6 < 0. The phase line shows an arrow pointing right for x < 3 and left for x > 3, indicating stability. Any initial condition approaches 3 as t increases.
示例 1：考虑 ẋ = –3x + 9。改写为 ẋ = –3(x – 3)。平衡点为 x* = 3，斜率 a = –3 < 0。(x, ẋ) 直线在 x = 3 处以负斜率穿过 x 轴。当 x = 0 时 ẋ = 9 > 0；当 x = 5 时 ẋ = –6 < 0。相线显示对 x < 3 箭头向右，对 x > 3 箭头向左，表明稳定。任何初始条件随 t 增加都会趋近 3。
Example 2: ẋ = 0.5x – 1. Here a = 0.5 > 0, b = –1, so x* = 2. The slope is positive, meaning the line rises through the x‑axis. For x = 0, ẋ = –1 < 0; for x = 4, ẋ = 1 > 0. Arrows on the phase line move away from 2, confirming instability. The exact solution x(t) = 2 + C e^(0.5t) grows exponentially if C ≠ 0.
示例 2：ẋ = 0.5x – 1。这里 a = 0.5 > 0，b = –1，故 x* = 2。斜率为正，意味着直线穿过 x 轴时上升。当 x = 0 时 ẋ = –1 < 0；当 x = 4 时 ẋ = 1 > 0。相线上的箭头从 2 向外指，证实不稳定。精确解 x(t) = 2 + C e^(0.5t) 在 C ≠ 0 时呈指数增长。
Example 3 (no equilibrium): ẋ = 0 × x + 4, i.e. ẋ = 4. The (x, ẋ) diagram is a horizontal line at ẋ = 4, entirely above the x‑axis, so ẋ > 0 for all x. The phase line is a single rightward arrow across the whole real line. The solution x(t) = x₀ + 4t grows without bound and has no equilibrium.
示例 3（无平衡点）：ẋ = 0 × x + 4，即 ẋ = 4。(x, ẋ) 图是一条位于 ẋ = 4 处的水平线，完全在 x 轴上方，因此对所有 x 都有 ẋ > 0。相线是整个实数轴上的一个右向箭头。解 x(t) = x₀ + 4t 无界增长且没有平衡点。
10. Local Stability Criterion and Linearisation Principle | 局部稳定性判据与线性化原理
For a general differentiable f(x), the equilibrium x* satisfies f(x*) = 0, and its stability is determined by f ‘(x*). If f ‘(x*) < 0, the equilibrium is stable; if f '(x*) > 0, it is unstable. This linear stability analysis works because close to x* the function can be approximated by its tangent line: f(x) ≈ f ‘(x*)(x – x*), which is exactly of the form ax + b with a = f ‘(x*) and b = –f ‘(x*) x*. Thus, every smooth equilibrium looks locally like our studied linear equation ẋ = a(x – x*).
对于一般的可微函数 f(x)，平衡点 x* 满足 f(x*) = 0，其稳定性由 f ‘(x*) 决定。若 f ‘(x*) < 0，平衡点稳定；若 f '(x*) > 0，则不稳定。此线性稳定性分析成立的原因是在 x* 附近函数可用其切线近似：f(x) ≈ f ‘(x*)(x – x*)，这恰好是我们所研究的线性方程 ẋ = a(x – x*) 的形式，其中 a = f ‘(x*)。因此，每个光滑的平衡点在局部看起来都如同这个线性方程。
For the linear equations themselves, f ‘(x*) = a at the single equilibrium, so the local and global behaviours coincide. This is why mastering the (x, ẋ) diagrams of ẋ = ax + b provides the foundation for understanding far more complex nonlinear phase portraits, such as those for logistic growth or other autonomous models in the IB syllabus.
对于线性方程本身，在唯一的平衡点处 f ‘(x*) = a，因此局部行为与全局行为一致。这就是为什么掌握 ẋ = ax + b 的 (x, ẋ) 图为理解更复杂的非线性相图（例如 IB 大纲中的 Logistic 增长或其他自治模型）奠定了基础。
11. Common Pitfalls and Misconceptions | 常见错误与误区
One frequent mistake is to equate the sign of b with the stability of the equilibrium. Stability is governed solely by a, the slope; the intercept b only sets the location of the fixed point. A large positive b does not guarantee stability if a > 0. For example, ẋ = 2x – 10 has a positive intercept? Wait, the ẋ‑intercept is b = –10, but the sign of b is irrelevant. Students must focus on evaluating f ‘(x*) = a.
一个常见错误是将 b 的符号与平衡点的稳定性等同起来。稳定性仅由斜率 a 决定；截距 b 只决定不动点的位置。若 a > 0，即使 b 为很大的正数也无法保证稳定。例如 ẋ = 2x – 10，其 ẋ 轴截距负值，但 b 的符号无关紧要。学生必须重点关注 f ‘(x*) = a 的值。
Another pitfall is forgetting to set ẋ = 0 to find the equilibrium when b ≠ 0. Some learners incorrectly assume the equilibrium is always at x = 0, overlooking the shift introduced by the inhomogeneous term. Always solve ax + b = 0 to locate x* before sketching the phase line.
另一个误区是在 b ≠ 0 时忘记设 ẋ = 0 来求平衡点。一些学习者错误地认为平衡点总在 x = 0，忽略了非齐次项带来的平移。在绘制相线之前，务必先求解 ax + b = 0 来确定 x*。
Furthermore, while drawing the (x, ẋ) diagram, students sometimes confuse the roles of axes. Remember that the horizontal axis represents the state variable x, and the vertical axis represents its rate of change. Arrows on the phase line are drawn on the x‑axis according to the sign of ẋ, not on the curve itself. The graph of f(x) is only a tool to read off signs.
此外，绘制 (x, ẋ) 图时，学生有时会混淆坐标轴的角色。记住横轴代表状态变量 x，纵轴代表其变化率。相线上的箭头是根据 ẋ 的符号画在 x 轴上的，而非画在曲线上。f(x) 的图形只是读取符号的工具。
12. Summary and Connections to IB Exam Skills | 总结及与 IB 考试技能的关联
The (x, ẋ) phase diagram for linear autonomous equations of the form ẋ = ax + b systematically reveals the equilibrium x* = –b/a (a ≠ 0) and its stability via the sign of the slope a. A negative slope yields a stable equilibrium acting as a sink, while a positive slope yields an unstable equilibrium acting as a source. When a = 0,
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