Conley Index Theory — Mathematics Knowledge System

Overview

The Conley index is a topological invariant of isolated invariant sets in continuous dynamical systems, generalising the Morse index to arbitrary compact invariant sets. Given an isolated invariant set S in a flow, the Conley index h(S) is the homotopy type of a pointed compact pair (N, L) where N is an isolating neighbourhood and L is its exit set. The Conley index is invariant under continuation (homotopies of the flow) and detects the existence of connecting orbits between invariant sets. It provides a powerful tool for proving the existence of periodic orbits, heteroclinic connections, and chaotic dynamics.

Intuition

At a hyperbolic fixed point of a vector field, the Morse index (dimension of the unstable manifold) tells you the 'shape' of the local dynamics -- k unstable directions means the index is a k-sphere. The Conley index generalises this: for any compact invariant set S, build an isolating neighbourhood N (a solid region containing only orbits that stay near S), and consider the exit set L (where orbits leave N). The homotopy type of N/L (quotient space) is the Conley index -- it captures the 'topological type' of the local dynamics around S.

Formal Definition

Definition

An index pair (N, L) for an isolated invariant set S in a flow phi_t is a compact pair with L subset N such that: S = Inv(N\L) (S is the maximal invariant set inside N\L), L is positively invariant relative to N (orbits exit N through L), and N\L is a neighbourhood of S. The Conley index h(S) is the homotopy type of the pointed space (N/L, [L]) (quotient with L collapsed to a point). This is well-defined and invariant under homotopies of the flow preserving the isolation.

h(S) = [N/L, [L]] \quad (\text{homotopy type of } N/L)

Conley index

\tilde{H}_*(N/L) = CH_*(S) \quad (\text{Conley homology})

Conley homology

\chi(h(S)) = \sum_k (-1)^k \dim CH_k(S)

Euler characteristic of Conley index

h(\text{hyperbolic fixed point of index } k) = [S^k, *]

Conley index of hyperbolic fixed point

Notation

Notation	Meaning
$h(S)$	Conley index of isolated invariant set S
$(N, L)$	Index pair for S
$CH_*(S)$	Conley homology of S
$\mathrm{Inv}(N)$	Maximal invariant set inside N

Theorems

Theorem 1: Continuation Invariance

If S_lambda is a continuous family of isolated invariant sets (indexed by lambda in [0,1]) with index pairs (N_lambda, L_lambda) varying continuously, then the Conley index h(S_0) is homotopy equivalent to h(S_1). The Conley index is invariant under continuation of the flow.

Theorem 2: Wazewski Property

If h(S) is not the trivial (one-point) homotopy type, then S is non-empty. In particular: if an isolating neighbourhood N has non-trivial Conley index, then the maximal invariant set inside N is non-empty. This provides existence proofs for invariant sets.

Theorem 3: Morse Decomposition and Connection Matrix

A Morse decomposition M = {M_1, ..., M_n} of a compact invariant set S (with a partial order on M_i) gives a connection matrix: a boundary operator on the direct sum of Conley homologies oplus CH_*(M_i) that computes the Conley homology of S. Non-zero off-diagonal entries detect connecting orbits between Morse sets.

Worked Examples

1
A saddle point in R^2 with one unstable direction (W^u is 1-dimensional) has an isolating neighbourhood N = small closed disk around the fixed point.
2
The exit set L is the portion of the boundary through which the unstable manifold exits -- two arcs on the boundary disk.
$L \subset \partial N,\; L \cong \text{two arcs (two points boundary of 1-disk)}$
3
N/L: collapsing L to a point in a disk with two boundary arcs identified to the basepoint gives S^1 (the circle).
$h = [S^1, *]$
4
Conley homology: CH_0 = 0, CH_1 = Z. This reflects one unstable direction.

✓ Answer

The Conley index of a saddle (index-1 hyperbolic fixed point in R^2) is h = [S^1, *], with CH_1 = Z.

Practice Problems

Hardfree response

Explain how the connection matrix in Conley's Morse decomposition detects heteroclinic orbits between equilibria.

Common Mistakes

Common Mistake

Thinking the Conley index depends on the choice of index pair.

A fundamental theorem states that the Conley index is independent of the choice of index pair (N, L) -- different valid index pairs give the same homotopy type. This independence is what makes it a well-defined invariant.

Quiz

The Conley index h(S) is defined as:

The Morse index (dimension of unstable manifold) of SThe homotopy type of the quotient N/L where (N,L) is an index pairThe number of periodic orbits inside SThe topological entropy of S

Historical Background

The Conley index was introduced by Charles Conley in his landmark 1978 monograph 'Isolated Invariant Sets and the Morse Index'. Conley, motivated by Morse theory and celestial mechanics, showed that the Morse index of a hyperbolic fixed point could be generalised to arbitrary isolated invariant sets using the homotopy type of an index pair. Robbin-Salamon (1992) further developed the theory, and Floer extended it to infinite-dimensional settings.

1978
Conley introduces the Conley index in his monograph
Charles Conley
1988
Floer uses Conley index ideas to develop Floer homology
Andreas Floer
1992
Robbin and Salamon axiomatise the Conley index
Joel Robbin, Dietmar Salamon

Summary

The Conley index h(S) = [N/L] is the homotopy type of an index pair for an isolated invariant set S.
It generalises the Morse index to arbitrary compact invariant sets.
Continuation invariance: h(S) is unchanged under homotopies of the flow preserving isolation.
Non-trivial Conley index proves existence of invariant sets (Wazewski); connection matrices detect connecting orbits.

References

BookConley, C. Isolated Invariant Sets and the Morse Index. AMS, 1978.
BookSmoller, J. Shock Waves and Reaction-Diffusion Equations. Springer, 1983.
WebsiteWikipedia -- Conley index theory