Conley index

A tool to analyze the dynamics of continuous or discrete dynamical systems. It can be used, for instance, to find special orbits like stationary, periodic or heteroclinic orbits, or to prove chaotic behaviour of the system. It has been applied to a wide range of problems, e.g. to find travelling-wave solutions of partial differential equations; to investigate the structure of global attractors of reaction-diffusion equations or delay equations; to find periodic solutions of Hamiltonian systems; to give a rigorous computer-assisted proof of chaos in Lorenz equations; to prove bifurcation and to analyze the set of bifurcating solutions in various settings. The original work of C. Conley and his school took place in the 1970s and early 1980s. Standard references for this work are [a3], [a12] and [a4]. A recent overview on the Conley index and its applications is [a8].

In order to describe the basic version of the Conley index, consider a flow $\varphi : \mathbf{R} \times X \rightarrow X$ on a locally compact metric space $X$ (cf. also Flow (continuous-time dynamical system)). A compact subset $S \subset X$ is called isolated invariant if there exists a compact neighbourhood $N$ of $S$ in $X$ such that $S$ is the invariant part of $N$:

\begin{equation*} S = \operatorname { inv } ( N ) : = \{ x \in N : \varphi ( t , x ) \in N \text { for all } t \in \bf R \}. \end{equation*}

In that case $N$ is said to be an isolating neighbourhood of $S$. The Conley index associates to an isolated invariant set $S$ the homotopy type of a pointed topological space in the following way. An index pair $( N , L )$ for an isolated invariant set $S$ consists of compact subsets $L \subset N$ of $X$ such that:

i) $\operatorname{clos} ( N \backslash L )$ is an isolating neighbourhood of $S$;

ii) $L$ is positively invariant in $N$: Given $x \in L$ and $t > 0$ with $\varphi ( [ 0 , t ] , x ) \subset N$, then $\varphi ( t , x ) \in L$;

iii) $L$ is an exit set for $N$: Given $x \in N$ and $t > 0$ with $\varphi ( t , x ) \notin N$, there exists a $t _ { 0 } \in [ 0 , t ]$ with $\varphi ( t _ { 0 } , x ) \in L$. Given an isolated invariant set $S$, it can be proved that index pairs exist. Moreover, if $( N , L )$ and $( N ^ { \prime } , L ^ { \prime } )$ are two index pairs for $S$, then the quotient spaces $N / L$ and $N ^ { \prime } / L ^ { \prime }$ are homotopy equivalent with base points $[ L ]$ and $[ L ^ { \prime } ]$ fixed (cf. also Homotopy). The Conley index $h ( S ) = h ( S , \varphi )$ of $S$ is by definition the homotopy type of the pointed space $( N / L , [ L ] )$, where $( N , L )$ is an index pair for $S$.

As an example, consider the flow $\varphi ( t , x ) = e ^ { t A } x$ on $X = \mathbf{R} ^ { n }$, where $A \in \mathcal{L} ( \mathbf{R} ^ { n } )$ has no eigenvalues on the imaginary axis; e.g. $A = \operatorname { diag } ( \lambda _ { 1 } , \dots , \lambda _ { n } )$ with

\begin{equation*} \lambda _ { 1 } \geq \ldots \geq \lambda _ { k } > 0 > \lambda _ { k + 1 } \geq \ldots \geq \lambda _ { n }. \end{equation*}

The origin is a hyperbolic stationary point of $\varphi$ and $S = \{ 0 \}$ is an isolated invariant set. Any compact neighbourhood of $S$ is an isolating neighbourhood of $S$. Suppose that the generalized eigenspace of $A$ corresponding to the eigenvalues with positive real part is spanned by $e _ { 1 } , \dots , e _ { k }$, and the complementary generalized eigenspace is spanned by $e_{k + 1} , \ldots , e _ { n }$. Then $( B ^ { k } \times B ^ { n - k } , S ^ { k - 1 } \times B ^ { n - k } )$ is an index pair for $S$; here $B ^ { l }$ is the unit ball in $\mathbf{R} ^ { l }$ with boundary $S ^ { l - 1 }$. Since $B ^ { n - k }$ is contractible (cf. also Contractible space), the Conley index of $S$ is equal to the homotopy type of $( B ^ { k } / S ^ { k - 1 } , [ S ^ { k - 1 } ] )$, which is the same as the homotopy type of $( S ^ { k } , * )$.

