Attractors of 3D Systems in Basic Models of Mechanics*
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International Applied Mechanics, Vol. 56, No. 5, September, 2020
ATTRACTORS OF 3D SYSTEMS IN BASIC MODELS OF MECHANICS*
N. V. Nikitina
The theorems (statements) on the existence of attractor are proved. A generalized Shilnikov theorem is formulated. In the expression for the saddle of a homoclinic loop, it includes an additional term that has a qualitative value in the formation of a strange attractor. A bifurcation program for the synchronization of 3D coupled identical generators is considered. The cause of new motions appearing in coupled generators is established. Keywords: two-disk dynamo model, attractor, bifurcation, strange attractor, symmetry principles, synchronization, multistable system Introduction. Research data on chaotic dynamics in [2, 3, 6, 8, 9, 14, 20, 28, 29] has shown a long-standing interest in the topics related to attractors of 3D systems of differential equations. Let us turn to the classical form of studying the dynamics of 3D systems, i.e., Shilnikov’s geometric theorem [29]. Based on this theorem, three theorems (with examples for each) will be formulated and proved here. As a result, we can formulate a generalized Shilnikov’s theorem. Since Shilnikov’s theorem is geometric (not proved, but true), the analytical statement of the (generalized) theorem distinguishes between the saddle value of a point sO and the saddle value of a loop s. The approaches resulted in the creation of standard forms of qualitative analytical information in the form of theorems, statements, and principles will be discussed. Let us consider bifurcations of 3D systems that lead to the emergence of regular and chaotic attractors. One approach to proving the theorems on the existence of attractors is related to the application of Shilnikov’s theorem and the extension of the theorem to a wider range of basic models under consideration. While developing the principle of symmetry for 3D systems, the system is divided into 2D subsystems, and the theory of 2D trajectory closure developed in the last century together with the principle of symmetry given in classical research [21] is applied. Mathematical models determined by 3D systems are models of generators of electromagnetic oscillations, connected solids, traffic flows, and other models of various physical phenomena in nature and technology [10, 12, 16, 17, 20, 27, 29]. The theorems presented here are based on a constructive method for identifying a negative saddle value and an analysis of the mechanisms of loss of orbital stability. The identification of a bifurcation process is related to the quality of motion that occurs in the motion field of the image (representative) point. The first studies of bifurcations of multidimensional systems do not contain statements in the form of theorems. This complicates the interpretation of the final results. The following issues are discussed herein: 1. Theorems on the existence of an attractor in 3D basic models. 2. The principle of symmetry in 3D systems. 3. Generalization of Shilnikov’s theorem. 4. Synchronization of cou
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