Lyapunov Functions, Invariant Sets, and Long-Time Behavior

How energy-like scalar functions, invariant regions, and invariance principles let us prove nonlinear stability and reason about what trajectories do for long times.
Modified

April 26, 2026

Keywords

Lyapunov function, invariant set, asymptotic stability, LaSalle, long-time behavior

1 Role

This is the fifth page of the ODEs and Dynamical Systems module.

Its job is to go one step beyond local linearization:

use a scalar certificate to trap trajectories, prove stability, and reason about what can happen as time goes to infinity

2 First-Pass Promise

Read this page after Phase Portraits, Equilibria, and Local Stability.

If you stop here, you should still understand:

  • why Lyapunov functions are energy-like stability certificates
  • what a positively invariant set is
  • why \dot V \le 0 is useful even when it is not strictly negative everywhere
  • how invariant-set reasoning helps explain long-time behavior

3 Why It Matters

Linearization is the first local tool near an equilibrium, but it has limits.

It can fail to decide borderline cases, and it does not automatically give global information.

What we often want instead is a quantity that behaves monotonically along trajectories.

If a scalar function V(y) decreases over time, then it becomes hard for trajectories to:

  • run away to infinity inside a bounded sublevel set
  • keep crossing level sets in arbitrary ways
  • oscillate freely unless some invariant structure allows it

This is why Lyapunov functions matter. They turn stability into a question about the sign of one derivative along the flow.

That viewpoint is central in:

  • nonlinear control
  • mechanical and dissipative systems
  • gradient-flow and optimization dynamics
  • stability proofs for algorithms and learning dynamics
  • long-time behavior arguments when explicit solutions are unavailable

4 Prerequisite Recall

  • phase portraits and equilibria organize qualitative behavior in autonomous systems
  • Jacobian linearization gives a first local test near hyperbolic equilibria
  • level sets of a scalar function are geometric regions in state space
  • uniqueness prevents trajectories from crossing through the same state

5 Intuition

5.1 A Lyapunov Function Is An Energy-Like Quantity

For an autonomous system

\[ y' = f(y), \]

a Lyapunov function is a scalar V(y) chosen so that V is small near the equilibrium and does not increase along trajectories.

The first-pass picture is:

V measures stored energy or distance-like potential, and the dynamics push the state downhill

5.2 Decreasing Level Sets Trap Motion

If V decreases along trajectories, then a solution that starts inside a sublevel set

\[ \{y: V(y)\le c\} \]

cannot climb back out through the boundary where V=c.

That is why invariant sets appear immediately in Lyapunov arguments.

5.3 Strict Decrease Is Strong, But Nonincrease Is Already Useful

If \dot V < 0 away from the equilibrium, we usually get asymptotic stability quite directly.

But often we only have

\[ \dot V \le 0. \]

That still tells us the trajectory cannot keep increasing V, but it does not by itself say where the trajectory finally goes.

5.4 Invariant Sets Explain The Remaining Freedom

If \dot V = 0 on a whole set, then long-time behavior can only live inside the part of that set that is actually invariant under the dynamics.

That is the first-pass idea behind LaSalle-type reasoning:

find the place where the Lyapunov decrease stops, then ask which motions can really stay there forever

6 Formal Core

Definition 1 (Definition: Lyapunov Function Near An Equilibrium) Let y_* be an equilibrium of

\[ y' = f(y). \]

A first-pass Lyapunov function is a scalar function V(y) such that:

  • V(y_*)=0
  • V(y)>0 for nearby y\ne y_*
  • the derivative of V along trajectories is nonpositive near y_*

The derivative along trajectories is

\[ \dot V(y)=\nabla V(y)\cdot f(y). \]

Definition 2 (Definition: Positively Invariant Set) A set S is positively invariant if every trajectory that starts in S stays in S for all future time for which the solution exists.

Sublevel sets of a decreasing Lyapunov function are the main first-pass examples.

Theorem 1 (Theorem Idea: Lyapunov’s Direct Method) If there is a Lyapunov function V near an equilibrium y_* with

\[ \dot V(y)\le 0, \]

then y_* is stable.

If in addition

\[ \dot V(y)<0 \]

for nearby y\ne y_*, then y_* is asymptotically stable.

At first pass, the key lesson is that stability can be proved without solving the ODE explicitly.

Theorem 2 (Theorem Idea: LaSalle-Type Invariance Principle) If V is nonincreasing on a bounded positively invariant region, then long-time behavior is pushed toward the largest invariant subset of the set where

\[ \dot V = 0. \]

This is the main bridge from Lyapunov decrease to long-time behavior.

7 A Small Worked Example

Consider the nonlinear system

\[ \begin{aligned} x' &= y,\\ y' &= -x - y - x^3. \end{aligned} \]

The origin (0,0) is an equilibrium.

7.1 Step 1: Choose A Natural Lyapunov Function

Take

\[ V(x,y)=\frac12 x^2+\frac12 y^2+\frac14 x^4. \]

This function is nonnegative and vanishes only at the origin.

It looks like an energy:

  • quadratic terms measure ordinary size
  • the quartic term matches the nonlinear restoring force x^3

7.2 Step 2: Differentiate Along Trajectories

Using the chain rule,

\[ \dot V = x x' + y y' + x^3 x'. \]

Substitute the dynamics:

\[ \dot V = x y + y(-x-y-x^3) + x^3 y = -y^2. \]

So

\[ \dot V \le 0. \]

7.3 Step 3: Read The Consequences

This tells us:

  • V never increases along trajectories
  • sublevel sets of V are positively invariant
  • the dynamics are dissipative rather than energy-preserving

But \dot V = 0 whenever y=0, not only at the origin, so strict Lyapunov decrease is not available.

7.4 Step 4: Use Invariant-Set Reasoning

Inside the set \{(x,y): \dot V = 0\} we have y=0.

To stay there for all future time, we also need

\[ y' = -x - x^3 = 0, \]

which forces x=0.

So the largest invariant subset of \{\dot V = 0\} is just the origin.

That is the first-pass LaSalle conclusion:

  • trajectories cannot wander forever inside a larger zero-dissipation set
  • the only state they can remain in forever is (0,0)

So the origin is asymptotically stable.

8 Computation Lens

Simulation can suggest that a trajectory spirals inward, but a Lyapunov argument explains why that behavior is structurally forced.

This matters computationally because:

  • numerics can hide or exaggerate long-time behavior
  • invariant regions tell us what qualitative behavior a solver should respect
  • dissipative systems are often judged by whether the numerical method preserves the expected decay pattern

This is why Time-Stepping for ODEs and Stability is an important companion page.

9 Application Lens

9.1 Nonlinear Control

Lyapunov functions are the standard language for proving that an equilibrium or tracking error is stable without solving the closed-loop dynamics.

9.2 Mechanics And Dissipation

Energy functions and damping terms naturally produce Lyapunov-style arguments for decay toward equilibrium.

9.3 Optimization And Learning Dynamics

Gradient flow, dissipative dynamics, and some convergence proofs are easiest to read through a decreasing scalar functional.

10 Stop Here For First Pass

If you can now explain:

  • what a Lyapunov function is trying to certify
  • what a positively invariant set is
  • why \dot V \le 0 already gives useful information
  • why long-time behavior is tied to the invariant part of \{\dot V=0\}

then this page has done its job.

11 Go Deeper

After this page, the next natural step is:

The strongest adjacent pages are:

12 Optional Deeper Reading After First Pass

The strongest current references connected to this page are:

13 Sources and Further Reading

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