General Phase Plane Analysis in IB Mathematics | IB数学：一般相平面分析

📚 General Phase Plane Analysis in IB Mathematics | IB数学：一般相平面分析

The phase plane is a powerful visual tool for understanding the behaviour of systems of two first-order differential equations. Instead of solving the equations exactly, we can study the qualitative picture formed by solution curves in the plane. This article introduces the key ideas of general phase plane analysis, with particular attention to what is needed for IB Mathematics Higher Level.

相平面是一种强有力的可视化工具，帮助我们理解两个一阶微分方程所组成系统的行为。我们不必精确求解方程，而是可以通过研究解曲线在平面中形成的图形来掌握系统的定性性质。本文将介绍一般相平面分析的核心概念，并重点关注IB数学高级水平中所需要的内容。

1. The Phase Plane and Trajectories | 相平面与轨线

Consider a system of two autonomous differential equations:

dx/dt = P(x, y), dy/dt = Q(x, y)

Because the right-hand sides do not depend explicitly on time, the system is called autonomous. A solution can be written as x = x(t), y = y(t), and as t varies the point (x(t), y(t)) moves along a curve in the xy-plane.

由于方程右端不显式依赖时间，我们称该系统为自治系统。解可以写成 x = x(t)，y = y(t)，随着 t 变化，点 (x(t), y(t)) 在 xy 平面中沿一条曲线运动。

This xy-plane is called the phase plane. Each curve that represents a solution is called a trajectory or orbit. The entire set of trajectories for different initial conditions is the phase portrait.

这个 xy 平面称为相平面。每一条代表解的曲线称为轨线或轨道。由不同初始条件得到的全部轨线构成了相图。

Importantly, autonomous systems have trajectories that do not cross. This follows from uniqueness of solutions for smooth functions P and Q. The only possible exceptions occur at equilibrium points.

重要的是，自治系统的轨线不会相交。这是因为对于光滑函数 P 和 Q，解具有唯一性；唯一的例外可能出现在平衡点处。

2. Equilibrium Points and Nullclines | 平衡点与零斜线

A point (x₀, y₀) is an equilibrium point if both derivatives vanish there:

P(x₀, y₀) = 0 and Q(x₀, y₀) = 0

At an equilibrium point the system is stationary: if the state starts exactly there, it will remain there forever.

在平衡点处，系统处于静止状态：如果状态从该点出发，它将永远停留在那里。

The nullclines help us locate and understand these points. The x-nullcline is the set of points where dx/dt = 0, i.e. P(x, y) = 0. On this curve, trajectories have vertical tangent direction. Similarly, the y-nullcline is where dy/dt = 0, and trajectories are horizontal there.

零斜线有助于我们找到并理解这些点。x 零斜线是满足 dx/dt = 0 的点集，即 P(x, y) = 0。在这条曲线上，轨线的切线方向是竖直的。类似地，y 零斜线是满足 dy/dt = 0 的点集，轨线在这条线上是水平的。

Equilibrium points are exactly the intersections of the two nullclines. By sketching the nullclines, we can divide the phase plane into regions where the signs of dx/dt and dy/dt are known.

平衡点恰好是两条零斜线的交点。通过绘制零斜线，我们可以把相平面划分为 dx/dt 和 dy/dt 符号已知的若干区域。

3. Linear Systems and Matrix Form | 线性系统与矩阵形式

It is often useful to write a linear system in matrix form:

d/dt [x; y] = A [x; y], A = [a b; c d]

This compact notation emphasizes that the system is linear in x and y. The matrix A encodes all the dynamics.

这种紧凑的记号强调系统关于 x 和 y 是线性的。矩阵 A 包含了全部动力学信息。

For a nonlinear system, we will later approximate the behaviour near an equilibrium by a linear system. This is why understanding linear systems is essential.

对于非线性系统，我们后面会在平衡点附近用线性系统做近似。因此，理解线性系统是至关重要的。

If we look for exponential solutions of the form [x; y] = e^{λt} [v₁; v₂], we obtain the eigenvalue problem Av = λv. The values of λ are found from the characteristic equation:

如果寻找形如 [x; y] = e^{λt} [v₁; v₂] 的指数解，我们就得到特征值问题 Av = λv。λ 的值可由特征方程求得：

λ² − (a + d)λ + (ad − bc) = 0

Here the trace of A is T = a + d, and the determinant is D = ad − bc. The equation is λ² − Tλ + D = 0.

