Nearly Linear Systems | 近似线性系统

📚 Nearly Linear Systems | 近似线性系统

In the study of differential equations and dynamical systems, many real-world phenomena are modelled by nonlinear systems. While linear systems have well-known analytical solutions and clear global behaviour, nonlinear systems can be far more complex. However, near an equilibrium point, a nonlinear system often behaves very much like a linear one. Such systems are called nearly linear systems, and the technique of linearisation allows us to understand their local stability and phase portrait without solving the full nonlinear equations. This article explores the theory, the Jacobian linearisation method, classification of equilibria, worked examples, and the limitations of the approach, all tailored for IB HL Mathematics students engaged with coupled differential equations and phase plane analysis.

在微分方程和动力系统的研究中，许多现实世界的现象都由非线性系统建模。线性系统具有已知的解析解和清晰的全局行为，而非线性系统则复杂得多。然而，在平衡点附近，非线性系统的行为往往与线性系统非常相似。这样的系统被称为近似线性系统，而线性化技术使我们无需完全求解非线性方程，即可了解其局部稳定性和相图。本文探讨其理论、雅可比线性化方法、平衡点分类、算例以及该方法的局限性，专门针对学习耦合微分方程和相平面分析的IB HL数学学生。

1. Introduction to Nonlinear Systems and Linearisation | 非线性系统与线性化导论

re generally nonlinear functions. Unlike linear systems with constant coefficients, nonlinear systems rarely admit simple closed-form solutions. However, the qualitative behaviour – such as whether trajectories spiral into a point, form closed orbits, or diverge – inear approximations, yielding a linear system whose solutions are much easier to analyse.

二维自治一阶微分方程组可以写为 dx/dt = f(x, y)，dy/dt = g(x, y)，其中 f 和 g 通常是非线性函数。与常系数线性系统不同，非线性系统很少有简单的闭式解。然而，轨迹是否螺旋进入一点、形成闭合轨道还是发散等定性行为，通常取决于平衡点的性质。线性化将平衡点附近的 f 和 g 替换为其线性近似，从而得到一个更容易分析的线性系统。

2. What is a Nearly Linear System? | 什么是近似线性系统？

inder tends to zero faster than (x² + y²)½. In mathematical terms, for an equilibrium (x₀, y₀), we set u = x − x₀, v = y − y₀, obtaining du/dt = a u + b v + R₁(u, v), dv/dt = c u + d v + R₂(u, v), where R₁, R₂ / √u²+v² ly linear if this holds and the matrix of coefficients [[a,b],[c,d]] has no eigenvalues with zero real part (i.e., the linearised system is hyperbolic).

如果通过适当的平移将平衡点移至原点后，非线性系统可以表示为一个线性部分和一个非线性余项之和，且该余项比到平衡点的距离更快地趋于零（即余项趋零的速度快于 (x² + y²)½），则称系统在该平衡点处是近似线性的。数学上，对于平衡点 (x₀, y₀)，令 u = x − x₀，v = y − y₀，得到 du/dt = a u + b v + R₁(u, v)，dv/dt = c u + d v + R₂(u, v)，其中当 (u,v) → (0,0) 时 R₁, R₂ / √u²+v² → 0。如果上述条件成立且系数矩阵 [[a,b],[c,d]] 没有实部为零的特征值（即线性化系统是双曲的），则该系统是近似线性的。

3. Equilibrium Points of Nonlinear Systems | 非线性系统的平衡点

ear system may have several isolated equilibria, each with its own local behaviour. To analyse the system globally, one must locate all equilibria and then examine their nature through linearisation. Typical examples include the origin in pendulum equations, coexistence and extinction points in population models, or rest positions in mechanical oscillators.

