Computational Homology

Kaczyński, Tomasz; Mischaikow, Konstantin; Mrożek, Marian

doi:10.1007/b97315

Cited by 370 publications

(407 citation statements)

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“…We use the Conley index theory to prove the claim. The reader not familiar with the Conley index may consult [17,19].…”

Section: Counting Periodic Orbitsmentioning

confidence: 99%

“…Since periodic points are of saddle type and hence are numerically unstable, we apply the subdivision algorithm [11] to find them. For each periodic orbit found numerically, we then construct a cubical index pair [17]. The existence of a periodic point in this index pair is then proved by the following Conley index version of Lefschetz fixed point theorem.…”

Section: Counting Periodic Orbitsmentioning

confidence: 99%

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On loops in the hyperbolic locus of the complex Hénon map and their monodromies

Arai

2016

Physica D: Nonlinear Phenomena

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Abstract. We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real Hénon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex Hénon map.

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“…We use the Conley index theory to prove the claim. The reader not familiar with the Conley index may consult [17,19].…”

Section: Counting Periodic Orbitsmentioning

confidence: 99%

Section: Counting Periodic Orbitsmentioning

confidence: 99%

On loops in the hyperbolic locus of the complex Hénon map and their monodromies

Arai

2016

Physica D: Nonlinear Phenomena

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“…For the cubical representation, we make the choice that the grid element B is in N + if and only if the vertex sign on the upper right hand corner of B, as determined by outer approximation based on interval arithmetic, is positive. The homology of the cubical set N + may now be computed using existing software [15,17].…”

Section: Case (D)mentioning

confidence: 99%

Probabilistic and numerical validation of homology computations for nodal domains

Day

Kalies

Mischaikow

et al. 2007

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. Homology has long been accepted as an important computable tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study, based on suitable discretizations. Such an approach immediately raises the question of how accurate the resulting homology computations are. In this paper we present a probabilistic approach to quantifying the validity of homology computations for nodal domains of random Fourier series in one and two space dimensions, which furnishes explicit probabilistic apriori bounds for the suitability of certain discretization sizes. In addition, we introduce a numerical method for verifying the homology computation using interval arithmetic.

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“…Throughout the paper, it is emphasized that the use of topological approach is crucial, and, in particular, an algebraic topological method known as the homological Conley index theory provides a powerful tool, especially that this index can be efficiently computed with the publically available CHomP software package [8] which is based on the theory introduced in [20,29,36] (see also references therein).…”

Section: Introductionmentioning

confidence: 99%

Recent development in rigorous computational methods in dynamical systems

Arai

Kokubu

Pilarczyk

2009

Japan J. Indust. Appl. Math.

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Abstract. We highlight selected results of recent development in the area of rigorous computations which use interval arithmetic to analyse dynamical systems. We describe general ideas and selected details of different ways of approach and we provide specific sample applications to illustrate the effectiveness of these methods. The emphasis is put on a topological approach, which combined with rigorous calculations provides a broad range of new methods that yield mathematically reliable results.

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Computational Homology

Cited by 370 publications

References 0 publications

On loops in the hyperbolic locus of the complex Hénon map and their monodromies

On loops in the hyperbolic locus of the complex Hénon map and their monodromies

Probabilistic and numerical validation of homology computations for nodal domains

Recent development in rigorous computational methods in dynamical systems

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