Isochronous and Period-Doubling Diagrams for Symplectic Maps of the Plane | Sciact - CRIS-система Института катализа

Isochronous and Period-Doubling Diagrams for Symplectic Maps of the Plane Научная публикация

Журнал

Chaos, Solitons & Fractals
ISSN: 0960-0779 , E-ISSN: 1873-2887

Вых. Данные

Год: 2025, Номер: 198, Номер статьи : 116513, Страниц : 28 DOI: 10.1016/j.chaos.2025.116513

Ключевые слова

Chaos, Integrability, Perturbation theory, Stability

Авторы

Zolkin T. ¹ , Nagaitsev S. ^2,3 , Morozov I. ^4,5 , Kladov S. ⁶ , Kim Y.-K. ⁶

Организации

1	Fermilab, PO Box 500, Batavia, IL 60510-5011, United States of America
2	Brookhaven National Laboratory, Upton, NY 11973
3	Old Dominion University, Norfolk, VA 23529
4	Synchrotron Radiation Facility ”SKIF”, Koltsovo 630559, Russia
5	Novosibirsk State Technical University, Novosibirsk 630073, Russia
6	University of Chicago, Chicago, IL 60637

Информация о финансировании (3)

1	United States Department of Energy	DE-AC02-07CH11359
2	United States Department of Energy	DE-AC05-06OR23177
3	Министерство науки и высшего образования Российской Федерации (с 15 мая 2018)	FWUR-2024-0041

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more thorough understanding of the underlying qualitative aspects. This paper aims to address this gap by revisiting the foundational concepts of reversibility and associated symmetries, first explored in the early works of G.D. Birkhoff. We extend the original framework proposed by Hénon by adding a period-doubling diagram to his isochronous diagram, which allows to represents the system’s bifurcations and the groups of symmetric periodic orbits that emerge in typical bifurcations of the fixed point. A qualitative and quantitative explanation of the main features of the region of parameters with bounded motion is provided, along with the application of this technique to other symplectic mappings, including cases of multiple reversibility. Modern chaos indicators, such as the Reversibility Error Method (REM) and the Generalized Alignment Index (GALI), are employed to distinguish between various dynamical regimes in the mixed space of variables and parameters. These tools prove effective in differentiating regular and chaotic dynamics, as well as in identifying twistless orbits and their associated bifurcations. Additionally, we discuss the application of these methods to real-world problems, such as visualizing dynamic aperture in accelerator physics, where our findings have direct relevance.

Zolkin T. , Nagaitsev S. , Morozov I. , Kladov S. , Kim Y.-K.
Isochronous and Period-Doubling Diagrams for Symplectic Maps of the Plane
Chaos, Solitons & Fractals. 2025. N198. 116513 :1-28. DOI: 10.1016/j.chaos.2025.116513 Scopus OpenAlex СКИФ ID

Поступила в редакцию:	10 янв. 2025 г.
Принята к публикации:	22 апр. 2025 г.
Опубликована online:	16 мая 2025 г.
Опубликована в печати:	1 сент. 2025 г.

Scopus:	2-s2.0-105005087709
OpenAlex:	W4405253310
СКИФ ID:	3915

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