IB Mathematics: Limit Cycles and Phase Plane Trajectories | IB数学：极限环与相平面轨迹

📚 IB Mathematics: Limit Cycles and Phase Plane Trajectories | IB数学：极限环与相平面轨迹

In IB Mathematics, especially when studying systems of differential equations, the phase plane gives us a visual way to understand how a system behaves over time. Instead of finding exact formulas for x(t) and y(t), we draw trajectories in the xy-plane that show the relationship between the two state variables.

在IB数学中，尤其是在学习微分方程组时，相平面为我们提供了理解系统随时间演化规律的可视化方法。我们不再需要求出 x(t) 和 y(t) 的精确公式，而是在 xy 平面中画出轨迹，以展现两个状态变量之间的关系。

1. What Is a Phase Plane? | 什么是相平面？

We study autonomous systems of the form

我们研究如下形式的自治系统：

dx/dt = f(x, y), dy/dt = g(x, y)

Because the right-hand sides do not contain t explicitly, every point (x, y) in the plane has a unique velocity vector (dx/dt, dy/dt). The xy-plane is called the phase plane, and a curve traced out by a solution is called a trajectory or orbit.

由于等号右侧不显含时间 t，平面上的每一点 (x, y) 都有唯一的速度向量 (dx/dt, dy/dt)。这个 xy 平面称为相平面，而解在平面上描绘出的曲线称为轨迹或轨线。

Example: the simple harmonic oscillator can be written as dx/dt = y and dy/dt = -x. Its phase-plane trajectories are circles centered at the origin.

例如，简谐振子可以写成 dx/dt = y，dy/dt = -x。它的相平面轨迹是以原点为圆心的圆。

2. Vector Fields and Trajectories | 向量场与轨迹

At each point of the phase plane, the system provides a tangent direction. Drawing these directions at many points creates a vector field. A solution trajectory must be tangent to the vector field at every point it passes through.

在相平面中的每个点上，系统都给出了一个切方向。在大量点上画出这些方向，就构成了向量场。一条解轨迹在其经过的每一点都必须与向量场相切。

If f(x, y) is not zero, the slope of a trajectory can be written as

如果 f(x, y) 不为零，轨迹的斜率可以写成

dy/dx = g(x, y) / f(x, y)

This equation removes time from the problem and lets us study the shape of trajectories geometrically.

这个方程消去了时间变量，使我们能从几何上研究轨迹的形状。

Remember: trajectories of an autonomous system do not intersect. If two trajectories met at a point, the direction at that point would not be unique.

请记住：自治系统的轨迹不会相交。如果两条轨迹在同一点相遇，那么该点的方向就不再唯一。

3. Nullclines and Equilibrium Points | 零斜线与平衡点

Nullclines are curves where one of the derivatives vanishes. The x-nullcline satisfies dx/dt = 0, so f(x, y) = 0. The y-nullcline satisfies dy/dt = 0, so g(x, y) = 0.

零斜线是某个导数等于零的曲线。x-零斜线满足 dx/dt = 0，即 f(x, y) = 0；y-零斜线满足 dy/dt = 0，即 g(x, y) = 0。

On the x-nullcline, trajectories cross with a vertical tangent (if dy/dt ≠ 0).
在 x-零斜线上，轨迹以竖直切线穿过（前提是 dy/dt ≠ 0）。
On the y-nullcline, trajectories cross with a horizontal tangent (if dx/dt ≠ 0).
在 y-零斜线上，轨迹以水平切线穿过（前提是 dx/dt ≠ 0）。

An equilibrium point, also called a fixed point or critical point, occurs where both derivatives are zero. These are the intersections of the x- and y-nullclines.

平衡点也称作不动点或临界点，出现在两个导数均为零的位置。它们就是 x-零斜线与 y-零斜线的交点。

4. Stability of Equilibrium Points | 平衡点的稳定性

To classify an equilibrium point, we linearize the system near it. The Jacobian matrix is

为了对平衡点进行分类，我们在其附近将系统线性化。雅可比矩阵为

J = [ [∂f/∂x, ∂f/∂y], [∂g/∂x, ∂g/∂y] ]

We then evaluate J at the equilibrium point and find its eigenvalues λ.

我们在平衡点处计算 J 并求出其特征值 λ。

Eigenvalues λ	Type of Equilibrium	轨迹类型
Real, both negative	Stable node	稳定结点
Real, both positive	Unstable node	不稳定结点
Real, opposite signs	Saddle point	鞍点
Complex with negative real part	Stable spiral	稳定螺线
Complex with positive real part	Unstable spiral	不稳定螺线
Purely imaginary	Centre	中心

This classification tells us whether nearby trajectories move toward the equilibrium, move away from it, or circulate around it.

