advanced36 min studyLesson 142 in Physics

What linear stability can and cannot prove

Use Jacobians, hyperbolicity, and Lyapunov functions to separate local decay, nonlinear stability, and the basin of attraction.

Before you begin

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The question for today

What you’ll understand

State the local assumptions behind linearization.
Identify nonhyperbolic cases where a Jacobian is inconclusive.
Use a scalar Lyapunov function to establish a region of attraction.

Linearization is a microscope: it enlarges a neighborhood of one state and discards higher powers of displacement. That is often exactly what an engineer needs after a small disturbance. It becomes misleading when its field of view is forgotten. A locally stable system may fail after a larger kick, while a zero eigenvalue can conceal either restoration or escape. Stability reasoning begins by stating which perturbations and which time behavior are being claimed.

A Taylor approximation with a domain

Shift a fixed point to the origin and call the displacement $\boldsymbol\xi$ . For a sufficiently smooth vector field, the neglected remainder is quadratic in displacement. The linearized equation therefore gives the first response to a sufficiently small perturbation. If every Jacobian eigenvalue has negative real part, nonlinear solutions starting sufficiently close decay to the fixed point. At least one positive-real-part eigenvalue makes the equilibrium unstable. These are local statements about the original autonomous differential equation.

A hyperbolic equilibrium has no eigenvalues on the imaginary axis. Its nearby orbit geometry is related to that of the linear flow, but the relation does not promise identical speeds or an exact trajectory formula. Nonnormal matrices can also temporarily amplify some perturbations even when all eigenvalues decay asymptotically. A physical tolerance may be exceeded during that transient. Asking only for eigenvalue signs can therefore miss an important finite-time design constraint.

\dot{\boldsymbol\xi}=J\boldsymbol\xi+O(\|\boldsymbol\xi\|^2),\qquad J=D\mathbf f(\mathbf z_*)

The quadratic remainder assumes sufficient differentiability near the equilibrium. Negative real parts yield local exponential attraction, with constants and neighborhood depending on the full system.↔ Scroll wide equations horizontally.

Zero linear information is still information

Compare $\dot x=-x^3$ and $\dot x=x^3$ . Both have derivative zero at the origin. Nevertheless the first points inward on both sides and is asymptotically stable, whereas the second points outward and is unstable. The Jacobian did not give a wrong answer; it gave no decisive answer. Leading nonlinear terms, center-manifold reductions, or an independent monotonic quantity must settle a nonhyperbolic case. Never label a nonlinear equilibrium a center solely from imaginary eigenvalues.

Distinguish remaining close from approaching. Lyapunov stability says a sufficiently small initial disturbance stays within a chosen neighborhood for all future times for which the solution exists. Asymptotic stability additionally requires convergence. The undamped ideal oscillator stays near the origin but does not approach it; a damped oscillator approaches. Neither definition automatically gives global existence for an arbitrary polynomial vector field, so an argument that traps trajectories in a bounded region also does useful existence work.

\dot x=-x^3\quad\Longrightarrow\quad x(t)=\frac{x_0}{\sqrt{1+2x_0^2t}}

This scalar solution approaches zero algebraically, not exponentially. Its linearization is zero, illustrating that nonhyperbolic stability may require nonlinear analysis and have a slower decay law.↔ Scroll wide equations horizontally.

An energy-like function can enlarge the argument

A Lyapunov function is positive away from equilibrium and decreases along trajectories in a stated domain. It need not be the system's mechanical energy. For a smooth positive-definite function $V$ , calculate its derivative using the actual vector field rather than a guessed damping law. A strictly negative derivative away from the equilibrium supports asymptotic attraction when the usual local regularity assumptions hold. A merely nonpositive derivative generally proves stability first; convergence requires further reasoning about the set where that derivative vanishes.

Sublevel sets provide a practical bridge from a local result to a certified region. If a compact sublevel set lies inside a domain where the derivative is negative, its boundary cannot be crossed outward. Initial states inside stay inside and tend toward the equilibrium under the appropriate strict-decay conditions. The largest certified set found by one trial function need not equal the true basin. Conversely, calling attraction global requires covering the full state domain and ruling out escape, not just drawing a large ellipse.

Work it through

Stable near zero, unsafe outside

Analyze $\dot x=-x+x^3$ near zero and find the attraction basin of zero on the real line.

The fixed points solve $x(x^2-1)=0$ : they are $-1,0,1$ . The derivative at zero is $-1$ , giving local exponential attraction. At either outer equilibrium the derivative is $2$ , giving instability.
Choose $V=x^2/2$ . Along the nonlinear flow, $\dot V=x(-x+x^3)=-x^2(1-x^2)$ , which is strictly negative for $0<|x|<1$ . Every smaller closed interval around zero is invariant.
For $x>1$ the velocity is positive; for $x<-1$ it is negative. These states move away, and $x=\pm1$ remain fixed. Therefore exactly the interval $(-1,1)$ tends to zero. The outer solutions can escape to infinity in finite time.

The result

The equilibrium is locally asymptotically stable, with basin (−1,1), and is not globally attracting. Linearizing at zero alone could not locate the two basin boundaries.

The living notebook

Explore the model

Interactive lab

Interactive lab · optimization

A discrete stability microscope

Set initial x₀ small and compare step sizes η below and above two. Predict whether the gradient-descent multiplier 1−η contracts or expands before stepping. This is exactly the discrete quadratic iteration xₙ=(1−η)ⁿx₀, not the nonlinear differential equation or its attraction basin.

ObjectiveDiscrete iterate

Position x0: 2
Objective x²/2: 2
Iteration behavior: Stable step

Step size η0.3Initial position x₀2

n = 0 · t = 0 s

Gradient descent on f(x) = x²/2: xₙ = (1 − η)ⁿx₀. Playback advances two discrete iterations per second. Values outside |x| ≤ 4 are explicitly reported. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA stability claim belongs to its actual evolution law; discrete multipliers and continuous eigenvalues use different criteria.

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Read the checkpoint explanation

Zero linear growth leaves nonlinear terms decisive. The equations ẋ = −x³ and ẋ = x³ share the same zero Jacobian but have opposite stability. Linearization is inconclusive, rather than a proof of neutrality.

Your turn

Try it yourself

For $\dot x=-x^3$ , verify that $V=x^2/2$ decreases and explain whether the decay is exponential.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Compare the derivative of V with the exact scalar solution above.

Compare with the solution

The derivative is $\dot V=-x^4<0$ for nonzero x. The exact solution exists for all forward times and tends to zero for any finite initial value, so this scalar equilibrium is globally asymptotically stable. Its magnitude behaves as $1/\sqrt{2t}$ at late times for nonzero initial data. That algebraic tail cannot be bounded by one fixed exponentially decaying envelope for all late times, so exponential stability does not follow.

Keep the idea

Take it with you

Linearization proves local attraction for strictly negative real parts, not an unrestricted basin.
Nonhyperbolic equilibria require nonlinear terms or independent stability tools.
A decreasing Lyapunov function must be paired with its domain and the precise convergence claim.

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