Fixed Points and Stability — Mathematics Knowledge System

Overview

Fixed points (equilibria) are the simplest long-term behaviors in a dynamical system: states where the system remains forever at rest. Their stability determines whether nearby trajectories return to the equilibrium (stable), move away (unstable), or neither (neutrally stable). The key tool is linearization: near a fixed point, the nonlinear system is well-approximated by its Jacobian matrix, whose eigenvalues classify the fixed point as a node, spiral, saddle, or center. This classification, due to Poincare, is complete for hyperbolic fixed points (no eigenvalue with zero real part) and forms the foundation for understanding more complex dynamics.

Intuition

Think of a marble on a surface: at a valley bottom it is stable (small nudges roll it back), at a hilltop it is unstable (any nudge sends it away), and at a saddle point it is stable in one direction but unstable in another. For a differential equation, linearization replaces the curved surface with its tangent plane at the equilibrium, and the eigenvalues of the Jacobian read off the curvatures in each direction. A negative real part means 'curving downward' (stable), a positive real part means 'curving upward' (unstable), and imaginary parts introduce rotation (spiraling or oscillating). A fixed point is stable if every small perturbation decays; asymptotically stable if perturbations both stay small AND die out completely.

Formal Definition

Definition

Let x* be an equilibrium of the system x' = F(x), and let J = DF(x*) be the Jacobian matrix evaluated at x*.

F(x^*) = 0

Definition of fixed point

J = DF(x^*) = \left.\frac{\partial F_i}{\partial x_j}\right|_{x=x^*}

Jacobian at fixed point

x(t) = x^* + e^{Jt} \xi + O(|\xi|^2), \quad \xi = x(0) - x^*

Linearized flow near x*

\text{tr}(J) = \lambda_1 + \lambda_2, \quad \det(J) = \lambda_1 \lambda_2

Trace and determinant determine eigenvalues in 2D

Notation

Notation	Meaning
$x^*$	Fixed point / equilibrium
$J = DF(x^*)$	Jacobian matrix of F evaluated at x*
$\lambda_1, \lambda_2$	Eigenvalues of the Jacobian
$\tau = \text{tr}(J)$	Trace of Jacobian (sum of eigenvalues)
$\Delta = \det(J)$	Determinant of Jacobian (product of eigenvalues)

Theorems

Theorem 1: Theorem 1

If x^* is a hyperbolic fixed point of \dot{ x } = F(x) (i.e., all eigenvalues of DF(x^*) have nonzero real part), then there exists a homeomorphism h defined in a neighborhood of x^* that conjugates the nonlinear flow to the linear flow e^{ Jt }. In particular, the topological type of the fixed point is completely determined by the eigenvalues of J.

Theorem 2: Theorem 2

For a planar system with Jacobian eigenvalues \lambda_{ 1,2 } = (\tau \pm \sqrt{ \tau^2 - 4\Delta })/2: if \Delta < 0 the fixed point is a saddle; if \Delta > 0 and \tau^2 < 4\Delta the eigenvalues are complex (spiral if \tau \neq 0, center if \tau = 0); if \Delta > 0 and \tau^2 > 4\Delta the eigenvalues are real (stable node if \tau < 0, unstable node if \tau > 0).

Theorem X: Theorem

See definition.

Worked Examples

1
Find fixed points: -x + x^3 = 0 gives x(x^2-1) = 0, so x = 0, ±1. dy/dt = -2y = 0 gives y = 0. Fixed points are (0,0), (1,0), (-1,0).
2
Compute Jacobian: J = [[-1+3x^2, 0],[0, -2]].
$J = \begin{pmatrix} -1+3x^2 & 0 \\ 0 & -2 \end{pmatrix}$
3
At (0,0): J = [[-1,0],[0,-2]], eigenvalues -1 and -2. Both negative, so (0,0) is a stable node.
4
At (±1, 0): J = [[2,0],[0,-2]], eigenvalues 2 and -2. One positive, one negative, so (±1,0) are saddle points.

✓ Answer

(0,0) is a stable node; (1,0) and (-1,0) are saddle points.

Practice Problems

Easyfree response

Find all fixed points of dx/dt = x - x*y, dy/dt = x*y - y, and classify each using the Jacobian.

EasyMultiple choice

A fixed point has Jacobian with trace tau = -3 and determinant Delta = 4. What type is it?

Common Mistakes

Common Mistake

Applying Hartman-Grobman to non-hyperbolic fixed points such as centers.

Hartman-Grobman requires hyperbolicity (no eigenvalue with zero real part); for centers (pure imaginary eigenvalues) the theorem does not apply and linearization can misclassify the nonlinear behavior.

Common Mistake

Confusing stability with asymptotic stability.

Stability means trajectories stay near x*; asymptotic stability means they also converge to x*. A center is stable but not asymptotically stable.

Common Mistake

Evaluating the Jacobian at the origin rather than at the fixed point.

Always compute J = DF(x*) evaluated at the specific fixed point x*, not at the origin unless x* = 0.

Common Mistake

Using tr and det to classify without verifying whether eigenvalues are real or complex.

Always check the discriminant tau^2 - 4*Delta: if negative, eigenvalues are complex conjugates (spiral/center); if positive, they are real (node/saddle).

Quiz

Historical Background

The classification of equilibria of planar vector fields was carried out systematically by Poincare in his 1881-1886 memoirs. Liapunov (Lyapunov) developed a complementary, rigorous stability theory in his 1892 doctoral dissertation 'The General Problem of the Stability of Motion,' introducing both the linearization method and the direct (energy-function) method that now bears his name. The complete classification for hyperbolic equilibria — sometimes called the Hartman-Grobman theorem — was proved rigorously in the 1960s, confirming that linearization exactly captures the local topology of the nonlinear flow.

1881
Poincare classifies equilibria of planar ODEs by eigenvalue type
Henri Poincare
1892
Lyapunov's doctoral dissertation establishes rigorous stability theory
Aleksandr Lyapunov
1959-1960
Hartman and Grobman independently prove the linearization theorem for hyperbolic fixed points
Philip Hartman, David Grobman

Summary

Fixed points satisfy F(x*)=0 and represent constant solutions; their stability is determined by the eigenvalues of the Jacobian DF(x*).
Planar fixed points are classified by trace and determinant: saddle (det<0), stable/unstable node or spiral (det>0), and center (det>0, tr=0).
The Hartman-Grobman theorem guarantees that for hyperbolic fixed points the nonlinear flow is topologically equivalent to the linearized flow.
Lyapunov's first method: all eigenvalues with negative real part implies asymptotic stability; any eigenvalue with positive real part implies instability.

References

BookStrogatz, S. H. Nonlinear Dynamics and Chaos. Westview Press, 2015. Chapters 5-6.
BookGuckenheimer, J. and Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1983. Chapter 1.
WebsiteWikipedia — Stability theory