IB Math: Nonlinear Differential Equations & Phase Plane Analysis | IB数学：非线性微分方程与相平面分析

📚 IB Math: Nonlinear Differential Equations & Phase Plane Analysis | IB数学：非线性微分方程与相平面分析

Nonlinear differential equations are among the most powerful and realistic models in mathematics. Unlike linear equations, they can describe population booms, pendulum swings, predator–prey cycles and even chaos. In this article we explore how to visualise and analyse their behaviour using the phase plane, a central technique for IB Mathematics higher-level and further mathematics students.

非线性微分方程是数学中最强大、最贴近现实的模型之一。与线性方程不同，它们可以描述种群爆发、单摆摆动、捕食者–猎物循环甚至混沌现象。本文将为IB高阶数学与进阶数学学生介绍如何借助相平面这一核心方法，来观察并分析非线性方程的行为。

1. Nonlinear Differential Equations | 非线性微分方程

A first-order ordinary differential equation is called nonlinear when the dependent variable or its derivative appears in a non-linear way, for example as x², sin(x), or x·dx/dt. The logistic equation dP/dt = rP(1 − P/K) is a classic nonlinear equation used in biology and economics.

所谓一阶常微分方程是“非线性的”，是指因变量或其导数以非线性的方式出现，例如 x²、sin(x) 或 x·dx/dt。逻辑斯蒂方程 dP/dt = rP(1 − P/K) 是生物学与经济学中经典的非线性方程。

Nonlinear equations rarely have closed-form solutions. Even when they do, their long-term behaviour is often analysed geometrically. This is why phase-plane methods, which study solutions in a state space rather than as time-plots, are indispensable.

非线性方程很少拥有闭式解。即便有，其长期行为也经常通过几何方法进行分析。因此，相平面方法提供了一种在状态空间中研究解的方法，而不是依赖时间曲线图，这一点至关重要。

2. Autonomous Systems and the Phase Plane | 自治系统与相平面

A system of two first-order autonomous differential equations has the form

ẋ = f(x, y), ẏ = g(x, y)

where ẋ and ẏ denote the time derivatives dx/dt and dy/dt and the functions f and g do not depend explicitly on t. Such systems are called autonomous because the rules do not change with time.

二元一阶自治微分方程组具有如下形式：

ẋ = f(x, y), ẏ = g(x, y)

其中 ẋ 与 ẏ 表示 dx/dt 与 dy/dt，而函数 f 和 g 不显含 t。这类系统称为“自治系统”，因为其演化规则不随时间变化。

Since the right-hand side depends only on (x, y), any solution (x(t), y(t)) traces a curve in the xy-plane called the phase plane. The plane itself is the state space, and each curve is a trajectory or orbit. The phase plane turns complicated differential equations into visual dynamical pictures.

由于右端只依赖于 (x, y)，任何解 (x(t), y(t)) 都会在 xy 平面中描绘出一条曲线，称为相平面。整个平面是状态空间，每条曲线称为轨迹或轨道。相平面把复杂的微分方程转化成直观的动态图像。

3. Nullclines and Direction Fields | 零斜线与时向场

To build a phase portrait by hand, we first draw the nullclines: the curves where ẋ = 0 or ẏ = 0. On the ẋ-nullcline, trajectories are vertical because x is not changing; on the ẏ-nullcline, trajectories are horizontal.

为了手工绘制相图，我们首先画出零斜线：即 ẋ = 0 或 ẏ = 0 的曲线。在 ẋ 零斜线上，轨迹是竖直的，因为 x 不发生变化；在 ẏ 零斜线上，轨迹是水平的。

For example, for the Lotka–Volterra model

ẋ = x(α − βy), ẏ = −y(γ − δx)

the nullclines are x = 0, y = α/β for ẋ = 0; and y = 0, x = γ/δ for ẏ = 0. These lines divide the plane into regions where the arrows are consistent, so we can sketch the overall flow.

例如，对于 Lotka–Volterra 模型

ẋ = x(α − βy), ẏ = −y(γ − δx)

其零斜线为 ẋ = 0：x = 0 与 y = α/β；ẏ = 0：y = 0 与 x = γ/δ。这些直线将平面划分为若干区域，同一区域内方向场变化一致，从而可以勾勒出整体流形。

Adding small direction arrows at selected points gives a direction field. Because the system is autonomous, the arrows are fixed in time; trajectories simply follow these arrows. This is the geometric equivalent of Euler’s method for a system.