其中 A 的迹为 T = a + d，行列式为 D = ad − bc。特征方程即 λ² − Tλ + D = 0。

4. Eigenvalues and Stability | 特征值与稳定性

The eigenvalues of the linear system determine the nature of the equilibrium point at the origin. Their real parts tell us about growth or decay in time.

线性系统在原点处的平衡点性质由特征值决定。特征值的实部告诉我们随时间增长还是衰减。

If both eigenvalues have negative real parts, the equilibrium is asymptotically stable: all nearby trajectories approach it as t → ∞.
如果两个特征值的实部均为负，则平衡点是渐近稳定的：所有邻近轨线在 t → ∞ 时趋近于它。

If at least one eigenvalue has positive real part, the equilibrium is unstable.
如果至少有一个特征值的实部为正，则平衡点是不稳定的。

If the real parts are zero, the equilibrium may be a centre, which is stable but not asymptotically stable.
如果实部为零，则平衡点可能是中心，它是稳定的但不是渐近稳定的。

This classification is often summarised in a stability diagram using the trace T and determinant D.

这种分类通常可以用以迹 T 和行列式 D 为坐标的稳定性图来总结。

5. Nodes and Saddles | 结点与鞍点

When the eigenvalues are real and distinct, the equilibrium is called a node or a saddle depending on their signs.

当特征值是互异的实数时，根据它们的符号，平衡点被称为结点或鞍点。

If both eigenvalues are real and have the same sign, the equilibrium is a node. When both are negative, trajectories move directly toward the origin; when both are positive, they move away from it. A stable node has two straight-line trajectories along the eigenvectors.

如果两个特征值都是实数且符号相同，则平衡点是结点。当两者都为负时，轨线直接趋向原点；当两者都为正时，轨线远离原点。稳定结点有两条沿特征向量方向的直线轨线。

If the eigenvalues are real and of opposite signs, the equilibrium is a saddle. It is always unstable. Trajectories approach along the eigenvector corresponding to the negative eigenvalue, and recede along the eigenvector corresponding to the positive eigenvalue. Most other trajectories swoop past the saddle without ever reaching it.

如果特征值是实数且符号相反，则平衡点是鞍点。它总是不稳定的。轨线沿负特征值对应的特征向量靠近，沿正特征值对应的特征向量远离。大多数其他轨线都会从鞍点附近掠过，而不会到达它。

6. Spirals and Centres | 螺旋点与中心

When the eigenvalues are complex, the equilibrium is either a spiral or a centre, depending on whether the real part is nonzero.

当特征值为复数时，平衡点要么是螺旋点，要么是中心，具体取决于实部是否为零。

Suppose λ = α ± iβ, with β ≠ 0. If α ≠ 0, the trajectories spiral around the equilibrium. If α < 0, the spiral converges to the equilibrium, so it is asymptotically stable; if α > 0, the spiral diverges and the equilibrium is unstable.

设 λ = α ± iβ，且 β ≠ 0。如果 α ≠ 0，轨线围绕平衡点螺旋运动。若 α < 0，螺旋向内收敛，因此平衡点渐近稳定；若 α > 0，螺旋向外发散，平衡点不稳定。

If α = 0, the eigenvalues are purely imaginary, λ = ± iβ. The trajectories form closed ellipses around the equilibrium, and the equilibrium is called a centre. Nearby trajectories neither approach nor leave it; instead they repeatedly orbit it.

如果 α = 0，特征值为纯虚数，λ = ± iβ。轨线形成围绕平衡点的闭合椭圆，该平衡点称为中心。邻近的轨线既不靠近也不远离它，而是不断绕行。

The direction of rotation can be determined by checking the sign of dy/dt or dx/dt at a convenient point, for example on the positive x-axis.

旋转方向可以通过在某个方便的点（例如正 x 轴上）检查 dy/dt 或 dx/dt 的符号来确定。

7. Linearization and Local Behaviour | 线性化与局部行为

For a nonlinear system, the behaviour near a non-degenerate equilibrium point is usually the same as that of the linearised system. This idea is the key to general phase plane analysis.

