Search results for: ARNOLD CONJECTURE, CONLEY INDEX, HAMILTONIAN SYSTEMS, COMPLEX PROJECTIVE SPACE, CUP-LENGTH.

• The E-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on T 2n

  Publication

  • M. Starostka
  • N. Waterstraat

  - ADVANCED NONLINEAR STUDIES - Year 2019

  We show that the E-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals on Hilbert spaces. When applied to the setting of the Arnold conjecture, this paves the way to a short proof on tori, where it was first shown by C. Conley and E. Zehnder in 1983.

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• The Arnold conjecture in $ \mathbb C\mathbb P^n $ and the Conley index

  Publication

  • L. Asselle
  • M. Izydorek
  • M. Starostka

  - DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B - Year 2023

  n this paper we give an alternative, purely Conley index based proof of the Arnold conjecture in CP^n asserting that a Hamiltonian diffeomorphism of CP^n endowed with the Fubini-Study metric has at least (n+1) fixed points.

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• The Conley index, cup-length and bifurcation

  Publication

  • Z. Dzedzej
  • K. Gęba
  • W. Uss

  - Journal of Fixed Point Theory and Applications - Year 2011

  Zastosowano strukturę modułu w indeksie kohomologicznym Conleya do dowodu twierdzenia o minimalnej ilości rozwiązań okresowych dla układów Hamiltonowskich. Wykazano też ogólne twierdzenia dotyczące nietrywialności struktury mudułu.

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• E-cohomological Conley index

  Publication

  • M. Starostka

  - Year 2017

  In this thesis we continue with developing the E-cohomological Conley index which was introduced by A.Abbondandolo. In particular, we generalize the index to non-gradient flows, we show that it an possesses additional multiplicative structure and we prove the continuation principle. Then, using continuation principle, we show how the computation of the E-cohomological Conley index can be reduced to the computation of the classical...

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• Morse cohomology in a Hilbert space via the Conley index

  Publication

  • M. Starostka

  - Journal of Fixed Point Theory and Applications - Year 2015

  The main theorem of this paper states that Morse cohomology groups in a Hilbert space are isomorphic to the cohomological Conley index. It is also shown that calculating the cohomological Conley index does not require finite-dimensional approximations of the vector field. Further directions are discussed.

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• Conley type index and hamiltonian inclusions

  Publication

  • Z. Dzedzej

  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica - Year 2010

  Opisano definicję i własności indeksu dla zbiorów niezmienniczych wielowartościowego układu dynamicznego w nieskończenie-wymiarowej przestrzeni Hilberta. Podano nowe przykłady zastosowań do twierdzeń o istnieniu nietrywialnych rozwiązań okresowych układów hamiltonowskichz prawą stroną niegładką.

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• The cohomological span of LS-Conley index

  Publication

  • J. Maksymiuk

  - JOURNAL OF DIFFERENTIAL EQUATIONS - Year 2015

  In this paper we introduce a new homotopy invariant – the cohomological span of LS-Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form Lx+K(x), where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula...

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• Conley type index applied to Hamiltonian inclusions

  Publication

  • Z. Dzedzej
  • W. Kryszewski

  - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS - Year 2008

  Podano dowód istnienia nietrywialnych rozwiązań okresowych dla inkluzji Hamiltonowskich, z potencjałem lokalnie Lipschitzowskim, okresowym, uogólniając klasyczne twierdzenie Ammana- Zehndera. Użyto techniki z teorii indeksu Conley'a dla wielowartościowych potoków w przestrzeni Hilberta.

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• Morse inequalities via Conley index theory

  Publication

  • M. Izydorek
  • M. Styborski

  - Year 2011

  The relation known as the Morse inequalities can be extended to a more general setting of flows on a locally compact metric spaces (Conley index) as well as dynamical systems on Hilbert spaces (LS-index). This paper is a discourse around this extension. Except the part concerning the LS-index the material is self-contained and has a character of a survey.

• Topological invariants for equivariant flows: Conley index and degree

  Publication

  • M. Styborski

  - Year 2010

  About forty years have passed since Charles Conley defined the homotopy index. Thereby, he generalized the ideas that go back to the calculus of variations work of Marston Morse. Within this long time the Conley index has proved to be a valuable tool in nonlinear analysis and dynamical systems. A significant development of applied methods has been observed. Later, the index theory has evolved to cover such areas as discrete dynamical...