在选定点添加小方向箭头便得到方向场。由于系统是自治的，箭头不随时间改变；轨迹只是沿着箭头行进。这相当于系统形式的欧拉方法的几何化表达。

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

An equilibrium point (x₀, y₀) satisfies f(x₀, y₀) = 0 and g(x₀, y₀) = 0. At such a point, neither x nor y changes, so the system stays at rest forever if not perturbed.

平衡点 (x₀, y₀) 满足 f(x₀, y₀) = 0 且 g(x₀, y₀) = 0。在该点处 x 与 y 都不变化，因此若不受扰动，系统将永远静止。

Stability depends on what happens after a small disturbance. If nearby trajectories move back to the equilibrium, it is asymptotically stable. If they move away, it is unstable. If they circle around without approaching, it is neutrally stable (centre).

稳定性取决于系统受到小扰动后的行为。若附近的轨迹回到平衡点，则为渐近稳定；若离开，则为不稳定；若围绕平衡点旋转而不靠近，则为中性稳定（中心型）。

For nonlinear systems, stability can often be determined locally by linearisation near the equilibrium, but some global phenomena, such as limit cycles, require a wider view.

对于非线性系统，通常可以在平衡点附近通过线性化来确定局部稳定性，但某些全局现象（如极限环）需要更广阔的视角。

5. Linearisation and the Jacobian Matrix | 线性化与雅可比矩阵

Consider an equilibrium point (x₀, y₀). Let u = x − x₀ and v = y − y₀. We expand f and g in a Taylor series and keep only the linear terms to obtain the linearised system

\dot u = fₓ(x₀,y₀)u + fᵧ(x₀,y₀)v,
\dot v = gₓ(x₀,y₀)u + gᵧ(x₀,y₀)v

where subscripts denote partial derivatives. The matrix of these partial derivatives is the Jacobian J, evaluated at the equilibrium.

考虑平衡点 (x₀, y₀)。令 u = x − x₀ 与 v = y − y₀。将 f 与 g 泰勒展开并只保留线性项，得到线性化系统

\dot u = fₓ(x₀,y₀)u + fᵧ(x₀,y₀)v,
\dot v = gₓ(x₀,y₀)u + gᵧ(x₀,y₀)v

其中下标表示偏导数。这些偏导数构成的矩阵即为雅可比矩阵 J，在平衡点处求值。

If the Jacobian has no eigenvalue with zero real part, the qualitative behaviour of the nonlinear system near the equilibrium is the same as that of the linearised system. This is the key content of the Hartman–Grobman theorem.

若雅可比矩阵没有实部为零的特征值，那么非线性系统在平衡点附近的定性行为与线性化系统相同。这正是 Hartman–Grobman 定理的核心内容。

6. Classification of Fixed Points | 平衡点的分类

Let λ₁ and λ₂ be the eigenvalues of the Jacobian matrix at the equilibrium. Their signs and natures determine the local phase portrait. The table below summarises the standard classification for real 2×2 systems.

设 λ₁ 与 λ₂ 是平衡点处雅可比矩阵的特征值。它们的符号与性质决定了局部的相图。下表总结了标准 2×2 实系统的常见分类。

Eigenvalues	Type	Stability
Real, λ₁ < λ₂ < 0	Stable node	Asymptotically stable
Real, λ₁ > λ₂ > 0	Unstable node	Unstable
Real, opposite signs	Saddle point	Unstable
Complex, negative real part	Stable spiral	Asymptotically stable
Complex, positive real part	Unstable spiral	Unstable
Purely imaginary	Centre	Neutrally stable

For a Jacobian matrix [[a, b], [c, d]], the trace is T = a + d and the determinant is D = ad − bc. Then λ² − Tλ + D = 0, so the discriminant Δ = T² − 4D determines whether the eigenvalues are real or complex.

