| 38 | 0 | 42 |
| 下载次数 | 被引频次 | 阅读次数 |
该文基于Kosambi-Cartan-Chern(KCC)理论对一类含阻尼的SD振子进行了全新的动力学分析.分别给出了SD振子在光滑和非光滑情况下的KCC几何不变量，结果表明，非光滑情况下的几何量不能由光滑情况下的几何量直接导出.基于这些几何量，分别分析了该SD振子在光滑和非光滑情况下轨迹任何一点处的KCC稳定性.数值分析结果表明，在偏离系统平衡点的某些区域，系统因参数的微小变化将呈现出复杂多变的动力学行为.该文的研究说明KCC理论也可以一定程度上分析非光滑系统.
Abstract：Based on Kosambi-Cartan-Chern(KCC) theory, a new dynamic analysis of a class of damped SD oscillators is presented in this paper.The KCC geometric invariants of SD oscillator in smooth and non-smooth cases are given respectively.The results show that the geometric quantities in the non-smooth case cannot be directly derived from the geometric quantities in the smooth case.Based on these geometric quantities, the KCC stability of the SD oscillator at any point of the trajectory in smooth and non-smooth cases is analyzed respectively.The numerical results show that in some regions that deviate from the equilibrium point of the system, the system will show complex dynamic behavior due to small changes in parameters.This study shows that KCC theory can also analyze non-smooth systems to a certain extent.
[1] THOMPSON J M T,HUNT G W.A General Theory of Elastic Stability[M].London:Wiley,1973.
[2] CAO Q,WIERCIGROCH M,PAVLOVSKAIA E,et al.An archetypal oscillator for smooth and discontinuous dynamics[J].Phys Rev E,2006(74):046218.
[3] CAO Q,WIERCIGROCH M,PAVLOVSKAIA E,et al.Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics[J].Philos Trans Roy Soc A:Math Phy Eng Sci,2008(266):635-652.
[4] CAO Q,WIERCIGROCH M,PAVLOVSKAIA E,et al.The limit case response of the archetypal oscillator for smooth and discontinuous dynamics[J].Int J Non-Lin Mech,2008(43):462.
[5] TIAN R,CAO Q,YANG S.The codimensiontwo bifurcation for the recent proposed SD oscillator[J].Nonlinear Dynam,2010,59:19-27.
[6] CHEN H,LLIBRE J,TANG Y.Global dynamics of a SD oscillator[J].Nonlinear Dynam,2018,91:1755-1777.
[7] CHEN H.Global analysis on the discontinuous limit case of a smooth oscillator[J].Int J Bifurcat Chaos,2016,26(4):165006.
[8] KOSAMBI D D.Parallelism and path-spaces[J].Math Z,1933,37(1):608-618.
[9] CARTAN E.Observations sur le mémoire précédent[J].Math Z,1933,37(1):619-622.
[10] CHERN S S.Sur la geometrie d'un systeme d'equations differentielles du second ordre[J].Bull Sci Math,1939,63:206-212.
[11] HARKO T,HO C Y,LEUNG C S,et al.Jacobi stability analysis of the Lorenz system[J].Int J Geom Methods M,2015(12):55-72.
[12] ABOLGHASEM H.Jacobi stability of circular orbits in a central force[J].J Dyn Syst Geom Theor,2012,10(2):197-214.
[13] ABOLGHASEM H.Jacobi stability of Hamiltonian system[J].Int J P Appl Math,2013:87(1):181-194.
[14] LIU Y.Analysis of global dynamics in an unusual 3D chaotic system[J].Nonlinear Dynam,2020,70:2203-2212.
[15] YAMASAKI K,YAJIMA T.Lotka-Volterra system and KCC theory:Difffferential geometric structure of competitions and predations[J].Nonlin Anal:Real World Appl,2013(14):1845-1853.
[16] YAMASAKI K,YAJIMA T.KCC analysis of the normal form of typical bifurcations in one-dimensional dynamical systems:Geometrical invariants of saddle-node,transcritical,and pitchfork bifurcations[J].Int J Bifurcat Chaos,2017,27(9):1750145.
[17] LIU Y,LI C,LIU A.Analysis of geometric invariants for three types of bifurcations in 2D differential systems[J].Int J Bifurcat Chaos,2021,31(7):2150105.
[18] BUZZI A,CLAUDIO M,JOAO C R,et al.Generic bifurcation of refracted systems[J].Adv Math,2012(234):653-666
[19] FILIPPOV A F.Jacobi analysis for an unusual 3D autonomous system[J].Math Sb,1960,93(1):99-128.
[20] JACQUEMARD A,TEIXEIRA M A.On singularities of discontinuous vector fields[J].Bull Sci Math,2003(127):611-633.
[21] LI S M.Phase portraits of planar piecewise linear refracting systems:Focus-saddle case[J].Nonlinear Anal:Real World Appl,2020,56(1):103153.
[22] GUPTA M K,YADAV C K.Jacobi stability analysis of Rikitake system[J].Int J Geom Meth Mod Phys,2016,13:1650098.
[23] GUPTA M K,YADAV C K.Jacobi stability analysis of modifified Chua circuit system[J].Int J Geom Meth Mod Phys,2017,14:1750089.
[24] GUPTA M K,YADAV C K.Jacobi stability analysis of Rossler system[J].Int J Bifurcat Chaos,2017,27:1750056.
[25] CHEN Y,YIN Z.The Jacobi stability of a Lorenz-type multistable hyperchaotic system with a curve of equilibria[J].Int J Bifurcat Chaos,2019,29:1950062.
[26] YAMASAKI K,YAJIMA T.KCC analysis of a one-dimensional system during catastrophic shift of the hill function:Douglas tensor in the nonequilibrium regione[J].Int J Bifurcat Chaos,2020,30(11):2020032.
[27] HUANG Q,LIU A,LIU Y.Jacobi stability analysis of the Chen system[J].Int J Bifurcat Chaos,2019,29(10):1950139.
[28] 张芷芬，丁同仁，黄文灶，等.微分方程定性理论[M].北京：科学出版社，1981.
[29] PERKO L.Differential Equations and Dynamical Systems[M].New York:Springer,2001.
[30] GUEKENHEIMER J,HOLMES P.Nonlinear Oscillations,Dynamical Systems and Bifurcation of Vectorfield[M].New York:Springer,1983:151-152.
[31] YAMASAKI K,YAJIMA T.Kosambi-Cartan-Chern stability in the intermediate nonequilibrium region of the brusselator mode[J].Int J Bifurcat Chaos,2022,22(2):32.
基本信息:
中图分类号:O19
引用信息:
[1]李润红,韦煜明.一类带阻尼SD振子的Jacobi分析[J].曲阜师范大学学报(自然科学版),2025,51(04):38-48.
基金信息:
广西自然科学基金(2023GXNSFAA026246)
2023-12-28
2023
2024-12-18
2024
2
2025-10-15
2025-10-15