这种分类告诉我们：附近的轨迹是趋向平衡点、远离平衡点，还是围绕它旋转。

5. Limit Cycles: A Special Closed Trajectory | 极限环：一种特殊的闭轨

A limit cycle is a closed, isolated trajectory. “Closed” means the system repeats exactly after some period T. “Isolated” means nearby trajectories are not closed; they spiral either toward the limit cycle or away from it.

极限环是一条封闭且孤立的轨迹。“封闭”意味着经过一段时间 T 后系统精确重复。“孤立”意味着附近的轨迹不是闭轨，它们要么螺旋趋向极限环，要么螺旋远离极限环。

This is different from a centre, where every nearby trajectory is also closed. In a centre, surrounding orbits are infinite in number. In a limit cycle, there is only one closed orbit at that location.

这一点与中心不同：在中心附近，每一条轨迹都是封闭的，周围存在无穷多条闭轨。而在极限环处，在这个位置上只有一条闭轨。

Limit cycles arise only in nonlinear systems. A linear system can have closed orbits, but they are never isolated.

极限环只出现在非线性系统中。线性系统可以拥有闭轨，但这些闭轨永远不会是孤立的。

6. The Van der Pol Oscillator | 范德波尔振荡器

The Van der Pol oscillator is the classic example of a system with a stable limit cycle. It has equations

范德波尔振荡器是稳定极限环的经典例子。其方程为

dx/dt = y, dy/dt = -x + μ(1 – x²)y

Here μ is a positive parameter. The term μ(1 – x²)y acts like nonlinear damping: it removes energy when |x| is large and adds energy when |x| is small.

其中 μ 为正参数。项 μ(1 – x²)y 的作用类似于非线性阻尼：当 |x| 较大时它耗散能量，当 |x| 较小时它输入能量。

For μ > 0, the equilibrium at the origin is an unstable spiral. Almost every trajectory spirals outward until it reaches a closed curve, after which it stays on that curve forever. That closed curve is a limit cycle.

当 μ > 0 时，原点处的平衡点是不稳定螺线。几乎所有轨迹都会向外螺旋，直到到达一条封闭曲线，此后永远停留在这条曲线上。这条封闭曲线就是极限环。

The limit cycle is stable because trajectories inside it spiral outward and trajectories outside it spiral inward.

该极限环是稳定的，因为环内的轨迹向外螺旋，环外的轨迹向内螺旋。

7. Existence and Nonexistence of Limit Cycles | 极限环的存在性与不存在性

Two important theorems help decide whether a limit cycle can exist in a given region.

两个重要定理可以帮助我们判断某个区域内是否存在极限环。

Poincaré-Bendixson Theorem: If a trajectory enters and remains in a closed bounded region of the plane, and if that region contains no equilibrium points, then the trajectory must approach a limit cycle.

庞加莱-本迪克森定理：如果一条轨迹进入并始终停留在平面上的某个封闭有界区域内，并且该区域内没有平衡点，那么这条轨迹必定趋近于某个极限环。

Bendixson’s Criterion: If in a simply connected region the expression

本迪克森判据：如果在某个单连通区域内，表达式

∂f/∂x + ∂g/∂y

does not change sign and is not identically zero, then no closed trajectory lies entirely inside that region.

不变号且不恒等于零，那么该区域内不存在完全位于其中的闭轨。

Bendixson’s criterion is often the quickest way to rule out limit cycles in an IB exam question.

在IB考试中，本迪克森判据通常是排除极限环最快的方法。

8. How to Sketch Phase-Plane Trajectories | 如何绘制相平面轨迹

Drawing an accurate phase portrait is a skill. Use a systematic procedure.

画出准确的相图是一项技能。请采用系统化的步骤。

Find all equilibrium points by solving f = 0 and g = 0.
借助 f = 0 和 g = 0 求出所有平衡点。
Draw the nullclines and mark where trajectories cross them vertically or horizontally.
画出零斜线，并标出轨迹在何处竖直或水平穿过它们。
Classify each equilibrium point using the Jacobian matrix.
用雅可比矩阵对每个平衡点进行分类。
Choose representative starting points and sketch the direction of motion.
选择有代表性的起点，画出运动方向。
Add arrows showing increasing time t.
添加箭头，表示时间 t 增大的方向。

Remember that trajectories along an unstable spiral leave the equilibrium, while trajectories along a stable spiral approach it. A saddle point has two special incoming curves and two special outgoing curves.