对于雅可比矩阵 [[a, b], [c, d]]，迹为 T = a + d，行列式为 D = ad − bc。于是 λ² − Tλ + D = 0，判别式 Δ = T² − 4D 决定特征值是实数还是复数。

In IB problems you may be asked to identify the type from T and D without solving the characteristic polynomial. Remember that if D < 0 then eigenvalues are real and opposite in sign, giving a saddle.

在IB题目中，可能要求不求解特征多项式，而直接由 T 与 D 判断类型。注意：若 D < 0 ，特征值必为异号实数，对应鞍点。

7. Example 1: The Nonlinear Pendulum | 例1：非线性单摆

A damped pendulum with angular displacement θ and angular velocity ω = θ̇ can be written as

θ̇ = ω, ω̇ = −(g/L) sin θ − bω

where g is gravity, L is length and b is the damping coefficient. This is a nonlinear autonomous system because of sin θ.

带阻尼的单摆可用角位移 θ 与角速度 ω = θ̇ 表示为

θ̇ = ω, ω̇ = −(g/L) sin θ − bω

其中 g 是重力加速度，L 是摆长，b 是阻尼系数。由于 sin θ 的出现，这是一个非线性自治系统。

The equilibrium points are (0, 0), (±π, 0), (±2π, 0), etc. At θ = 0 the Jacobian has complex eigenvalues with negative real part when b > 0, so the pendulum’s lowest position is a stable spiral. At θ = ±π the pendulum is inverted; that equilibrium is a saddle, which means an inverted pendulum is unstable unless actively controlled.

平衡点为 (0, 0)、(±π, 0)、(±2π, 0) 等。在 θ = 0 处，当 b > 0 时雅可比矩阵具有负实部复特征值，因此最低位置是稳定螺线点。在 θ = ±π 处对应倒置状态；该平衡点为鞍点，表明倒立摆不稳定，除非施加主动控制。

8. Example 2: Lotka–Volterra Predator–Prey | 例2：Lotka–Volterra 捕食者–猎物模型

The classic predator–prey equations are

ẋ = x(α − βy), ẏ = −y(γ − δx)

where x is the prey population, y is the predator population, and α, β, γ, δ are positive constants. This is nonlinear due to the product terms x y.

经典的捕食者–猎物方程为

ẋ = x(α − βy), ẏ = −y(γ − δx)

其中 x 为猎物数量，y 为捕食者数量，α、β、γ、δ 均为正常数。由于乘积项 xy 的存在，这是非线性系统。

There are two equilibria: (0, 0), a saddle, and (x*, y*) = (γ/δ, α/β). The Jacobian at the nonzero equilibrium is purely imaginary, so the linearisation predicts a centre. In fact the nonlinear system has closed trajectories encircling this point, giving periodic oscillations that match real ecological boom–bust cycles.

系统有两个平衡点：(0, 0) 为鞍点，非零平衡点为 (x*, y*) = (γ/δ, α/β)。非零平衡点的雅可比矩阵特征值为纯虚数，因此线性化预测为中心点。事实上，非线性系统确实有环绕该点的闭合轨迹，产生周期振荡，与真实生态中的爆发–崩溃循环相符。

The trajectories in the positive quadrant are closed curves, meaning that both populations repeat exactly a cycle of rise and fall. This geometric insight is difficult to obtain from the differential equations alone.

第一象限内的轨迹是闭合曲线，意味着两个种群严格重复着上升与下降的周期循环。这种几何洞察仅靠微分方程本身很难获得。

9. Limit Cycles and Relaxation Oscillations | 极限环与张弛振荡

Not all closed curves in a phase plane are centres. A special solution called a limit cycle is an isolated closed trajectory: nearby trajectories spiral toward it or away from it. The Van der Pol oscillator,

ẍ − μ(1 − x²)ẋ + x = 0

is the most famous example. In state-space form with y = ẋ, we have ẋ = y and ẏ = μ(1 − x²)y − x.

相平面中并非所有闭合曲线都是中心点。一类特殊的解称为极限环，它是“孤立”的闭合轨迹：附近的轨迹螺旋趋向它或远离它。范德波尔振荡器

ẍ − μ(1 − x²)ẋ + x = 0

是最著名的例子。令 y = ẋ，状态空间形式为 ẋ = y，ẏ = μ(1 − x²)y − x。

When μ > 0, the position x = y = 0 is an unstable spiral, but all trajectories are attracted to a stable limit cycle. The emergence of a limit cycle from a change of stability is a classic example of a Hopf bifurcation.

