Imperfect Bifurcation of One-Dimensional Vector Field with a Time-Dependent Parameter
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摘要: 提出新的简便的方法研究一维向量场含线性时变参数的非完全分岔问题. 建立了量级平衡的基本方法,并结合微分方程解对参数和时间的连续性,讨论时变分岔方程的分岔转迁滞后和对分岔图的影响. 对3种不同类型的分岔模型方程作了具体的分析,给出它们分岔转迁的量级关系,定性分析结果与数值结果基本一致. 研究表明:存在非完全分岔参数的临界值,使得当非完全分岔参数分别小于、等于和大于临界值时,相应的时变分岔分别出现滞后、与定常分岔几乎一致和提前现象.Abstract: A new and simple method is proposed for studying an imperfect bifurcation problem of one-dimensional vector field with a time-dependent parameter. A basic method of scaling balance is established. By combining this method with the continuity of the solutions of ordinary differential equations with respect to time and parameters, the delayed bifurcation transition and the influences on the bifurcation diagrams are discussed. Three different types of model bifurcation equations are analyzed concretely, the scaling relations and the intervals of bifurcation transition are given. The qualitative analysis results coincide with the numerical ones. Our studies imply that there exists a critical value of imperfect parameter such that if the imperfect parameter is respectively smaller than, equal to and larger than the critical value, the corresponding time dependent bifurcation is delayed, almost the same as and advanced the steady one respectively.
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Key words:
- bifurcation /
- time varying /
- vectors
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