Linearizing a nonlinear equation means replacing it with a simpler, straight-line approximation that behaves almost identically near a chosen point. The core idea is the same whether you’re working with a single equation or a full system of differential equations: pick a point, compute the slope (derivative) there, and use that slope to build a linear version. The approximation works well close to your chosen point and gets worse as you move away from it.
The Core Formula: First-Order Taylor Expansion
Every linearization technique traces back to the same mathematical foundation. For a function f(x), the linear approximation around a point x = a is:
f(x) ≈ f(a) + f'(a)(x − a)
This is the first-order Taylor polynomial. You evaluate the function at your chosen point a to get f(a), then add the derivative at that point multiplied by how far x is from a. The result is the equation of the tangent line at x = a, and it serves as your linearized model.
For a function of two variables, f(x, y), the same idea extends naturally. Around the point (a, b):
f(x, y) ≈ f(a, b) + fₓ(a, b)(x − a) + f_y(a, b)(y − b)
Here fₓ and f_y are the partial derivatives with respect to x and y. Instead of a tangent line, you get a tangent plane, but the principle is identical: hold the first-order terms, discard everything higher.
Choosing the Right Operating Point
The point you linearize around, often called the operating point or equilibrium point, determines everything about the quality of your approximation. In physical systems, the natural choice is an equilibrium: a state where the system isn’t changing. Mathematically, an equilibrium point satisfies f(x) = 0, meaning the system has settled to a steady state.
Choosing an equilibrium simplifies the math because the constant term drops out. If f(x*) = 0, then the linearization becomes just f'(x*)(x − x*), a clean proportional relationship between the deviation from equilibrium and the rate of change. If you linearize around a non-equilibrium point, the constant term remains and your approximation includes a steady offset that you need to carry through all subsequent calculations.
Some systems have multiple equilibrium points, and you may need to linearize around each one separately to understand the system’s behavior in different regimes.
The Classic Example: A Pendulum
The simple pendulum is the textbook case of linearization in action. The exact equation of motion for a pendulum involves sin(θ), where θ is the angle of swing:
d²θ/dt² = −(g/L) sin(θ)
This is nonlinear because of the sine term. But for small angles (roughly below 15 degrees), sin(θ) ≈ θ when θ is measured in radians. This is just the Taylor expansion of sin(θ) around θ = 0, keeping only the first-order term. The equation becomes:
d²θ/dt² = −(g/L) θ
That’s the equation for simple harmonic motion, which has a clean, exact solution. The linearized version tells you the period depends only on the length of the pendulum and gravity, not on the amplitude. This prediction holds well for small swings and gradually breaks down as the angle increases, exactly as you’d expect from a local approximation.
Linearizing Systems With the Jacobian Matrix
When you have a system of two or more nonlinear equations, each involving multiple variables, the single-variable derivative gets replaced by a matrix of partial derivatives called the Jacobian. Consider a system of two differential equations:
dx/dt = f(x, y)
dy/dt = g(x, y)
First, find the equilibrium points by solving f(x*, y*) = 0 and g(x*, y*) = 0 simultaneously. Then compute all four partial derivatives of f and g with respect to x and y, evaluated at the equilibrium. These form the Jacobian matrix:
J = [ fₓ(x*, y*) f_y(x*, y*) ]
[ gₓ(x*, y*) g_y(x*, y*) ]
Now define new variables that measure the deviation from equilibrium: u = x − x* and v = y − y*. The linearized system is:
du/dt = fₓ u + f_y v
dv/dt = gₓ u + g_y v
Or in compact matrix form: du/dt = Ju. This converts your coupled nonlinear system into a linear system you can solve with standard matrix techniques. The eigenvalues of J tell you whether the equilibrium is stable (solutions decay back toward it) or unstable (solutions diverge away), and whether the system oscillates or moves monotonically.
Step-by-Step Process
- Find equilibria: Set all right-hand sides to zero and solve for the state variables.
- Compute partial derivatives: Take every partial derivative of every equation with respect to every state variable.
- Evaluate at the equilibrium: Plug the equilibrium coordinates into each partial derivative.
- Assemble the Jacobian: Arrange these values into a matrix.
- Shift coordinates: Rewrite the system in terms of deviations from equilibrium.
For a system with n equations and n variables, the Jacobian is an n×n matrix. The process scales directly, though the algebra gets heavier.
Log Transforms: A Different Kind of Linearization
Not all linearization relies on Taylor series. When an equation follows an exponential or power-law relationship, taking logarithms can convert it into a linear form directly.
If your data follows an exponential relationship like y = e^(β₀ + β₁x), taking the natural log of both sides gives log(y) = β₀ + β₁x. That’s a straight line in log(y) versus x. The independent variable x has a multiplicative rather than additive effect on y: each unit increase in x multiplies y by e^β₁.
For power-law relationships like y = axᵇ, taking the log of both sides gives log(y) = log(a) + b·log(x). This is linear in log(y) versus log(x), and the slope b is the power-law exponent. In this “log-log” model, a 1% increase in x corresponds to roughly a b% increase in y.
These transforms are especially common in biology, economics, and data fitting. Unlike Taylor-based linearization, which only works locally, a log transform can capture the global relationship between variables, provided the underlying equation truly is exponential or power-law in form.
How Far the Approximation Reaches
A linearization is a local approximation, and it degrades as you move away from the operating point. The error for a first-order Taylor approximation is governed by the remainder term:
|R₁(x)| ≤ M · |x − a|² / 2
Here M is the maximum value of the second derivative between your point of interest and the operating point a. The error grows with the square of the distance from a, so doubling your distance from the operating point roughly quadruples the error. This is why the small-angle approximation for a pendulum works beautifully at 5 degrees but starts to visibly deviate by 20 or 30 degrees.
For linearized systems of differential equations, a result called the Hartman-Grobman theorem provides a formal guarantee. It states that near a “hyperbolic” equilibrium point (one where the Jacobian’s eigenvalues all have nonzero real parts), the linearized system captures the true qualitative behavior of the nonlinear system. Solutions that spiral inward in the linearized version also spiral inward in the real system, and unstable equilibria in the linearized version are genuinely unstable. The theorem breaks down at non-hyperbolic equilibria, where at least one eigenvalue has a zero real part. In those cases, the linearization alone can’t tell you whether the system is stable or not.
Software Tools for Complex Models
For real-world systems with dozens of interacting variables, hand-computing a Jacobian is impractical. Engineering software automates the process. MATLAB’s linmod function, for example, extracts a linear state-space model from a Simulink system model around a specified operating point. You provide the system name and the operating point values, and it returns the system matrices (commonly labeled A, B, C, D) that define the linearized input-output relationship. It can also return the result in transfer function form.
Python’s scientific computing ecosystem offers similar capabilities through libraries like SciPy, and symbolic math tools like SymPy can compute Jacobians analytically. These tools handle the bookkeeping of partial derivatives and matrix assembly, letting you focus on choosing good operating points and interpreting the results rather than grinding through algebra.