平衡点（或临界点）通过联立求解 f(x, y) = 0 和 g(x, y) = 0 得到。非线性系统可能存在多个孤立的平衡点，每个点具有各自的局部行为。为了进行全局分析，需要找出所有平衡点，再通过线性化研究其性质。典型的例子包括摆方程中的原点、种群模型中的共存点和灭绝点，或机械振荡器中的静止位置。

4. Linearisation Using the Jacobian Matrix | 利用雅可比矩阵进行线性化

valuated at the equilibrium. For a system dx/dt = f(x,y), dy/dt = g(x,y), the Jacobian is:

在平衡点 (x₀, y₀) 附近的线性近似由在该点计算的雅可比矩阵 J 构造。对于系统 dx/dt = f(x,y)，dy/dt = g(x,y)，雅可比矩阵为：

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

u + A₂₂ v. The eigenvalues of A determine the local stability and type of the equilibrium, provided the equilibrium is hyperbolic (no eigenvalue has zero real part).

令 A = J(x₀, y₀)。则在平移变量 (u, v) 下的线性化系统为 du/dt = A₁₁ u + A₁₂ v，dv/dt = A₂₁ u + A₂₂ v。只要平衡点是双曲的（没有特征值实部为零），A 的特征值就决定了该平衡点的局部稳定性和类型。

5. Classification of Equilibrium Points in Linear Systems | 线性系统中平衡点的分类

near system with trace τ = a + d and determinant Δ = ad − bc, the eigenvalues are λ = (τ ± √τ² − 4Δ)/2. The nature of the equilibrium at the origin is classified according to the signs and nature of the eigenvalues. The standard classification includes:

对于迹 τ = a + d 和行列式 Δ = ad − bc 的 2×2 线性系统，特征值为 λ = (τ ± √τ² − 4Δ)/2。原点的平衡点类型根据特征值的符号和性质进行分类。标准分类包括：

• Stable Node: both eigenvalues real and negative; trajectories approach the origin.
• Unstable Node: both eigenvalues real and positive; trajectories move away.
• Saddle point: one positive and one negative real eigenvalue; trajectories approach along one direction and leave along another.
• Stable Focus (spiral): complex eigenvalues with negative real part; trajectories spiral inwards.
• Unstable Focus (spiral): complex eigenvalues with positive real part; trajectories spiral outwards.
• Center: purely imaginary eigenvalues; closed orbits around the equilibrium (structurally unstable).

• 稳定结点：两个实特征值均为负；轨迹趋近原点。
• 不稳定结点：两个实特征值均为正；轨迹远离原点。
• 鞍点：一个正实特征值和一个负实特征值；轨迹沿某一方向趋近，而沿另一方向离开。
• 稳定焦点（螺旋）：复特征值且实部为负；轨迹向内螺旋。
• 不稳定焦点（螺旋）：复特征值且实部为正；轨迹向外螺旋。
• 中心：纯虚数特征值；围绕平衡点的闭合轨道（结构不稳定）。

These types can also be determined by the trace-determinant diagram, a useful tool in qualitative analysis.

这些类型也可以通过迹-行列式图来判断，这是定性分析中的一个有用工具。

6. The Hartman–Grobman Theorem and Local Behaviour | 哈特曼–格罗布曼定理与局部行为

The Hartman–Grobman theorem assures us that if a system is nearly linear at a hyperbolic equilibrium, then there exists a continuous transformation (a homeomorphism) that maps the trajectories of the nonlinear system onto those of its linearisation in a neighbourhood of the equilibrium. In other words, the phase portrait of a nonlinear system near a hyperbolic equilibrium is topologically equivalent to that of the linearised system. This justifies the use of linear stability analysis for nearly linear systems.

哈特曼–格罗布曼定理告诉我们，如果系统在双曲平衡点处是近似线性的，则存在一个连续变换（同胚），将非线性系统的轨迹映射为其线性化在平衡点邻域内的轨迹。换句话说，非线性系统在双曲平衡点附近的相图与其线性化系统是拓扑等价的。这为对近似线性系统使用线性稳定性分析提供了理论依据。

广告 / Sponsored

7. How Linearisation Approximates Nearly Linear Systems | 线性化如何近似近似线性系统

2) For each equilibrium, compute the Jacobian matrix and evaluate it at that point to obtain matrix A. (3) Find the eigenvalues of A. (4) Classify the nature of the linearised system at the origin. (5) Transfer the qualitative picture back to the original equilibrium, remembering a nonlinear system may have multiple equilibria and global bifurcations. The local behaviour – stability and type – is preserved if the eigenvalues do not lie on the imaginary axis.

