308:: 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.
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is independent of the choice of the index pair. In the special case of the negative gradient flow of a smooth function, the Conley index of a nondegenerate (Morse) critical point of index
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Some of the most important properties of the index are direct consequences of its definition, inheriting properties from homology and homotopy. Some of them include the following:
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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
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We remark that also Conley showed that the Conley index is independent of the choice of an index pair, so that the index is well defined.
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is nonempty. This principle can be amplified to establish existence of fixed points and periodic orbits inside
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1667:. CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978
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Conley shows that every isolating invariant set admits an index pair. For an isolated invariant set
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Notice that, a Morse set is an isolated invariant set, so that the conley index is defined for it.
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55:. It has an enormous range of applications to the study of dynamics, including existence of
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768:{\displaystyle \phi (,x)\subset N_{1}\Rightarrow \phi (,x)\subset N_{2}}
251:. Charles Conley showed that index pairs exist and that the index of
910:{\displaystyle \phi (t,x)\not \in N_{1}\Rightarrow \exists t'\in }
1383:{\displaystyle CH_{\bullet }(S,\phi )=H_{\bullet }(N_{1}/N_{2},)}
327:
We build the Conley Index from the concept of a index pair.
91:. Conley index theory formed the basis for development of
1638:{\displaystyle CH_{k}(S)=\oplus _{i=1}^{n}CH_{k}(M_{i})}
47:, which describes the topological structure of a closed
31:, analyzes topological structure of invariant sets of
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A key role in the theory is played by the notions of
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On Conley’s fundamental theorem of dynamical systems
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1777:Conley's fundamental theorem of dynamical systems
1715:Dynamical systems. Examples of complex behaviour
1717:. Universitext. Springer-Verlag, Berlin, 2005
39:. It is a far-reaching generalization of the
8:
1703:, vol 1, part 1, pp 547–598, Elsevier 2002
1665:Isolated invariant sets and the Morse index
538:{\displaystyle S={\text{Inv}}(N_{1}/N_{2})}
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1185:the homotopy type of the quotient space
1764:Separation of Topological Singularities
1650:The Conley index is homotopy invariant.
1252:, seen as a topological pointed space.
1727:Konstantin Mischaikow, Marian Mrozek,
1519:{\displaystyle S=\cup _{i=1}^{n}M_{i}}
304:A deep theorem due to Conley asserts
182:of a space built from a certain pair
7:
963:{\displaystyle \phi (t',x)\in N_{2}}
291:is the pointed homotopy type of the
1737:, vol 2, pp 393–460, Elsevier 2002
1553:is an isolated invariant set, then
16:Theorem in dynamical systems theory
1454:
878:
476:{\displaystyle N_{2}\subset N_{1}}
14:
1695:John Franks, Michal Misiurewicz,
1766:(Wolfram Demonstrations Project)
1460:{\displaystyle S\neq \emptyset }
1697:Topological methods in dynamics
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1245:{\displaystyle (N_{1}/N_{2},)}
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1174:{\displaystyle h(S,\phi ):=)]}
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69:partial differential equations
1:
1734:Handbook of Dynamical Systems
1701:Handbook of Dynamical Systems
1060:and the we define, then, the
1033:{\displaystyle (N_{1},N_{2})}
436:{\displaystyle (N_{1},N_{2})}
221:{\displaystyle (N_{1},N_{2})}
994:, we choose some index pair
81:delay differential equations
77:reaction–diffusion equations
51:by means of a nondegenerate
1685:Encyclopedia of Mathematics
580:{\displaystyle N_{1}/N_{2}}
126:and isolated invariant set
1818:
1802:Fixed points (mathematics)
1434:{\displaystyle h(S)\neq 0}
804:{\displaystyle x\in N_{1}}
636:{\displaystyle x\in N_{2}}
397:is a pair of compact sets
228:of compact sets called an
87:in dynamical systems, and
1257:(co)homology Conley index
1678:Thomas Bartsch (2001) ,
21:dynamical systems theory
306:continuation invariance
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105:isolating neighborhood
71:, structure of global
1792:Differential topology
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1546:{\displaystyle M_{i}}
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1279:is the chain complex
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1062:homotopy Conley index
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587:is a neighborhood of
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53:gradient vector field
1797:Topological dynamics
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171:{\displaystyle h(S)}
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61:Hamiltonian systems
25:Conley index theory
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89:bifurcation theory
41:Hopf index theorem
1743:978-0-444-50168-4
1723:978-3-540-22908-7
1709:978-0-444-82669-5
1272:{\displaystyle S}
1255:Analogously, the
1077:{\displaystyle S}
1053:{\displaystyle S}
987:{\displaystyle S}
600:{\displaystyle S}
502:
390:{\displaystyle S}
346:{\displaystyle S}
284:{\displaystyle N}
264:{\displaystyle S}
243:{\displaystyle S}
139:{\displaystyle S}
119:{\displaystyle N}
99:Short description
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1663:Charles Conley,
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85:chaotic behavior
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1731:. Chapter 9 in
1699:. Chapter 7 in
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65:travelling wave
57:periodic orbits
33:diffeomorphisms
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1758:External links
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1747:M. R. Razvan,
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1680:"Conley index"
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93:Floer homology
67:solutions for
35:and of smooth
29:Charles Conley
27:, named after
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1713:JĂĽrgen Jost,
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1673:0-8218-1688-8
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180:homotopy type
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1729:Conley index
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329:
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323:Construction
316:
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305:
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298:
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229:
148:Conley index
147:
102:
45:Morse theory
24:
18:
83:, proof of
1786:Categories
1658:References
1398:Properties
917:such that
375:index pair
353:in a flow
73:attractors
1690:EMS Press
1587:⊕
1487:∪
1455:∅
1452:≠
1426:≠
1326:∙
1312:ϕ
1298:∙
1107:ϕ
948:∈
925:ϕ
890:∈
879:∃
876:⇒
845:ϕ
789:∈
753:⊂
723:ϕ
720:⇒
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