In this example one recovers the Morse index of the hyperbolic fixed point. Therefore the Conley index can be interpreted as a generalized Morse index. In applications one usually first has a set $N$ which is an isolating neighbourhood of some a priori unknown isolated invariant set $S : = \operatorname { inv } ( N )$. Then one tries to compute $h ( S )$ or to obtain some information, like its homology groups. For this computation the invariance of the Conley index under certain deformations of the flow, the continuation invariance, is very useful — in analogy to the homotopy invariance of the Brouwer degree. Finally one can use the knowledge about $h ( S )$ in order to investigate the invariant set $S$ itself. Whereas one can immediately deduce that $S$ is not empty if $h ( S )$ is not trivial, additional information on the flow inside $N$ is needed in order to obtain more detailed results about $S$, for example that $S$ contains a periodic orbit.

The original version of the Conley index has been refined and extended in several directions. For an equivariant version together with a product structure on the cohomology level, see [a6]. The Conley index and this additional structure played an important role in Floer's work on the Arnol'd conjecture and in the development of Floer homology. In [a11] and [a2] the Conley index has been generalized to semi-flows on metric spaces which need not be locally compact. This has been applied to parabolic differential equations and delay differential equations. Discrete dynamical systems are being considered in [a10] and [a9], multi-valued discrete dynamical systems in [a7]. The multi-valued version is the basis for rigorous numerical computations of the Conley index for concrete dynamical systems, since it allows one to incorporate interval arithmetic. Parametrized versions of the Conley index have been defined in [a1] and [a5]; an abstract categorical approach is given in [a13].

References

[a1]	T. Bartsch, "The Conley index over a space" Math. Z. , 209 (1992) pp. 167–177
[a2]	V. Benci, "A new approach to the Morse–Conley theory and some applications" Ann. Mat. Pura Appl. (4) , 4 (1991) pp. 231–305
[a3]	C. Conley, "Isolated invariant sets and the Morse index" , CBMS Regional Conf. Ser. , 38 , Amer. Math. Soc. (1978)
[a4]	J. Smoller, "Shock waves and reaction-diffusion equations" , Springer (1983)
[a5]	M. Mrozek, J. Reineck, R. Srzednicki, "The Conley index over a base" Trans. Amer. Math. Soc. , 352 (2000) pp. 4171–4194
[a6]	A. Floer, "A refinement of the Conley index and an application to the stability of hyperbolic invariant sets" Ergod. Th. Dynam. Syst. , 7 (1987) pp. 93–103
[a7]	T. Kaczyński, M. Mrozek, "Conley index for discrete multivalued dynamical systems" Topol. Appl. , 65 (1995) pp. 83–96
[a8]	K. Mischaikow, M. Mrozek, "Conley index theory" B. Fiedler (ed.) G. Iooss (ed.) N. Kopell (ed.) , Handbook of Dynamical Systems III: Towards Applications , Elsevier (to appear)
[a9]	M. Mrozek, "Leray functor and the cohomological Conley index for discrete dynamical systems" Trans. Amer. Math. Soc. , 318 (1990) pp. 149–178
[a10]	J. Robbin, D. Salamon, "Dynamical systems, shape theory and the Conley index" Ergod. Th. Dynam. Syst. , 8 (1988) pp. 375–393
[a11]	K. Rybakowski, "The homotopy index and partial differential equations" , Springer (1987)
[a12]	D. Salamon, "Connected simple systems and the Conley index of isolated invariant sets" Trans. Amer. Math. Soc. , 291 (1985) pp. 1–41
[a13]	A. Szymczak, "The Conley index for discrete dynamical systems" Topol. Appl. , 66 (1995) pp. 215–240