应用线性化需遵循系统步骤：（1）通过求解 f = 0，g = 0 找到所有平衡点。（2）对每个平衡点计算雅可比矩阵并在该点求值得到矩阵 A。（3）求 A 的特征值。（4）对线性化系统在原点处的类型进行分类。（5）将定性图形转移回原平衡点，记住非线性系统可能有多个平衡点和全局分岔。若特征值不在虚轴上，则局部行为（稳定性和类型）得以保留。

8. Worked Example 1: Predator–Prey Model (Lotka–Volterra) | 案例1：捕食者-猎物模型（Lotka–Volterra）

Consider the classical Lotka– e coexistence point (x*, y*) = (c/d, a/b), J = [[0, −bc/d],[ad/b, 0]] with eigenvalues ± i√ac. These are purely imaginary, so the linearised system has a centre. However, the full nonlinear system possesses a family of closed orbits around this equilibrium, and the linearisation fails to be structurally stable; this is a borderline case where Hartman–Grobman does not apply because the eigenvalues lie on the imaginary axis. The system is nearly linear but not hyperbolic, so linearisation alone does not guarantee the nature of orbits.

考虑经典的 Lotka–Volterra 方程 dx/dt = x(a − by)，dy/dt = y(−c + dx)，其中 a, b, c, d > 0。平衡点为 (0,0) 和 (c/d, a/b)。在 (0,0) 处，J = [[a, 0],[0, −c]]，特征值为 a > 0 和 −c < 0，故原点是鞍点，意味着单一物种灭绝是不稳定的。在共存点 (x*, y*) = (c/d, a/b) 处，J = [[0, −bc/d],[ad/b, 0]]，特征值为 ± i√ac，是纯虚数，因此线性化系统具有中心。然而，完整的非线性系统围绕该平衡点存在一族闭合轨道，线性化在结构上不稳定；这是一个边缘情形，哈特曼–格罗布曼定理不适用，因为特征值位于虚轴上。该系统是近似线性的，但不是双曲的，因此单独使用线性化无法保证轨道的性质。

9. Worked Example 2: Competing Species | 案例2：竞争物种模型

re −1 and 1−β. If β > 1, then 1−β < 0, so both eigenvalues are negative → stable node, meaning species x dominates. If β < 1, then one eigenvalue is positive, giving a saddle; species y can invade. At the interior equilibrium, linearisation yields a 2×2 matrix, and by computing trace and determinant we can classify it as a node or focus, leading to coexistence or exclusion depending on parameters. The linearisation accurately predicts local behaviour because the eigenvalues are not purely imaginary.

一个简单的两个物种竞争模型为 dx/dt = x(1 − x − αy)，dy/dt = y(1 − y − βx)，其中 α, β > 0 代表种间竞争。平衡点：(0,0)，(1,0)，(0,1)，以及如果 α, β 条件允许，可能有一个内点。对于点 (1,0)，我们得到 J = [[−1, −α],[0, 1−β]]，特征值为 −1 和 1−β。若 β > 1，则 1−β < 0，两个特征值均为负 → 稳定结点，意味着物种 x 占优势。若 β < 1，则一个特征值为正，得到鞍点；物种 y 可以入侵。在内平衡点处，线性化得到一个 2×2 矩阵，通过计算迹和行列式可将其归类为结点或焦点，进而决定是共存还是排斥，具体取决于参数。线性化能准确预测局部行为，因为特征值不是纯虚数。

10. When Linearisation Fails: Non-Hyperbolic Equilibria | 线性化失效的情形：非双曲平衡点

es, the nonlinear system may exhibit a spiral, a centre, or a more complicated behaviour that is not captured by the linear terms. For example, the equation of a simple pendulum d²θ/dt² + (g/L) sinθ = 0, written as a system, has equilibria at (0,0) and (π,0). At (0,0) the linearised system has eigenvalues ± i√g/L (centre), and the true motion consists of closed orbits; at (π,0) the eigenvalues are ±√g/L (a saddle). Although linearisation correctly predicts the saddle, the centre prediction is only structurally stable because of energy conservation; small damping turns it into a focus, showing the sensitivity of non-hyperbolic points.

