Linearized Stability

Linearized Stability analyses the stability of a one-dimensional dynamic system linearly approximated around the equilibrium points. The study of linearized stability provides information about how quickly nearby trajectories converge to or diverge from the points of equilibrium or fixed points. Before understanding linearized stability, we shall first explore the notion of stability.

Learn about Trajectories.

Stability of an Equilibrium Point

Now consider a trough filled with some liquid of density d, as shown in the figure below.

As a ball is dropped inside it, we find three positions of equilibrium, where if x represents the position of the ball at any instant t, at these points dx/dt = 0. Suppose small perturbations are introduced at points 1, 2, and 3. In that case, it is evident that even after such perturbations remain at position 1 or 3, but for point 2, the ball might slide into position 1 or 3 in whichever direction the perturbation is produced.

Thus, we notice:

• Positions 1 and 3 are characterised by the fact that small perturbations about these positions cause the ball to return to the same situation.
• Position 2 is characterised by the fact that small perturbations about its positions do not cause the ball to return to the same situation.

Hence, we can say that positions 1 and 3 are stable equilibrium points, whereas position 2 is a point of unstable equilibrium.

Linearized Stability Analysis of Differential Equations

Now we shall discuss the stability of the one-dimensional differential equation given by

dx/dt = f(x) at fixed points (equilibrium points).

To analyse the linear stability, we expand the given function f(x) around the fixed point and use the linear approximation to determine the nature of the fixed points.

In other words, we linearise the expansion of f(x) around fixed points.

Let there be a one-dimensional dynamical system in which the following differential equation describes the position x of a particle,

dx/dt = f(x) ……(1)

Now suppose x_f be the fixed point of f, that is, f(x_f) = 0.

Consider a very small perturbation from the fixed point x_f given by 𝜀, such that 𝜀 = x – x_f.

With given x_f, we have to determine how a very small perturbation 𝜀 change with time t.

If the perturbations grow with time, then the particle’s position diverges from the fixed point; this implies x_f is an unstable point. However, if those perturbations decay with time, the particle’s position converges to the fixed point, which means x_f is a stable point.

Here,

Since x_f is constant, so dx_f/dt = 0. Then

Now, x = 𝜀 + x_f. So,

To perform linearization of the above differential equation, we take Taylor’s series expansion of f(𝜀 + x_f) and truncate that Taylor’s series expansion at the linear term.

We have,

Or,

Where O(𝜀²) represent all the term in the Taylor expansion where the power of (𝜀 – x_f + x_f ) is equal to or greater than 2.

Since 𝜀 is very small, the higher powers of 𝜀 can be neglected.

We get, d𝜀/dt ≈ f(x_f) + f’(x_f) 𝜀

But f(x_f) = 0, being a fixed point.

Therefore, we have d𝜀/dt = f’(x_f) 𝜀 ……(2)

The solution of the above differential equation provided that such linear approximation holds for 𝜀 sufficiently small.

𝜀(t) = 𝜀_o e ^{f’(xf) t} ………….(3)

Now, if f’(x_f) is greater than zero, then 𝜀 will increase with time, so in regions close to fixed point x_f, the solution will tend to diverge, and as a result, x_f will be an unstable fixed point.

When f’(x_f) is less than zero, the 𝜀 will decrease with time; as a result, trajectories will converge to x_f, and x_f will be a stable fixed point.

How to Find the Nature of Equilibrium Points

For the given one-dimensional differential equation dx/dt = f(x):

• Find the equilibrium points or fixed points by equating f(x) to zero.
• Differentiate f(x) with respect to x.
• Determine whether f’(x_f) is positive or negative.
• If f’(x_f) > 0, then x_f is unstable point. If f’(x_f) < 0, then x_f is stable point.
• If f’(x) = 0, then x_f is a neutral point or the particle is at rest. In some cases, sign of higher order derivatives are considered.

Related Articles

• Differential Equations
• Taylor’s Series
• Ordinary Differential Equations
• Linear Differential Equations

Solved Examples of Linearized Stability

Example 1:

Find the nature of the fixed points of dx/dt = cos x.

Solution:

For the fixed points, cos x = 0

Therefore, x_f = (2n + 1) 𝜋/2 for n ∈ N

Now, f’(x) = – sin x

f’(x_f) = – 1 < 0; for x_f = (4k + 1) 𝜋/2 for k ∈ N

That is, points 𝜋/2, 5𝜋/2, 9𝜋/2, … are stable points.

And f’(x_f) = 1 > 0; for x_f = (4k + 1) 𝜋/2 for k ∈ N

That is, points –𝜋/2, 3𝜋/2, 7𝜋/2, … are unstable points.

Example 2:

Find the nature of the fixed points of dx/dt = x – x².

Solution:

For the fixed points, x – x² = 0

Therefore, x_f = 0 and 1

Now, f’(x) = 1 – 2x

f’(0) = 1 > 0, x_f = 0 is an unstable point

f’(1) = – 1 < 0, x_f = 1 is a stable point.