Linearization near Equilibria

The Central Idea

A nonlinear system may be complicated globally but nearly linear when viewed sufficiently close to an equilibrium.

The strategy is:

We then study the linear system using matrices, eigenvalues, and eigenvectors.

Why Linearization Works

The idea comes from linear approximation in calculus.

For a differentiable function \(f(x)\), near \(x=a\),

The derivative provides the best first-order approximation to the function near the reference point.

Dynamical-systems linearization applies the same idea to evolution laws.

One-Dimensional Continuous Systems

Consider

with equilibrium

Near \(x^\ast\),

Since

we obtain

Introduce a Perturbation Variable

Let

Then

Because \(x^\ast\) is constant,

The local dynamics become

The Local Linear Model

The linearized equation

has solution

Therefore:

• if \(f'(x^\ast)<0\), perturbations decay;
• if \(f'(x^\ast)>0\), perturbations grow;
• if \(f'(x^\ast)=0\), the first-order model is inconclusive.

Example: Linearizing a Nonlinear Scalar System

Consider

The equilibria satisfy

Factor:

Therefore,

Linearization at \(x^\ast=0\)

Let

Then

At

we have

Thus the local perturbation equation is

Perturbations grow, so the equilibrium is unstable.

Linearization at \(x^\ast=1\)

At

we obtain

The linearized equation is

Since perturbations decay exponentially, \(x^\ast=1\) is locally asymptotically stable.

Linearization at \(x^\ast=-1\)

At

we again have

Therefore,

The equilibrium \(x^\ast=-1\) is also locally asymptotically stable.

Discrete Linearization

For a discrete system

with fixed point

write

Then

The Discrete Local Multiplier

The quantity

is the local multiplier.

Iterating the linearized perturbation gives

Therefore:

• if \(|f'(x^\ast)|<1\), perturbations shrink;
• if \(|f'(x^\ast)|>1\), perturbations grow;
• if \(|f'(x^\ast)|=1\), linearization alone is inconclusive.

Example: Discrete Nonlinear Map

Consider

The fixed points satisfy

so

Since

the local multipliers are easy to evaluate.

Classifying the Discrete Fixed Points

At

we have

Since

the fixed point is locally asymptotically stable.

At

we have

Therefore \(x^\ast=1\) is unstable.

Multivariable Linear Approximation

Now consider a system

where

Near an equilibrium \(\mathbf{x}^\ast\),

Since

only the derivative matrix remains.

The Jacobian Matrix

The derivative matrix of a vector-valued function is called the Jacobian matrix.

For

the Jacobian is

The Linearized Vector System

Define the perturbation

Near the equilibrium,

The nonlinear system has therefore been replaced locally by a linear system whose matrix is the Jacobian evaluated at the equilibrium.

Why the Jacobian Matters

The Jacobian records how each component of the vector field changes with respect to each state variable.

Near an equilibrium, those partial derivatives determine the first-order response to small perturbations.

Eigenvalues and eigenvectors of the Jacobian then reveal local rates, directions, contraction, expansion, and rotation.

Example: Computing a Jacobian

Consider

Define

and

We compute the four required partial derivatives.

Jacobian of the Example

The partial derivatives are

Therefore,

Find the Equilibria First

Linearization is performed around an equilibrium, so the equilibrium must be found before evaluating the Jacobian.

For

equilibrium requires

and

Equilibria of the Example

From

either

or

Combining with

gives the equilibria

and

Linearization at \((0,0)\)

Evaluate

at

We obtain

The eigenvalues are

Interpretation at \((0,0)\)

One eigenvalue is positive and one is negative.

Therefore the linearized system has one expanding direction and one contracting direction.

The equilibrium is locally a

Because the eigenvalues have nonzero real parts, the nonlinear system has the same basic local saddle character.

Linearization at \((1,0)\)

Evaluate the Jacobian at

We obtain

Its eigenvalues are

The zero eigenvalue means the standard first-order classification is inconclusive.

Hyperbolic Equilibria

A continuous-time equilibrium is called hyperbolic when no eigenvalue of the Jacobian has zero real part.

Thus,

is the hyperbolicity condition for a continuous system.

Why Hyperbolicity Is Important

At a hyperbolic equilibrium, the local nonlinear dynamics generally have the same qualitative structure as the linearized system.

