Zero-Hopf bifurcations and chaos of quadratic jerk systems

B. Sang, Rizgar H. Salih, Ning Wang
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引用次数: 2

Abstract

The purpose of this paper is to propose some coefficient conditions, characterizing the stability of periodic solutions bifurcated from zero-Hopf bifurcations of the general quadratic jerk system, and apply these theoretical results to a special jerk system in order to predict chaos. First, we characterize the zero-Hopf bifurcations of the general quadratic jerk system in $\mathbb{R}^3$. The coefficient conditions on stability of periodic solutions are obtained via the averaging theory of first order. Next, we apply the theoretical results to a two-parameter jerk system. Finally special attention is paid to a jerk system with one non-negative parameter $\epsilon$ and one non-linearity. By studying the continuation of periodic solution initiating at the zero-Hopf bifurcation, we numerically find a sequence of period doubling bifurcations which leads to the creation of chaotic attractor.
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二次急动系统的零Hopf分岔与混沌
本文的目的是提出一些系数条件,表征一般二次加加系统从零Hopf分支分叉的周期解的稳定性,并将这些理论结果应用于一个特殊的加加系统,以预测混沌。首先,我们刻画了$\mathbb{R}^3$中一般二次加加系统的零Hopf分岔。利用一阶平均理论得到了周期解稳定性的系数条件。接下来,我们将理论结果应用于一个双参数急动系统。最后,特别注意具有一个非负参数$\epsilon$和一个非线性的急动系统。通过研究起始于零Hopf分岔的周期解的连续性,我们在数值上找到了一个导致混沌吸引子产生的倍周期分岔序列。
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来源期刊
CiteScore
2.40
自引率
0.00%
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0
期刊介绍: Journal of Nonlinear Functional Analysis focuses on important developments in nonlinear functional analysis and its applications with a particular emphasis on topics include, but are not limited to: Approximation theory; Asymptotic behavior; Banach space geometric constant and its applications; Complementarity problems; Control theory; Dynamic systems; Fixed point theory and methods of computing fixed points; Fluid dynamics; Functional differential equations; Iteration theory, iterative and composite equations; Mathematical biology and ecology; Miscellaneous applications of nonlinear analysis; Multilinear algebra and tensor computation; Nonlinear eigenvalue problems and nonlinear spectral theory; Nonsmooth analysis, variational analysis, convex analysis and their applications; Numerical analysis; Optimal control; Optimization theory; Ordinary differential equations; Partial differential equations; Positive operator inequality and its applications in operator equation spectrum theory and so forth; Semidefinite programming polynomial optimization; Variational and other types of inequalities involving nonlinear mappings; Variational inequalities.
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