请记住：不稳定螺线附近的轨迹离开平衡点，稳定螺线附近的轨迹则趋向平衡点。鞍点有两条特殊进入曲线和两条特殊离开曲线。

9. Applications of Limit Cycles | 极限环的应用

Limit cycles appear in biology, chemistry, and engineering. In predator-prey models, the Lotka-Volterra system produces closed orbits, but those orbits are centres, not limit cycles. In real ecosystems, nonlinear effects can turn those centres into stable limit cycles, meaning populations can settle into a regular periodic rhythm.

极限环出现在生物学、化学和工程学中。在捕食者-猎物模型中，Lotka-Volterra 系统产生闭轨，但这些闭轨是中心，而非极限环。在真实生态系统中，非线性效应可以把这些中心变为稳定极限环，意味着种群数量可以进入规则的周期节律。

In chemistry, the Belousov-Zhabotinsky reaction shows colour changes that repeat periodically. Its mathematical model has a stable limit cycle, so the reaction oscillates without stopping.

在化学中，Belousov-Zhabotinsky 反应呈现周期性重复的颜色变化。其数学模型具有稳定极限环，因此反应会持续振荡而不会停止。

In engineering, electronic circuits such as the Van der Pol oscillator use limit cycles to generate stable signals. The amplitude of the signal is fixed by the limit cycle, not by initial conditions.

在工程学中，范德波尔振荡器等电子电路利用极限环来产生稳定信号。信号的振幅由极限环决定，而不是由初始条件决定。

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

When solving phase-plane questions in IB exams, clarity and organisation matter. Label axes, mark the direction of increasing t, and state which curves are nullclines.

在解答IB考试中的相平面问题时，清晰与条理非常重要。要标注坐标轴，标出时间 t 增大的方向，并说明哪些曲线是零斜线。

Do not confuse a centre with a limit cycle. A centre is surrounded by closed orbits; a limit cycle is isolated.
不要把中心与极限环混淆。中心周围全是闭轨，而极限环是孤立的。
Do not assume that a closed curve in a sketch is a limit cycle. Check whether nearby trajectories are also closed.
不要以为图中一条封闭曲线就是极限环。要检查附近轨迹是否也是封闭的。
Do not evaluate the Jacobian at the origin if the equilibrium is elsewhere. Substitute the equilibrium coordinates first.
如果平衡点不在原点，不要用原点去计算雅可比矩阵。先把平衡点坐标代入。
Remember that Bendixson’s criterion only rules out closed curves; it does not prove that a limit cycle exists.
注意本迪克森判据只能排除闭轨，并不能证明极限环一定存在。

In an exam, if you are asked to “show that a limit cycle exists,” state the Poincaré-Bendixson theorem explicitly and verify its conditions.

在考试中，如果要求你“证明存在极限环”，要明确写出庞加莱-本迪克森定理，并验证其条件。

11. Worked Example Idea | 典型例题思路

Consider the system

考虑如下系统

dx/dt = y, dy/dt = -x + (1 – x²)y

The only equilibrium point is found by setting y = 0 and -x + (1 – x²)y = 0. This gives x = 0, y = 0. The Jacobian at the origin is

令 y = 0 且 -x + (1 – x²)y = 0，可以求出唯一的平衡点，即 x = 0, y = 0。原点处的雅可比矩阵为

J = [ [0, 1], [-1, 1] ]

The eigenvalues are λ = (1 ± i√3)/2. Their real part is positive, so the origin is an unstable spiral. The surrounding trajectories spiral outward, and Bendixson’s criterion is not sufficient to rule out a limit cycle here. In fact, this is the Van der Pol system with μ = 1, and a stable limit cycle exists.

其特征值为 λ = (1 ± i√3)/2。由于实部为正，原点是不稳定螺线。周围的轨迹向外螺旋，而本迪克森判据在这里不足以排除极限环。实际上，这正是 μ = 1 时的范德波尔系统，存在稳定极限环。

For revision, practice by drawing nullclines, classifying equilibria, and then comparing your sketch with the actual vector field.

复习时，请练习画零斜线、对平衡点分类，然后把自己的草图与真实向量场进行比较。

Mastering limit cycles and phase-plane trajectories helps you understand why nonlinear systems can oscillate forever without external input. In IB Mathematics, being able to interpret a phase portrait is just as important as solving equations symbolically.

掌握极限环与相平面轨迹，有助于你理解非线性系统为什么可以在没有外部输入的情况下永远振荡。在IB数学中，能够解读相图与能够符号化求解方程同样重要。

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IB Mathematics: Limit Cycles and Phase Plane Trajectories | IB数学：极限环与相平面轨迹