Scientific Heritage of L. P. Shilnikov. Part II. Homoclinic Chaos

Sergey V. Gonchenko; Lev M. Lerman; Andrey L. Shilnikov; Dmitry V. Turaev

doi:10.1134/S1560354725020017

Scientific Heritage of L. P. Shilnikov. Part II. Homoclinic Chaos

Sergey V. Gonchenko, Lev M. Lerman, Andrey L. Shilnikov, Dmitry V. Turaev

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We review the works initiated and developed by L. P. Shilnikov on homoclinic chaos, highlighting his fundamental contributions to Poincaré homoclinics to periodic orbits and invariant tori. Additionally, we discuss his related findings in non-autonomous and infinite-dimensional systems. This survey continues our earlier review [1], where we examined Shilnikov’s groundbreaking results on bifurcations of homoclinic orbits — his extension of the classical work by A. A. Andronov and E. A. Leontovich from planar to multidimensional autonomous systems, as well as his pioneering discoveries on saddle-focus loops and spiral chaos.

Original language	English
Article number	Paper No. 010402
Pages (from-to)	155-173
Number of pages	19
Journal	Regular and Chaotic Dynamics
Volume	30
Issue number	2
DOIs	https://doi.org/10.1134/S1560354725020017
State	Published - 1 Apr 2025
Externally published	Yes

Keywords

Banach space
Poincaré homoclinic orbit
exponential dichotomy
hyperbolic set
integral curve
nonautonomous system
saddle periodic orbit
symbolic dynamics

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematics (miscellaneous)
Modeling and Simulation
Mathematical Physics
Mechanical Engineering
Applied Mathematics

Access to Document

10.1134/S1560354725020017

Cite this

@article{f9f6201b2f62471288b03aa3f893c21a,

title = "Scientific Heritage of L. P. Shilnikov. Part II. Homoclinic Chaos",

abstract = "We review the works initiated and developed by L. P. Shilnikov on homoclinic chaos, highlighting his fundamental contributions to Poincar{\'e} homoclinics to periodic orbits and invariant tori. Additionally, we discuss his related findings in non-autonomous and infinite-dimensional systems. This survey continues our earlier review [1], where we examined Shilnikov{\textquoteright}s groundbreaking results on bifurcations of homoclinic orbits — his extension of the classical work by A. A. Andronov and E. A. Leontovich from planar to multidimensional autonomous systems, as well as his pioneering discoveries on saddle-focus loops and spiral chaos.",

keywords = "Banach space, Poincar{\'e} homoclinic orbit, exponential dichotomy, hyperbolic set, integral curve, nonautonomous system, saddle periodic orbit, symbolic dynamics",

author = "Gonchenko, \{Sergey V.\} and Lerman, \{Lev M.\} and Shilnikov, \{Andrey L.\} and Turaev, \{Dmitry V.\}",

note = "Publisher Copyright: {\textcopyright} Pleiades Publishing, Ltd. 2025.",

year = "2025",

month = apr,

day = "1",

doi = "10.1134/S1560354725020017",

language = "English",

volume = "30",

pages = "155--173",

journal = "Regular and Chaotic Dynamics",

issn = "1560-3547",

publisher = "Pleiades Publishing",

number = "2",

}

TY - JOUR

T1 - Scientific Heritage of L. P. Shilnikov. Part II. Homoclinic Chaos

AU - Gonchenko, Sergey V.

AU - Lerman, Lev M.

AU - Shilnikov, Andrey L.

AU - Turaev, Dmitry V.

N1 - Publisher Copyright: © Pleiades Publishing, Ltd. 2025.

PY - 2025/4/1

Y1 - 2025/4/1

N2 - We review the works initiated and developed by L. P. Shilnikov on homoclinic chaos, highlighting his fundamental contributions to Poincaré homoclinics to periodic orbits and invariant tori. Additionally, we discuss his related findings in non-autonomous and infinite-dimensional systems. This survey continues our earlier review [1], where we examined Shilnikov’s groundbreaking results on bifurcations of homoclinic orbits — his extension of the classical work by A. A. Andronov and E. A. Leontovich from planar to multidimensional autonomous systems, as well as his pioneering discoveries on saddle-focus loops and spiral chaos.

AB - We review the works initiated and developed by L. P. Shilnikov on homoclinic chaos, highlighting his fundamental contributions to Poincaré homoclinics to periodic orbits and invariant tori. Additionally, we discuss his related findings in non-autonomous and infinite-dimensional systems. This survey continues our earlier review [1], where we examined Shilnikov’s groundbreaking results on bifurcations of homoclinic orbits — his extension of the classical work by A. A. Andronov and E. A. Leontovich from planar to multidimensional autonomous systems, as well as his pioneering discoveries on saddle-focus loops and spiral chaos.

KW - Banach space

KW - Poincaré homoclinic orbit

KW - exponential dichotomy

KW - hyperbolic set

KW - integral curve

KW - nonautonomous system

KW - saddle periodic orbit

KW - symbolic dynamics

UR - https://www.scopus.com/pages/publications/105003391162

U2 - 10.1134/S1560354725020017

DO - 10.1134/S1560354725020017

M3 - Article

AN - SCOPUS:105003391162

SN - 1560-3547

VL - 30

SP - 155

EP - 173

JO - Regular and Chaotic Dynamics

JF - Regular and Chaotic Dynamics

IS - 2

M1 - Paper No. 010402

ER -

Scientific Heritage of L. P. Shilnikov. Part II. Homoclinic Chaos

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this