只有当平衡点是双曲的（即雅可比矩阵没有实部为零的特征值）时，线性化才能提供可靠的定性信息。如果特征值是纯虚数（中心）或零，泰勒展开中的高阶项将主导动力学行为。在这些情况下，非线性系统可能展现螺旋、中心或更复杂的行为，而这些行为无法由线性项捕捉。例如，单摆方程 d²θ/dt² + (g/L) sinθ = 0 写为系统后，平衡点在 (0,0) 和 (π,0)。在 (0,0) 处线性化系统的特征值为 ± i√g/L（中心），而真实运动由闭合轨道组成；在 (π,0) 处特征值为 ±√g/L（鞍点）。尽管线性化对鞍点的预测正确，但中心的预测仅是结构稳定的，因为有能量守恒；微小的阻尼就会将其变为焦点，显示了非双曲点的敏感性。

11. Step-by-Step Procedure for Analysing a Nearly Linear System | 分析近似线性系统的分步流程

To master problems on nearly linear systems, students should adopt a structured approach:

要掌握近似线性系统的问题，学生应采用结构化的方法：

• Identify the system: write it in standard autonomous first-order form.
• Determine all equilibrium points by setting f(x,y)=0 and g(x,y)=0.
• y − y₀ (conceptual step; in practice work directly with the Jacobian at the equilibrium).
• Compute the Jacobian matrix J(x,y) and evaluate A = J(x₀, y₀).
• Calculate eigenvalues λ of A, or use trace τ and determinant Δ to classify.
• Classify the equilibrium based on λ: node, saddle, spiral, or centre. Note stability (attracting if all Re(λ) < 0).
• Sketch the local phase portrait around the equilibrium.
• Combine information from all equilibria and use nullclines/vector fields to construct a global phase portrait.

• 识别系统：将其写为标准自治一阶形式。
• 通过令 f(x,y)=0 和 g(x,y)=0，确定所有平衡点。
• 对每个平衡点 (x₀, y₀)，通过 u = x − x₀, v = y − y₀ 平移至原点（概念步骤；实际中直接使用该点处的雅可比矩阵）。
• 计算雅可比矩阵 J(x,y) 并求值 A = J(x₀, y₀)。
• 计算 A 的特征值 λ，或使用迹 τ 和行列式 Δ 进行分类。
• 根据 λ 对平衡点分类：结点、鞍点、螺旋或中心。注意稳定性（如果所有 Re(λ) < 0 则为吸引子）。
• 绘制平衡点附近的局部相图。
• 结合所有平衡点的信息，并利用零倾线/向量场构建全局相图。

12. Summary and Key Takeaways | 总结与要点

Nearly linear systems form the bridge between nonlinear reality and linear tractability. By evaluating the Jacobian at equilibrium points, we obtain a linearised system whose qualitative behaviour locally mirrors the original, provided the equilibrium is hyperbolic. The classification of equilibria into nodes, saddles, spirals and centers gives deep insight into the dynamics of interacting populations, mechanical oscillators, chemical reactions, and more. IB students working with coupled differential equations and phase planes should become comfortable with linearisation and the Hartman–Grobman theorem’s message: 'locally, a nonlinear system looks like its linearisation.' However, always check eigenvalues; if they lie on the imaginary axis, the system is not hyperbolic, and higher-order analysis is required.

近似线性系统构成了非线性现实与线性可解性之间的桥梁。通过在平衡点处计算雅可比矩阵，我们得到一个线性化系统，只要平衡点是双曲的，其局部定性行为就与原系统相符。将平衡点分类为结点、鞍点、螺旋和中心，可以深入理解种群相互作用、机械振荡、化学反应等动力学问题。学习耦合微分方程和相平面的IB学生应熟练掌握线性化方法，并领会哈特曼–格罗布曼定理的要旨：“局部上，非线性系统看起来像其线性化”。然而，务必检查特征值；如果它们位于虚轴上，系统就不是双曲的，需要进行高阶分析。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）