一类特殊三维自治动力学系统隐藏多吸引子的数值仿真与硬件实验研究
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摘要
研究了在一类特殊的、同时具有保守性和耗散性的三维自治动力学系统中隐藏多吸引子的共存现象.随着控制参数的变化,系统的平衡点从无平衡点演变为非零平衡点进而再演变为无平衡点,或者从非零平衡点演变为无平衡点.定性探讨了系统平衡点的演化与稳定性分布,并采用分岔图、李雅普诺夫指数谱和相轨图等动力学方法,开展了不同初始条件下随控制参数变化的分岔分析.设计并制作了硬件电路,实验结果验证了共存多吸引子的真实性.
This paper investigated a kind of special three-dimensional autonomous dynamical system which possesses conservativeness and dissipativeness simultaneously.With the variations of the control parameters,the equilibrium points of the system had a transition from no equilibrium point,to non-zero equilibrium points,and then to no equilibrium point,or from non-zero equilibrium points to no equilibrium point.The evolution rules and the stability distributions of the system equilibrium point were explored qualitatively.By utilizing the dynamical methods of bifurcation diagrams,Lyapunov exponent spectra,and phase portraits,bifurcation analyses with the variations of the control parameters were performed under different initial conditions.Meanwhile,a hardware circuit is designed and fabricated,and thereby the truth of the coexisting multiple attractors was verified by the experimental results.
This paper investigated a kind of special three-dimensional autonomous dynamical system which possesses conservativeness and dissipativeness simultaneously.With the variations of the control parameters,the equilibrium points of the system had a transition from no equilibrium point,to non-zero equilibrium points,and then to no equilibrium point,or from non-zero equilibrium points to no equilibrium point.The evolution rules and the stability distributions of the system equilibrium point were explored qualitatively.By utilizing the dynamical methods of bifurcation diagrams,Lyapunov exponent spectra,and phase portraits,bifurcation analyses with the variations of the control parameters were performed under different initial conditions.Meanwhile,a hardware circuit is designed and fabricated,and thereby the truth of the coexisting multiple attractors was verified by the experimental results.
引文
[1] LI C,SPROTT J C.Coexisting hidden attractors in a 4-D simplified Lorenz system[J].International Journal of Bifurcation and Chaos,2014,24(3):1450034.
[2] KUZNETSOV A P,KUZNETSOV S P,MOSEKILDE E,STANKEVICH N V.Co-existing hidden attractors in a radio-physical oscillator system[J].Journal of Physics A:Mathematical and Theoretical,2015,48(12):125101(1-8).
[3] LEONOV G A,KUZNETSOV N V,MOKAEV T N.Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity[J].Communications Nonlinear Science and Numerical Simulation,2015,28(1-3):166-174.
[4] LI C,SPROTT J C.Multistability in the Lorenz system:a broken butterfly[J].International Journal of Bifurcation and Chaos,2014,24(10):1450131.
[5] PISARCHIK A N,FEUDEL U.Control of multistability[J].Physics Reports,2014,540:167-218.
[6] YUAN FANG,WANG GUANGYI,WANG XIAOWEi.Extreme multistability in a memristor-based multi-scroll hyper-chaotic system[J].Chaos,2016,26(7):073107.
[7]李春彪,刘保彬,郑晓晨.一个新的超混沌系统及其线性反馈同步[J].计算机应用研究,2009,26(9):3304-3306.LI C BIAO,LIU B B,ZHENG X C.New hyperchaotic system and its linear feedback synchronization[J].Application Research of Computers,2009,26(9):3304-3306.(Ch).
[8]包伯成,王其红,胡文,等.基于Colpitts振荡器模型的网格涡卷混沌系统[J].四川大学学报(工程科学版),2011,43(3):145-149.BAO B C,WANG Q H,HU W,et al.Grid-scroll chaotic system based on colpitts oscillator model[J].Journal of Sichuan University Engineering Science Edition,2011,43(3):145-149.(Ch).
[9] DUDKOWSKI D,JAFARI S,KAPITANIAK T,et al.Hidden attractors in dynamical systems[J].Physics Reports,2016,637:1-50.
[10] SPROTT J C.A dynamical system with a strange attractor and invariant tori[J].Physics Letters A,2014,378:1361-1363.
[11]包涵,包伯成,林毅,等.忆阻自激振荡系统的隐藏吸引子及其动力学特性研究[J].物理学报,2016,65(18):180503.BAO H,BAO B C,LIN Y,et al.Hidden attractor and its dynamical characteristic in memristive self-oscillating system[J].Acta Phys Sin,2016,65(18):180501(1-12).(Ch).
[2] KUZNETSOV A P,KUZNETSOV S P,MOSEKILDE E,STANKEVICH N V.Co-existing hidden attractors in a radio-physical oscillator system[J].Journal of Physics A:Mathematical and Theoretical,2015,48(12):125101(1-8).
[3] LEONOV G A,KUZNETSOV N V,MOKAEV T N.Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity[J].Communications Nonlinear Science and Numerical Simulation,2015,28(1-3):166-174.
[4] LI C,SPROTT J C.Multistability in the Lorenz system:a broken butterfly[J].International Journal of Bifurcation and Chaos,2014,24(10):1450131.
[5] PISARCHIK A N,FEUDEL U.Control of multistability[J].Physics Reports,2014,540:167-218.
[6] YUAN FANG,WANG GUANGYI,WANG XIAOWEi.Extreme multistability in a memristor-based multi-scroll hyper-chaotic system[J].Chaos,2016,26(7):073107.
[7]李春彪,刘保彬,郑晓晨.一个新的超混沌系统及其线性反馈同步[J].计算机应用研究,2009,26(9):3304-3306.LI C BIAO,LIU B B,ZHENG X C.New hyperchaotic system and its linear feedback synchronization[J].Application Research of Computers,2009,26(9):3304-3306.(Ch).
[8]包伯成,王其红,胡文,等.基于Colpitts振荡器模型的网格涡卷混沌系统[J].四川大学学报(工程科学版),2011,43(3):145-149.BAO B C,WANG Q H,HU W,et al.Grid-scroll chaotic system based on colpitts oscillator model[J].Journal of Sichuan University Engineering Science Edition,2011,43(3):145-149.(Ch).
[9] DUDKOWSKI D,JAFARI S,KAPITANIAK T,et al.Hidden attractors in dynamical systems[J].Physics Reports,2016,637:1-50.
[10] SPROTT J C.A dynamical system with a strange attractor and invariant tori[J].Physics Letters A,2014,378:1361-1363.
[11]包涵,包伯成,林毅,等.忆阻自激振荡系统的隐藏吸引子及其动力学特性研究[J].物理学报,2016,65(18):180503.BAO H,BAO B C,LIN Y,et al.Hidden attractor and its dynamical characteristic in memristive self-oscillating system[J].Acta Phys Sin,2016,65(18):180501(1-12).(Ch).