对于非线性系统，在非退化平衡点附近的局部行为通常与线性化系统相同。这一思想是一般相平面分析的关键。

Suppose (x₀, y₀) is an equilibrium of a nonlinear system. Define small deviations u = x − x₀ and v = y − y₀. Using Taylor expansion and ignoring higher-order terms, we get the linearised system:

设 (x₀, y₀) 是非线性系统的一个平衡点。定义小偏差 u = x − x₀ 和 v = y − y₀。利用泰勒展开并忽略高阶项，我们得到线性化系统：

d/dt [u; v] = J [u; v]

Here J is the Jacobian matrix evaluated at the equilibrium:

其中 J 是在平衡点处计算的雅可比矩阵：

J = [∂P/∂x ∂P/∂y; ∂Q/∂x ∂Q/∂y]

If J has two real, distinct eigenvalues, or a pair of complex eigenvalues with nonzero real part, then the local phase portrait of the nonlinear system is qualitatively the same as that of the linear system. If J has a double eigenvalue or purely imaginary eigenvalues, higher-order terms can matter and the linear analysis may be inconclusive.

如果 J 有两个互异的实特征值，或一对实部非零的复特征值，那么非线性系统的局部相图在定性上与线性系统相同。如果 J 有二重特征值或纯虚特征值，则高阶项可能有影响，线性分析可能无法给出确定的结论。

8. Steps for Constructing a Phase Portrait | 构造相图的步骤

We can systematically sketch the phase portrait of a two-dimensional autonomous system by following these steps.

我们可以按以下步骤系统地画出二维自治系统的相图。

Find the equilibrium points. Solve P(x, y) = 0 and Q(x, y) = 0 simultaneously.
找到平衡点。联立求解 P(x, y) = 0 与 Q(x, y) = 0。

Draw the nullclines. Sketch the curves where dx/dt = 0 and dy/dt = 0. Mark the direction of motion in each region.
画出零斜线。绘制 dx/dt = 0 与 dy/dt = 0 的曲线，并在各区域标出运动方向。

Classify each equilibrium. Compute the Jacobian at each equilibrium and determine the signs of the eigenvalues or the trace-determinant pair.
对每个平衡点分类。在每个平衡点处计算雅可比矩阵，并根据特征值符号或迹-行列式对进行分类。

Sketch straight-line trajectories if they exist. For real distinct eigenvalues, trajectories along eigenvectors are straight lines.
若存在直线轨线则画出它们。对于互异实特征值，沿特征向量的轨线是直线。

Fill in nearby trajectories. Use continuity and the direction field to sketch smooth curves consistent with the classification.
填入附近的轨线。利用连续性和方向场，画出与分类一致的光滑曲线。

Always check the direction arrows by evaluating dx/dt and dy/dt at a few test points.

始终通过在若干测试点计算 dx/dt 和 dy/dt 来检查箭头方向。

9. Application: Predator-Prey Model | 应用：捕食者-被捕食者模型

A classic example is the Lotka–Volterra predator-prey model. Let x represent the prey population and y the predator population. A typical form is:

一个经典例子是 Lotka–Volterra 捕食者-被捕食者模型。设 x 表示被捕食者种群数量，y 表示捕食者种群数量。一个典型的形式为：

dx/dt = ax − bxy, dy/dt = −cy + dxy

with a, b, c, d positive constants. The nonzero equilibrium is found by setting both derivatives to zero:

其中 a、b、c、d 为正数。令两个导数为零，可得非零平衡点：

(x₀, y₀) = (c/d, a/b)

The Jacobian at this point has purely imaginary eigenvalues, so the equilibrium is a centre. This means the populations oscillate in a periodic cycle that depends on the initial conditions.

该点处的雅可比矩阵具有纯虚特征值，因此平衡点是中心。这意味着种群数量会以依赖于初始条件的周期形式振荡。

However, real ecosystems rarely display perfect periodic cycles. Adding a small damping term or external disturbances can turn the centre into a stable or unstable spiral, which better reflects reality.