Nodes remain node-like, saddles remain saddle-like, and spiraling attraction or repulsion is preserved locally.

This is one of the central reasons linearization is so powerful.

Hartman-Grobman Idea

The Hartman-Grobman theorem formalizes the local relationship between a nonlinear system and its linearization near a hyperbolic equilibrium.

Informally,

The trajectories need not be identical, but their local qualitative organization is preserved.

Nonhyperbolic Equilibria

An equilibrium is nonhyperbolic when at least one continuous-time eigenvalue has zero real part.

Examples include

and

In such cases, nonlinear terms that were discarded during linearization may determine the actual behavior.

Example: Linearization Can Be Inconclusive

Consider

The equilibrium is

Since

the linearized equation is

This approximation does not reveal the true asymptotic stability of the nonlinear system.

Why Higher-Order Terms Matter

Near \(x=0\),

is smaller than a typical linear term, but if the linear term vanishes, the cubic term becomes the leading contribution.

Thus a term that normally looks negligible can become decisive when first-order information disappears.

Continuous Linearization Classification

For a two-dimensional continuous system with Jacobian eigenvalues \(\lambda_1,\lambda_2\):

• both real parts negative \(\Rightarrow\) local asymptotic stability;
• at least one real part positive \(\Rightarrow\) instability;
• opposite-sign real eigenvalues \(\Rightarrow\) saddle;
• complex eigenvalues with negative real part \(\Rightarrow\) stable spiral;
• complex eigenvalues with positive real part \(\Rightarrow\) unstable spiral;
• zero real parts \(\Rightarrow\) linearization may be inconclusive.

Discrete Multivariable Linearization

Consider a discrete nonlinear system

Let \(\mathbf{x}^\ast\) satisfy

With

the local approximation is

Discrete Eigenvalue Criterion

For the linearized discrete system,

stability is controlled by eigenvalue magnitudes.

• If every \(|\lambda_i|<1\), perturbations decay.
• If some \(|\lambda_i|>1\), perturbations grow.
• If some \(|\lambda_i|=1\), first-order analysis may be inconclusive.

Hyperbolicity in Discrete Time

A discrete fixed point is hyperbolic when no eigenvalue of the Jacobian lies on the unit circle.

The unit circle is therefore the critical stability boundary for discrete linearization.

Stable and Unstable Subspaces

Eigenvectors associated with contracting eigenvalues span locally stable directions.

Eigenvectors associated with expanding eigenvalues span locally unstable directions.

These linear subspaces provide first approximations to stable and unstable manifolds of nonlinear systems.

Example: Continuous Saddle Directions

Suppose the Jacobian at an equilibrium is

The eigenvalue

corresponds to contraction, while

corresponds to expansion.

The associated eigenvectors identify the stable and unstable local directions.

Example: Spiral Linearization

Suppose

The eigenvalues are

The negative real part causes contraction.

The imaginary part produces rotation.

Therefore the local phase portrait is a stable spiral.

Trace and Determinant

For a two-dimensional matrix

define the trace

and determinant

These quantities help classify two-dimensional linearizations.

Characteristic Equation

The eigenvalues satisfy

Therefore,

The discriminant

determines whether the eigenvalues are real or complex.

Trace-Determinant Classification

For a continuous two-dimensional linearization:

• \(\Delta<0\) indicates a saddle;
• \(\Delta>0\) and \(\tau<0\) often indicate local attraction;
• \(\Delta>0\) and \(\tau>0\) often indicate local repulsion;
• \(\tau^2-4\Delta<0\) gives complex eigenvalues and rotational behavior.

Boundary cases require additional care.

Example: Trace and Determinant

Suppose

Then

and

The discriminant is

The eigenvalues are complex with negative real part, indicating a stable spiral.

Linearization Is Local

A linearized system describes behavior near the equilibrium.

It does not automatically describe the entire state space.

A nonlinear system may have:

• multiple equilibria;
• different basins of attraction;
• limit cycles;
• separatrices;
• large-amplitude nonlinear behavior;
• chaotic attractors.

None of these global structures is guaranteed to be captured by one local linearization.

Local versus Global Approximation

The approximation

becomes better as

Far from the equilibrium, higher-order nonlinear terms may dominate and produce very different behavior.

