Conley index theory
Encyclopedia
In dynamical systems theory
, Conley index theory, named after Charles Conley, analyzes topological structure of invariant sets of diffeomorphism
s and of smooth flows
. It is a far-reaching generalization of the Hopf index theorem that predicts existence of fixed points of a flow inside a planar region in terms of information about its behavior on the boundary. Conley's theory is related to Morse theory
, which describes the topological structure of a closed manifold
by means of a nondegenerate gradient vector field. It has an enormous range of applications to the study of dynamics, including existence of periodic orbits in Hamiltonian system
s and travelling wave solutions for partial differential equation
s, structure of global attractor
s for reaction-diffusion equations and delay differential equation
s, proof of chaotic behavior in dynamical systems, and bifurcation theory
. Conley index theory formed the basis for development of Floer homology
.
N and isolated invariant set S. The Conley index h(S) is the homotopy type of a certain pair (N1, N2) of compact subsets of N, called an index pair. Charles Conley showed that index pairs exist and that the index of S is independent of the choice of an isolated neighborhood N and the index pair. In the special case of a nondegenerate (Morse) critical point of index k, the Conley index is the pointed homotopy type of the k-sphere Sk.
A deep theorem due to Conley asserts continuation invariance: Conley index is invariant under certain deformations of the dynamical system. Computation of the index can, therefore, be reduced to the case of the diffeomorphism or a vector field whose invariant sets are well understood.
If the index is nontrivial then the invariant set S is nonempty. This principle can be amplified to establish existence of fixed points and periodic orbits inside N.
Dynamical systems theory
Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...
, Conley index theory, named after Charles Conley, analyzes topological structure of invariant sets of diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
s and of smooth flows
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
. It is a far-reaching generalization of the Hopf index theorem that predicts existence of fixed points of a flow inside a planar region in terms of information about its behavior on the boundary. Conley's theory is related to Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
, which describes the topological structure of a closed manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
by means of a nondegenerate gradient vector field. It has an enormous range of applications to the study of dynamics, including existence of periodic orbits in Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
s and travelling wave solutions for partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s, structure of global attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
s for reaction-diffusion equations and delay differential equation
Delay differential equation
In mathematics, delay differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times....
s, proof of chaotic behavior in dynamical systems, and bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
. Conley index theory formed the basis for development of Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
.
Short description
A key role in the theory is played by the notions of isolating neighborhoodIsolating neighborhood
In the theory of dynamical systems, an isolating neighborhood is a compact set in the phase space of an invertible dynamical system with the property that any orbit contained entirely in the set belongs to its interior. This is a basic notion in the Conley index theory...
N and isolated invariant set S. The Conley index h(S) is the homotopy type of a certain pair (N1, N2) of compact subsets of N, called an index pair. Charles Conley showed that index pairs exist and that the index of S is independent of the choice of an isolated neighborhood N and the index pair. In the special case of a nondegenerate (Morse) critical point of index k, the Conley index is the pointed homotopy type of the k-sphere Sk.
A deep theorem due to Conley asserts continuation invariance: Conley index is invariant under certain deformations of the dynamical system. Computation of the index can, therefore, be reduced to the case of the diffeomorphism or a vector field whose invariant sets are well understood.
If the index is nontrivial then the invariant set S is nonempty. This principle can be amplified to establish existence of fixed points and periodic orbits inside N.