Nonlinear Differential Equations and the Phase Plane | 非线性微分方程与相平面

📚 Nonlinear Differential Equations and the Phase Plane | 非线性微分方程与相平面

Nonlinear differential equations describe systems where the rate of change depends on the variables in a non-proportional way. Such equations rarely admit closed-form analytical solutions, so we rely on qualitative methods to understand long-term behaviour. The phase plane provides a powerful geometric framework for analysing autonomous systems of two first-order ODEs, revealing equilibrium points, stability, and global dynamics without explicit integration.

非线性微分方程描述的是变化率以非线性方式依赖于变量的系统。这类方程鲜有封闭形式的解析解，因此我们依赖定性方法来理解其长期行为。相平面为分析由两个一阶常微分方程组成的自治系统提供了强大的几何框架，无需显式积分即可揭示平衡点、稳定性和全局动力学。

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1. Introduction to Nonlinear Differential Equations | 非线性微分方程简介

In an IB context, a nonlinear ODE is one where the dependent variable or its derivatives appear in products, powers (other than 1), or inside transcendental functions. For example, the logistic equation dx/dt = rx(1 − x/K) is nonlinear because of the x² term. Unlike linear equations, superposition does not hold, and small changes in initial conditions can lead to vastly different trajectories. This richness makes them ideal for modelling real-world phenomena such as population growth, chemical reactions, and mechanical oscillations.

在IB课程中，非线性常微分方程是指因变量或其导数以乘积、幂（非一次）或超越函数形式出现的方程。例如，逻辑斯谛方程 dx/dt = rx(1 − x/K) 由于 x² 项而是非线性的。与线性方程不同，叠加原理不成立，初始条件的微小变化可能导致截然不同的轨迹。这种丰富的动力学特性使其成为模拟种群增长、化学反应和机械振荡等现实世界现象的理想工具。

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

An autonomous system of two first-order ODEs takes the form dx/dt = f(x, y), dy/dt = g(x, y), where time t does not appear explicitly in f or g. The xy-plane is called the phase plane. A solution (x(t), y(t)) traces a directed curve known as a trajectory or orbit. The vector field (f(x,y), g(x,y)) attaches an arrow to every point, and the collection of representative orbits forms the phase portrait. By sketching the portrait, we can determine whether populations coexist, oscillate, or become extinct without solving the equations analytically.

自治系统由两个一阶常微分方程构成，形式为 dx/dt = f(x, y), dy/dt = g(x, y)，其中时间 t 显式不出现在 f 或 g 中。xy 平面称为相平面。解 (x(t), y(t)) 描绘出一条有向曲线，称为轨迹或轨道。向量场 (f(x,y), g(x,y)) 为每个点附加一个箭头，所有代表性轨道构成的图形称为相图。通过绘制相图，我们无需解析求解方程即可判断种群是共存、振荡还是灭绝。

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3. Equilibrium Points and Nullclines | 平衡点与零等倾线

Equilibrium points (also called fixed or critical points) occur where f(x,y) = 0 and g(x,y) = 0 simultaneously. They correspond to constant solutions because dx/dt = dy/dt = 0. To locate them, we solve the algebraic system. The x-nullcline is the curve where f(x,y) = 0; along it, trajectories move vertically because dx/dt = 0. Similarly, the y-nullcline is where g(x,y) = 0, giving horizontal motion. Equilibrium points lie at the intersections of these nullclines, which also partition the phase plane into regions where the direction of flow is consistent.

平衡点（也称为不动点或临界点）出现在 f(x,y) = 0 和 g(x,y) = 0 同时成立的位置。它们对应常数解，因为 dx/dt = dy/dt = 0。为了找到它们，我们需要求解该代数方程组。x 零等倾线是满足 f(x,y) = 0 的曲线；沿着该线，轨迹垂直移动，因为 dx/dt = 0。类似地，y 零等倾线满足 g(x,y) = 0，给出水平运动。平衡点位于这些零等倾线的交点处，它们还将相平面划分为流动方向保持一致的区域。

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

To determine the local behaviour near an equilibrium point (x₀, y₀), we linearise the nonlinear system. Let u = x − x₀, v = y − y₀ be small perturbations. Using a Taylor expansion and keeping only first-order terms, we obtain the linear system du/dt ≈ ∂f/∂x·u + ∂f/∂y·v, dv/dt ≈ ∂g/∂x·u + ∂g/∂y·v, where all partial derivatives are evaluated at (x₀, y₀). The Jacobian matrix J is

J = [[∂f/∂x, ∂f/∂y], [∂g/∂x, ∂g/∂y]] at (x₀,y₀).

