The Third Type of Dynamics and Poincaré Homoclinic Trajectories

IF 0.8 4区 地球科学 Q4 ENGINEERING, ELECTRICAL & ELECTRONIC Radiophysics and Quantum Electronics Pub Date : 2024-07-08 DOI:10.1007/s11141-024-10329-4
S. V. Gonchenko, A. S. Gonchenko, K. E. Morozov
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Abstract

We present a review of some fundamental results in the theory of dynamical systems, which have led to the discovery of dynamical chaos and its three forms, namely, two classical forms, such as conservative chaos and dissipative chaos, as well as the completely new third form, the so-called mixed dynamics in which the sets of attractors and repellers have non-empty intersection. The major part of the work is devoted to homoclinic Poincaré trajectories, i.e., doubly asymptotic trajectories to saddle periodic ones, as the main elements of dynamical chaos. Using simple examples, we show the appearance of such trajectories during periodic perturbations of two-dimensional conservative systems. As is known, the homoclinic trajectories were discovered by Poincaré. In this work, we discuss the problem (the planar circular restricted three-body problem) solving which this discovery was made. Some interesting facts concerning its history are given in the appendix.

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第三类动力学和普恩卡莱同线性轨迹
我们回顾了动力学系统理论中的一些基本成果,这些成果导致了动力学混沌及其三种形式的发现,即两种经典形式,如保守混沌和耗散混沌,以及全新的第三种形式,即所谓的混合动力学,其中吸引子集和排斥子集具有非空交集。研究的主要内容是同室波恩卡莱轨迹,即鞍状周期轨迹的双渐近轨迹,这是动力学混沌的主要元素。通过简单的例子,我们展示了这种轨迹在二维保守系统的周期性扰动过程中的出现。众所周知,均线轨迹是由波恩卡莱发现的。在这部著作中,我们讨论了这一发现所解决的问题(平面圆受限三体问题)。有关其历史的一些有趣事实见附录。
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来源期刊
Radiophysics and Quantum Electronics
Radiophysics and Quantum Electronics ENGINEERING, ELECTRICAL & ELECTRONIC-PHYSICS, APPLIED
CiteScore
1.10
自引率
12.50%
发文量
60
审稿时长
6-12 weeks
期刊介绍: Radiophysics and Quantum Electronics contains the most recent and best Russian research on topics such as: Radio astronomy; Plasma astrophysics; Ionospheric, atmospheric and oceanic physics; Radiowave propagation; Quantum radiophysics; Pphysics of oscillations and waves; Physics of plasmas; Statistical radiophysics; Electrodynamics; Vacuum and plasma electronics; Acoustics; Solid-state electronics. Radiophysics and Quantum Electronics is a translation of the Russian journal Izvestiya VUZ. Radiofizika, published by the Radiophysical Research Institute and N.I. Lobachevsky State University at Nizhnii Novgorod, Russia. The Russian volume-year is published in English beginning in April. All articles are peer-reviewed.
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