Invariants of weak equivalence in primitive matrices

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

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Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

Modern Physics Letters B ◽

10.1142/s0217984918501555 ◽

2018 ◽

Vol 32 (15) ◽

pp. 1850155 ◽

Cited By ~ 6

Author(s):

Chengwei Dong

Keyword(s):

Nonlinear System ◽

Variational Method ◽

Periodic Orbits ◽

Symbolic Dynamics ◽

Chaotic Systems ◽

Building Blocks ◽

Topological Classification ◽

One Dimensional ◽

Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

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Topological classification of one-dimensional attractors and repellers ofA-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports

Journal of Mathematical Sciences ◽

10.1007/bf02169053 ◽

1999 ◽

Vol 95 (5) ◽

pp. 2523-2545 ◽

Cited By ~ 7

Author(s):

V. Z. Grines

Keyword(s):

Topological Classification ◽

Fundamental Groups ◽

One Dimensional

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Homeomorphic classification of certain inverse limit spaces with open bonding maps

Pacific Journal of Mathematics ◽

10.2140/pjm.1982.103.589 ◽

1982 ◽

Vol 103 (2) ◽

pp. 589-601 ◽

Cited By ~ 19

Author(s):

William Watkins

Keyword(s):

Inverse Limit ◽

Inverse Limit Spaces

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A classification of inverse limit spaces of tent maps with finite critical orbit

Topology and its Applications ◽

10.1016/j.topol.2007.03.003 ◽

2007 ◽

Vol 154 (11) ◽

pp. 2265-2281 ◽

Cited By ~ 6

Author(s):

Sonja Štimac

Keyword(s):

Inverse Limit ◽

Critical Orbit ◽

Inverse Limit Spaces ◽

Tent Maps

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On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers

Sbornik Mathematics ◽

10.1070/sm1997v188n04abeh000216 ◽

1997 ◽

Vol 188 (4) ◽

pp. 537-569 ◽

Cited By ~ 5

Author(s):

V Z Grines

Keyword(s):

Topological Classification ◽

One Dimensional ◽

Structurally Stable

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Organization of the periodic orbits in the Rössler flow

International Journal of Modern Physics B ◽

10.1142/s0217979218502272 ◽

2018 ◽

Vol 32 (21) ◽

pp. 1850227 ◽

Cited By ~ 5

Author(s):

Chengwei Dong

Keyword(s):

Periodic Orbits ◽

Symbolic Dynamics ◽

Building Blocks ◽

Topological Classification ◽

Systematic Classification ◽

New Approach ◽

One Dimensional ◽

Surface Of Section ◽

Chaotic Flow

In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.

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