然而，真实生态系统很少呈现完美的周期循环。加入小的阻尼项或外部扰动会使中心变为稳定或不稳定的螺旋点，这更能反映实际情况。

10. Application: Damped Pendulum | 应用：阻尼摆

The motion of a damped pendulum can be written as a second-order equation:

阻尼摆的运动可以写成二阶方程：

d²θ/dt² + c dθ/dt + (g/L) sin θ = 0

Let x = θ and y = dθ/dt. Then the system becomes:

令 x = θ，y = dθ/dt。则系统变为：

dx/dt = y, dy/dt = −(g/L) sin x − c y

Equilibria occur when y = 0 and sin x = 0, so at x = nπ. The point x = 0 corresponds to the pendulum hanging straight down; it is stable. The point x = π corresponds to the upside-down position; it is unstable.

平衡点出现在 y = 0 且 sin x = 0 时，即 x = nπ。x = 0 对应摆垂直向下悬挂，是稳定的；x = π 对应倒立位置，是不稳定的。

The phase portrait shows closed loops near the stable equilibrium when damping is zero, or inward spirals when damping is positive. The unstable equilibrium at x = π appears as a saddle.

相图中，当阻尼为零时，稳定平衡点附近是闭合环；当阻尼为正时，则呈向内的螺旋。x = π 处的不稳定平衡点表现为鞍点。

This example demonstrates how phase plane analysis enables us to understand a physical system without solving its complicated equation explicitly.

这个例子展示了相平面分析如何使我们在不显式求解复杂方程的情况下理解一个物理系统。

11. Using the Trace-Determinant Plane | 利用迹-行列式平面

For a linear system with coefficient matrix A, the trace T and determinant D provide a quick way to classify the equilibrium.

对于系数矩阵为 A 的线性系统，迹 T 和行列式 D 提供了快速分类平衡点的方法。

Condition	Type of equilibrium
D < 0	Saddle
D > 0 and T² − 4D > 0	Node (stable if T < 0, unstable if T > 0)
D > 0 and T² − 4D < 0	Spiral (stable if T < 0, unstable if T > 0)
D > 0 and T = 0	Centre
T² − 4D = 0	Borderline case: repeated eigenvalue

The condition T² − 4D is the discriminant of the characteristic equation. Its sign distinguishes real from complex eigenvalues.

T² − 4D 是特征方程的判别式，其符号可区分实特征值与复特征值。

12. Common Pitfalls in IB Exams | IB考试中的常见误区

Students often confuse the stability of a nonlinear equilibrium with the stability of its linearised system. Always remember that linearisation is only a local approximation, valid near the equilibrium point in the generic case.

同学们经常混淆非线性平衡点的稳定性与其线性化系统的稳定性。请记住，线性化只是局部近似，在一般情形下仅当接近平衡点时才是有效的。

Another frequent mistake is omitting the direction of motion on trajectories. For example, a stable spiral must be drawn with arrows pointing towards the origin, while an unstable spiral must point away from it.

另一个常见错误是漏掉轨线上的运动方向。例如，稳定螺旋的箭头必须指向原点，而不稳定螺旋的箭头必须远离原点。

When sketching nullclines, be careful: the x-nullcline is where dx/dt = 0, not where x = 0. Similarly, the y-nullcline is where dy/dt = 0, not the y-axis.

绘制零斜线时要小心：x 零斜线是 dx/dt = 0 的地方，而不是 x = 0。类似地，y 零斜线是 dy/dt = 0 的地方，而不是 y 轴。

Finally, do not forget to check whether the system is autonomous before using the phase plane method. If time appears explicitly in P or Q, trajectories may cross and the phase plane analysis is no longer applicable.

最后，在使用相平面方法前，请确认系统是否为自治系统。如果 P 或 Q 中显式出现时间，轨线可能相交，相平面分析就不再适用。

By mastering equilibrium classification, nullclines and linearisation, you can unlock a deep understanding of dynamical systems. The phase plane is not just a tool for exams; it is a window into the behaviour of countless real-world systems.

通过掌握平衡点分类、零斜线和线性化，你可以深入理解动力系统。相平面不仅是考试的工具，更是观察无数现实世界系统行为的一扇窗口。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

General Phase Plane Analysis in IB Mathematics | IB数学：一般相平面分析