Linearization and Phase Portraits

Near a hyperbolic equilibrium, the eigenstructure of the Jacobian provides a local template for the nonlinear phase portrait.

It predicts whether nearby trajectories:

• approach directly;
• move away directly;
• spiral inward;
• spiral outward;
• approach in some directions and depart in others.

Linearization and Timescales

Eigenvalues also reveal local rates of motion.

A mode with

decays much faster than one with

Systems with very different eigenvalue magnitudes may contain multiple characteristic timescales.

Fast and Slow Directions

Suppose a stable linearization has eigenvalues

and

Motion associated with \(\lambda_2\) decays much more rapidly.

After a short transient, the slower \(\lambda_1\)-direction may dominate the visible trajectory.

Linearization and Parameter Changes

The Jacobian itself may depend on a parameter.

As that parameter changes, an eigenvalue may cross a stability boundary.

In continuous time, important boundaries include

In discrete time, the corresponding boundary is

Such crossings are often associated with bifurcations.

Example: Parameter-Dependent Scalar System

Consider

The equilibrium

exists for every \(\mu\).

Since

the linearization at zero is

Stability Changes at a Critical Parameter

For

the equilibrium is locally stable when

and unstable when

At

the linearization becomes nonhyperbolic.

This signals that a qualitative change may occur.

Linearization and Control

In control systems, nonlinear plant dynamics are often linearized near a desired operating point.

The resulting local model can be used to:

• design feedback controllers;
• estimate disturbance response;
• analyze local stability;
• identify dominant modes;
• determine local response speeds.

The controller must still be interpreted within the region where the linear approximation is valid.

Linearization in Mechanical Systems

Mechanical systems often contain nonlinear force laws.

Near an equilibrium configuration, those forces can frequently be approximated by their first derivatives.

This produces linear oscillators, damped systems, or coupled matrix models that approximate small motions.

Small-Angle Pendulum as a Familiar Example

The nonlinear pendulum equation is

Near

we use

The linearized equation becomes

This approximation is accurate for sufficiently small angular displacements.

Common Error: Linearizing Before Finding the Equilibrium

The Jacobian can be computed anywhere, but local equilibrium classification requires evaluating it at an equilibrium.

The correct workflow is:

Common Error: Forgetting to Shift the Equilibrium

If an equilibrium is

the natural perturbation is

Linearization describes deviations from the equilibrium, not necessarily the original coordinate itself.

Common Error: Treating Linearization as Exact

A nonlinear system and its linearization are generally not identical.

The linearization neglects second-order and higher terms.

It should therefore be interpreted as a local approximation, not an exact global replacement.

Common Error: Trusting a Nonhyperbolic Linearization Too Far

If a continuous Jacobian has an eigenvalue with zero real part, or a discrete Jacobian has an eigenvalue on the unit circle, first-order classification may fail.

Higher-order nonlinear terms may decide stability, periodic behavior, or bifurcation structure.

Common Error: Using Continuous Eigenvalue Rules for Discrete Systems

In continuous time, stability is linked to negative eigenvalue real parts.

In discrete time, stability is linked to eigenvalue magnitudes less than \(1\).

Thus

is stable for the scalar continuous equation

but unstable for the scalar discrete update

Common Error: Ignoring the Model Domain

A locally valid mathematical trajectory may extend into states that are not meaningful for the application.

Population, concentration, financial, and physical models often impose constraints such as

Linearized conclusions should be interpreted within the admissible state space.

A Reliable Procedure for One-Dimensional Linearization

1. Write the dynamical equation.
2. Find all equilibria.
3. Choose one equilibrium \(x^\ast\).
4. Compute \(f'(x)\).
5. Evaluate \(f'(x^\ast)\).
6. Introduce \(u=x-x^\ast\).
7. Write the local linear equation \(u'=f'(x^\ast)u\) or \(u_{n+1}=f'(x^\ast)u_n\).
8. Classify the local behavior.
9. If the linear coefficient lies on a critical boundary, use higher-order analysis.