为了确定平衡点 (x₀, y₀) 附近的局部行为，我们对非线性系统进行线性化。令 u = x − x₀, v = y − y₀ 为微小扰动。使用泰勒展开并仅保留一阶项，我们得到线性系统 du/dt ≈ ∂f/∂x·u + ∂f/∂y·v, dv/dt ≈ ∂g/∂x·u + ∂g/∂y·v，其中所有偏导数均在 (x₀, y₀) 处计算。雅可比矩阵 J 为

J = [[∂f/∂x, ∂f/∂y], [∂g/∂x, ∂g/∂y]] 在 (x₀,y₀) 处。

The eigenvalues λ₁ and λ₂ of J determine the nature and stability of the equilibrium. The trace τ = ∂f/∂x + ∂g/∂y and determinant Δ = (∂f/∂x)(∂g/∂y) − (∂f/∂y)(∂g/∂x) characterise the linearised dynamics through the characteristic equation.

J 的特征值 λ₁ 和 λ₂ 决定了平衡点的性质和稳定性。迹 τ = ∂f/∂x + ∂g/∂y 和行列式 Δ = (∂f/∂x)(∂g/∂y) − (∂f/∂y)(∂g/∂x) 通过特征方程刻画了线性化动力学。

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5. Classification of Equilibrium Points | 平衡点的分类

The classification of an isolated equilibrium point depends on the eigenvalues of J. The following table summarises the main types for a two-dimensional system. We assume Δ ≠ 0 so that the equilibrium is isolated and the linearisation is valid.

孤立平衡点的分类取决于 J 的特征值。下表总结了二维系统的主要类型。我们假设 Δ ≠ 0，以确保平衡点是孤立的且线性化有效。

Eigenvalues	Name	Stability
Real, both negative	Stable node	Asymptotically stable
Real, both positive	Unstable node	Unstable
Real, opposite signs	Saddle point	Unstable
Complex with negative real part	Stable spiral	Asymptotically stable
Complex with positive real part	Unstable spiral	Unstable
Purely imaginary	Centre	Stable (not asymptotically)

特征值	类型名称	稳定性
两个负实数	稳定结点	渐近稳定
两个正实数	不稳定结点	不稳定
异号实数	鞍点	不稳定
实部为负的复数	稳定螺旋	渐近稳定
实部为正的复数	不稳定螺旋	不稳定
纯虚数	中心	稳定（非渐近）

For a centre, the linearised system predicts closed orbits, but the nonlinear system may exhibit a true centre, a stable spiral, or an unstable spiral depending on higher-order terms.

对于中心点，线性化系统预测出闭合轨道，但非线性系统根据高阶项可能表现出真正的中心、稳定螺旋或不稳定螺旋。

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6. Phase Portraits: Nodes and Saddle Points | 相图：结点与鞍点

When eigenvalues are both real and negative, trajectories approach the equilibrium along two special straight-line trajectories determined by the eigenvectors. The node is asymptotically stable: regardless of the initial condition, the system eventually settles at this point. If both eigenvalues are real and positive, the arrows reverse, creating an unstable node from which all trajectories diverge.

当特征值均为负实数时，轨迹沿着由特征向量决定的两条特殊直线轨迹逼近平衡点。该结点是渐近稳定的：无论初始条件如何，系统最终都会停在这个点。如果两个特征值均为正实数，箭头方向反转，形成一个所有轨迹均远离的不稳定结点。

A saddle point arises when eigenvalues are real but have opposite signs. One direction (corresponding to the negative eigenvalue) attracts, while the orthogonal direction (positive eigenvalue) repels. The stable and unstable manifolds are curves that organise the global dynamics. Saddle points are always unstable, but they act as gateways between different basins of attraction in nonlinear systems.

