Abstract

This paper studies the periodic solutions of a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance 1 : n at the origin. Considering arbitrary positive integers n and N with n ≤ N and N ≥ 2 , the new explicit expression of displacement function for the four-dimensional system is detected by introducing the technique on power trigonometric integrals. Then some precise and detailed results in comparison with the existing works, including the existence condition, the exact number, and the parameter control conditions of periodic solutions, are obtained, which can provide a new theoretical description and mechanism explanation for the phenomena of emergence and disappearance of periodic solutions. Results obtained in this paper improve certain existing results under some parameter conditions and can be extensively used in engineering applications. To verify the applicability and availability of the new theoretical results, as an application, the periodic solutions of a circular mesh antenna model are obtained by theoretical method and numerical simulations.

Highlights

  • Many problems in the fields of engineering and science can be described by nonlinear polynomial systems

  • For a four or higher dimensional N-degree perturbation system, the upper bound of the number of periodic solutions bifurcating from the center singular point in certain resonance

  • It is remarkable that the explicit expression h(z) of System (1) for arbitrary positive integers n and N with n ≤ N and N ≥ 2 is obtained by introducing the technique on the exact formulas of power trigonometric integrals, which is new and plays an important role in detecting the exact number and parameter control conditions of the periodic solutions of System (1)

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Summary

Introduction

Many problems in the fields of engineering and science can be described by nonlinear polynomial systems. The study of the existence and number of periodic solutions for high-dimensional polynomial systems is an important and hot topic in bifurcation theory that can help scientists better comprehend and analyze the complex periodic vibration phenomena exhibited in systems from different fields. For a four or higher dimensional N-degree perturbation system, the upper bound of the number of periodic solutions bifurcating from the center singular point in certain resonance.

Preliminaries
Displacement Function
Periodic Solutions
Application
Conclusions

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