当 μ > 0 时，平衡点 x = y = 0 是不稳定螺线点，但所有轨迹都被吸引到稳定极限环上。稳定性变化产生极限环，是 Hopf 分岔的经典例子。

Limit cycles cannot appear in linear systems, so they are a hallmark of nonlinearity. Recognising them in a phase portrait is a useful skill for IB extended essays and university-level work.

线性系统中不会出现极限环，因此它是非线性的标志性特征。在相图中识别极限环，对IB扩展论文及大学阶段的学习都是非常有用的技能。

10. Qualitative Analysis without Solving | 不求解方程的定性分析

In IB problems, you may be given a differential equation you cannot solve analytically. Phase-plane techniques allow you to answer questions about steady states, stability and periodic behaviour by combining nullclines, vector fields and linearisation.

在IB题目中，你可能会遇到无法解析求解的微分方程。相平面技术让你结合零斜线、向量场和线性化，就能回答关于稳态、稳定性与周期行为的问题。

A common question is to sketch the phase portrait near an equilibrium. Follow these steps: find the equilibrium points; compute the Jacobian at each; classify each using eigenvalues; draw local arrows; then fill in global directions using nullclines and intuition about conservation or dissipation.

常见题型是在平衡点附近画出相图。可按以下步骤：先求平衡点；在每个平衡点计算雅可比矩阵；用特征值分类；画出局部箭头；再借助零斜线以及对守恒或耗散的理解，补全全局方向。

Another typical task is to determine whether a population model can become extinct. If the origin is a saddle or an unstable node, extinction is not inevitable; if it is asymptotically stable, the population may collapse.

另一类典型任务是判断种群模型是否可能灭绝。若原点是鞍点或不稳定结点，则灭绝不是必然；若原点是渐近稳定的，则种群可能崩溃。

11. IB Exam Tips and Common Mistakes | IB考试技巧与常见错误

Always check whether the system is autonomous before drawing a phase plane. If t appears explicitly, the plane may not be stationary.

在画相平面之前务必确认系统是否为自治系统。若方程中显含 t，则相平面可能不是定常的。
Do not confuse the phase plane with a slope field for a single differential equation. Phase plane uses state variables (x, y), not (t, x).

不要将相平面与单变量微分方程的斜率场混淆。相平面使用状态变量 (x, y)，而不是 (t, x)。
When linearising, evaluate the Jacobian at the exact equilibrium point. Common errors include using non-equilibrium points or losing a negative sign in partial derivatives.

线性化时，必须在精确的平衡点上计算雅可比矩阵。常见错误包括代入非平衡点，或在偏导数中丢掉负号。
For a saddle point, remember that the stable and unstable manifolds are tangent to the eigenvectors of the Jacobian, but near a nonlinear system the curves bend.

对于鞍点，稳定流形与不稳定流形与雅可比矩阵的特征向量相切，但在非线性系统中曲线会弯曲。
In examination sketches, clearly label nullclines, equilibrium points and at least a few directions arrows. Marks are often awarded for these labels.

在考试绘图中，请明确标注零斜线、平衡点和若干方向箭头。这些标注往往是得分点。

12. Conclusion | 总结

Nonlinear differential equations require a blend of analytic, numeric and geometric thinking. Phase-plane analysis is a cornerstone of that approach, offering immediate insight into stability and oscillation that algebra cannot. Mastery of nullclines, linearisation and the classification of equilibria will serve you well not only in IB mathematics, but in physics, engineering, economics and life sciences.

非线性微分方程需要分析、数值与几何思维的结合。相平面分析正是这种方法的基石，它能直观揭示稳定性与振荡行为，而这些是纯代数难以展现的。熟练掌握零斜线、线性化与平衡点分类，不仅对IB数学考试很有帮助，在物理、工程、经济学与生命科学中也同样重要。

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IB Math: Nonlinear Differential Equations & Phase Plane Analysis | IB数学：非线性微分方程与相平面分析