鞍点出现在特征值为异号实数时。一个方向（对应负特征值）具有吸引性，而与之正交的方向（正特征值）具有排斥性。稳定流形和不稳定流形是组织全局动力学的曲线。鞍点总是不稳定的，但在非线性系统中它们起着不同吸引盆之间门户的作用。

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7. Phase Portraits: Spirals and Centres | 相图：螺旋与中心

Complex eigenvalues λ = α ± iβ, with α < 0, produce a stable spiral. Trajectories wind around the equilibrium point infinitely many times while approaching it, creating an inward spiral. If α > 0, the spiral is unstable and unwinds outward. The sign of α is determined by the trace τ (for a 2×2 matrix, τ = 2α), so τ controls the exponential envelope of the oscillation.

复特征值 λ = α ± iβ，当 α < 0 时产生稳定螺旋。轨迹围绕平衡点旋转无穷多次并逐渐逼近它，形成向内螺旋。如果 α > 0，螺旋不稳定并向外旋开。α 的符号由迹 τ 决定（对于 2×2 矩阵，τ = 2α），因此 τ 控制振荡的指数包络。

When eigenvalues are purely imaginary (τ = 0 and Δ > 0), the linearised system has a centre. Orbits are concentric ellipses or circles, indicating neutral stability. In nonlinear systems, the true behaviour may be a centre, but small nonlinearities can transform it into a weak spiral. Detecting a genuine centre requires additional conserved quantities or symmetry arguments, such as a first integral.

当特征值为纯虚数（τ = 0 且 Δ > 0）时，线性化系统具有中心点。轨道为同心椭圆或圆，表明中性稳定性。在非线性系统中，真实行为可能是中心，但微小的非线性项可能将其转变为弱螺旋。检验真正的中心需要额外的守恒量或对称性论证，例如首次积分。

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8. Stability Analysis and the Trace-Determinant Plane | 稳定性分析与迹-行列式平面

The characteristic equation for J is λ² − τλ + Δ = 0. The discriminant D = τ² − 4Δ determines whether eigenvalues are real (D ≥ 0) or complex (D < 0). The τ-Δ plane organises all possible linear behaviours: the parabola τ² = 4Δ separates nodes from spirals; the positive Δ-axis hosts centres when τ = 0; and the region Δ < 0 corresponds to saddle points. A stable equilibrium requires τ < 0 and Δ > 0, summarising the Routh–Hurwitz criteria for two dimensions.

J 的特征方程为 λ² − τλ + Δ = 0。判别式 D = τ² − 4Δ 决定特征值是实数 (D ≥ 0) 还是复数 (D < 0)。τ-Δ 平面组织所有可能的线性行为：抛物线 τ² = 4Δ 将结点与螺旋分开；正 Δ 轴上 τ = 0 处为中心点；Δ < 0 的区域对应鞍点。稳定平衡点要求 τ < 0 且 Δ > 0，这概括了二维情况下的劳斯–赫尔维茨准则。

This geometric classification helps in parameter studies. By tracking how τ and Δ change as a parameter varies, one can predict bifurcations—qualitative changes in the phase portrait, such as a stable node becoming a stable spiral or a saddle emerging when Δ switches sign.

这种几何分类有助于参数研究。通过追踪 τ 和 Δ 随参数变化的情况，可以预测分岔 —— 相图的定性改变，例如稳定结点变为稳定螺旋，或当 Δ 变号时出现鞍点。

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9. Limit Cycles and Nonlinear Phenomena | 极限环与非线性现象

A limit cycle is an isolated closed trajectory that attracts or repels nearby orbits. It is a genuinely nonlinear phenomenon that cannot occur in linear systems. If all nearby trajectories spiral towards the cycle as t → ∞, it is a stable limit cycle; if they spiral away, it is unstable. Limit cycles model self-sustained oscillations, such as the beating of a heart, predator-prey cycles, and electronic oscillators. The Poincaré–Bendixson theorem provides sufficient conditions for the existence of a limit cycle in a planar region, a key tool for IB HL exploration.