A Reliable Procedure for Continuous Vector Systems

1. Write \(\mathbf{x}'=\mathbf{F}(\mathbf{x})\).
2. Find every equilibrium \(\mathbf{x}^\ast\).
3. Compute the Jacobian \(J(\mathbf{x})\).
4. Evaluate \(J(\mathbf{x}^\ast)\).
5. Compute its eigenvalues and, when useful, eigenvectors.
6. Determine whether the equilibrium is hyperbolic.
7. Classify the local linear phase portrait.
8. Transfer that qualitative classification to the nonlinear system when justified.
9. Use nonlinear methods when the equilibrium is nonhyperbolic.

A Reliable Procedure for Discrete Vector Systems

1. Write \(\mathbf{x}_{n+1}=\mathbf{F}(\mathbf{x}_n)\).
2. Find fixed points from \(\mathbf{F}(\mathbf{x}^\ast)=\mathbf{x}^\ast\).
3. Compute the Jacobian of \(\mathbf{F}\).
4. Evaluate it at the fixed point.
5. Compute the eigenvalues.
6. Compare every eigenvalue magnitude with \(1\).
7. Classify contracting and expanding directions.
8. Use additional analysis when eigenvalues lie on the unit circle.

Practice 1: Scalar Linearization

Consider

Find the equilibria and compute \(f'(x^\ast)\) at each one.

Practice 2: Classify the Scalar Equilibria

Using

classify the equilibria found in Practice 1.

Practice 3: Discrete Linearization

Consider

Show that \(x^\ast=0\) is a fixed point and determine its local multiplier.

Practice 4: Compute a Jacobian

For

compute the Jacobian matrix.

Practice 5: Evaluate a Jacobian

Using the Jacobian from Practice 4, evaluate it at

Practice 6: Classify from Eigenvalues

A continuous system has Jacobian eigenvalues

Classify the local equilibrium.

Practice 7: Saddle Linearization

A continuous system has Jacobian eigenvalues

Classify the equilibrium.

Practice 8: Spiral Linearization

A continuous system has eigenvalues

What local phase portrait should be expected?

Practice 9: Discrete Eigenvalues

A discrete fixed point has Jacobian eigenvalues

Classify the fixed point locally.

Practice 10: Inconclusive Linearization

Consider

Explain why linearization at

is inconclusive, and use sign analysis to determine the true local stability.

Practice Answers

Practice 1

Let

The equilibria are

and

Since

we obtain

and

Practice 2

At

the derivative is positive:

Therefore \(0\) is unstable.

At

we have

Therefore \(2\) is locally asymptotically stable.

Practice 3

Let

Since

the origin is a fixed point.

Also,

Thus

Because

the fixed point is locally asymptotically stable.

Practice 4

Let

and

Then

Therefore,

Practice 5

At

the Jacobian becomes

Practice 6

Both eigenvalues are real and negative.

Therefore the equilibrium is locally asymptotically stable and has node-like behavior.

Practice 7

One eigenvalue is negative and one is positive.

Therefore,

Practice 8

The eigenvalues have negative real part and nonzero imaginary part.

Therefore,

Practice 9

Both eigenvalues satisfy

Therefore,

The negative eigenvalue corresponds to alternating behavior in its associated direction.

Practice 10

For

we have

Therefore,

so the first-order linearization is

and is inconclusive.

If \(x>0\), then

so the state moves farther right.

If \(x<0\), then

so the state moves farther left.

Therefore,

What to Remember

Linearization replaces a nonlinear system by its first-order approximation near an equilibrium.

For a scalar continuous system,

For a continuous vector system,

For a discrete system,

• the Jacobian contains the first partial derivatives of the evolution law;
• eigenvalues determine local contraction, expansion, and rotation;
• hyperbolic equilibria are especially well described by linearization;
• continuous stability depends on eigenvalue real parts;
• discrete stability depends on eigenvalue magnitudes;
• nonhyperbolic cases require additional nonlinear analysis;
• linearization is local, not a global replacement for the nonlinear system.

The method works because sufficiently close to an equilibrium, derivatives often reveal the dominant structure of the dynamics.

Where We Go Next

Linearization allows us to classify the local behavior of an equilibrium for a fixed set of parameter values.

But equilibria and their eigenvalues can change as parameters vary. A stable equilibrium can become unstable, new equilibria can appear, existing ones can disappear, and periodic behavior can emerge.

These qualitative transitions are called bifurcations.

The next lesson, Bifurcations, studies how changes in parameters reorganize the structure and long-term behavior of dynamical systems.