极限环是一个孤立的闭合轨迹，它吸引或排斥邻近的轨道。这是一种真正的非线性现象，在线性系统中不可能出现。如果当 t → ∞ 时所有邻近轨迹都螺旋趋向该环，则为稳定极限环；如果螺旋远离，则为不稳定极限环。极限环模拟自持振荡，例如心脏跳动、捕食者-猎物周期和电子振荡器。庞加莱–本迪克森定理为平面区域中极限环的存在提供了充分条件，是IB高水平学生探索的关键工具。

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10. Population Models: The Lotka–Volterra System | 种群模型：Lotka–Volterra 系统

The classic Lotka–Volterra model for a predator (y) and prey (x) takes the form dx/dt = ax − bxy, dy/dt = −cy + dxy with a, b, c, d > 0. The equilibrium points are (0,0) and (c/d, a/b). The Jacobian at (0,0) is [[a, 0], [0, −c]], giving a saddle point. At the interior equilibrium, J = [[0, −bc/d], [ad/b, 0]]; the trace is 0 and the determinant is ac > 0, so the linearisation predicts a centre. The nonlinear system indeed possesses a family of closed orbits around the interior equilibrium, corresponding to periodic population oscillations whose amplitude depends on initial conditions.

经典的 Lotka–Volterra 捕食者 (y) – 猎物 (x) 模型形式为 dx/dt = ax − bxy, dy/dt = −cy + dxy，其中 a, b, c, d > 0。平衡点为 (0,0) 和 (c/d, a/b)。在 (0,0) 处的雅可比矩阵为 [[a, 0], [0, −c]]，给出鞍点。在内部平衡点处，J = [[0, −bc/d], [ad/b, 0]]；迹为 0，行列式为 ac > 0，因此线性化预测为中心点。该非线性系统确实在内部平衡点周围拥有一族闭合轨道，对应于周期性的种群振荡，其振幅取决于初始条件。

In a modified version, such as the competition or cooperation models, the signs of the interaction terms change, leading to nodes, saddles, or spirals, which can be analysed using the phase plane techniques discussed above.

在修改版本中，例如竞争或合作模型，相互作用项的符号改变，可能导致结点、鞍点或螺旋，这些都可以用上述相平面技术进行分析。

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11. Sketching Phase Portraits from Systems | 从系统绘制相图

A systematic approach to sketching a phase portrait involves several steps: (i) Find all equilibrium points by solving f = g = 0. (ii) Compute the Jacobian and classify each equilibrium using eigenvalues. (iii) Draw the nullclines and indicate the direction of the vector field on them. (iv) Determine the signs of dx/dt and dy/dt in each region bounded by nullclines to infer the quadrant direction of the flow arrows. (v) Sketch the stable and unstable manifolds of saddle points, as they often form boundaries of basins. (vi) Add a few typical trajectories that start in different regions to capture the global structure. For IB assessments, one must be able to interpret given phase portraits and match them to systems of ODEs.

绘制相图的系统方法包括以下步骤：(i) 通过解 f = g = 0 求出所有平衡点。(ii) 计算雅可比矩阵并使用特征值对每个平衡点进行分类。(iii) 画出零等倾线并在其上标明向量场的方向。(iv) 确定零等倾线所划分的每个区域内 dx/dt 和 dy/dt 的符号，以推断流动箭头的象限方向。(v) 绘制鞍点的稳定流形和不稳定流形，因为它们通常形成吸引盆的边界。(vi) 添加几条从不同区域出发的典型轨迹，以捕捉全局结构。对于IB评估，学生必须能够解读给定的相图并将其与常微分方程组匹配。

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12. Applications and Concluding Remarks | 应用与结语

Phase plane analysis is not merely an abstract mathematical exercise. It is applied in physics (simple and damped pendulums converted into first-order systems), chemistry (reaction kinetics), biology (epidemiology), and economics (competing firms). Through linearisation and qualitative reasoning, one can predict long-term behaviour, sensitivity to initial conditions, and the effect of parameter changes. In the IB syllabus, this topic strengthens the link between calculus and

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