Recently,large-scale trapped ion systems have been realized in experiments for quantum simulation and quantum computation.They are the simplest systems for dynamical stability and parametric resonance.In this model,th...Recently,large-scale trapped ion systems have been realized in experiments for quantum simulation and quantum computation.They are the simplest systems for dynamical stability and parametric resonance.In this model,the Mathieu equation plays the most fundamental role for us to understand the stability and instability of a single ion.In this work,we investigate the dynamics of trapped ions with the Coulomb interaction based on the Hamiltonian equation.We show that the many-body interaction will not influence the phase diagram for instability.Then,the dynamics of this model in the large damping limit will also be analytically calculated using few trapped ions.Furthermore,we find that in the presence of modulation,synchronization dynamics can be observed,showing an exchange of velocities between distant ions on the left side and on the right side of the trap.These dynamics resemble that of the exchange of velocities in Newton's cradle for the collision of balls at the same time.These dynamics are independent of their initial conditions and the number of ions.As a unique feature of the interacting Mathieu equation,we hope this behavior,which leads to a quasi-periodic solution,can be measured in current experimental systems.Finally,we have also discussed the effect of anharmonic trapping potential,showing the desynchronization during the collision process.It is hoped that the dynamics in this many-body Mathieu equation with damping may find applications in quantum simulations.This model may also find interesting applications in dynamics systems as a pure mathematical problem,which may be beyond the results in the Floquet theorem.展开更多
基于开闭环控制的思想,设计了一类由外激力与线性误差反馈组成的开闭环控制器,研究了Mathieu-Duffing振子混沌轨道至任意目标周期轨道的控制问题;同时,利用Liapunov稳定性理论与二阶常微分方程初值问题的一个比较定理,证明了上述开闭环...基于开闭环控制的思想,设计了一类由外激力与线性误差反馈组成的开闭环控制器,研究了Mathieu-Duffing振子混沌轨道至任意目标周期轨道的控制问题;同时,利用Liapunov稳定性理论与二阶常微分方程初值问题的一个比较定理,证明了上述开闭环控制夹带盆(basin of entrainment)的全局性.最后,利用数值模拟,验证了理论结果的正确性.展开更多
为了研究大幅值参数激励M ath ieu方程,通过引入新的变换,将强参数激励系统转化为弱参数激励系统,利用改进的变形参数法求解无阻尼项的M ath ieu方程,分别得到具有π和2π方程的周期解和相应的过渡曲线,并利用过渡曲线得到方程的不稳定...为了研究大幅值参数激励M ath ieu方程,通过引入新的变换,将强参数激励系统转化为弱参数激励系统,利用改进的变形参数法求解无阻尼项的M ath ieu方程,分别得到具有π和2π方程的周期解和相应的过渡曲线,并利用过渡曲线得到方程的不稳定区域和稳定域.结果表明,该方法与文献[7]得到的过渡曲线基本吻合,通过数值法验证了用该方法得到稳定域比文献[7]的结果更为合理.展开更多
本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mat...本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应,得到了其非线性频率响应曲线;再利用Floquet理论对周期解进行稳定性分析,得到了两种类型的分岔及它们的位置.然后,基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的,另一个基频事先是未知的),推导了相应的两时间尺度IHB法,精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值,尤其在临界点附近处的准周期运动响应.得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合.最后,研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.展开更多
基金supported by the Innovation Program for Quantum Science and Technology(Grant Nos.2021ZD0301200,2021ZD0303200,and 2021ZD0301500)the Alliance of International Science Organizations(ANSO)。
文摘Recently,large-scale trapped ion systems have been realized in experiments for quantum simulation and quantum computation.They are the simplest systems for dynamical stability and parametric resonance.In this model,the Mathieu equation plays the most fundamental role for us to understand the stability and instability of a single ion.In this work,we investigate the dynamics of trapped ions with the Coulomb interaction based on the Hamiltonian equation.We show that the many-body interaction will not influence the phase diagram for instability.Then,the dynamics of this model in the large damping limit will also be analytically calculated using few trapped ions.Furthermore,we find that in the presence of modulation,synchronization dynamics can be observed,showing an exchange of velocities between distant ions on the left side and on the right side of the trap.These dynamics resemble that of the exchange of velocities in Newton's cradle for the collision of balls at the same time.These dynamics are independent of their initial conditions and the number of ions.As a unique feature of the interacting Mathieu equation,we hope this behavior,which leads to a quasi-periodic solution,can be measured in current experimental systems.Finally,we have also discussed the effect of anharmonic trapping potential,showing the desynchronization during the collision process.It is hoped that the dynamics in this many-body Mathieu equation with damping may find applications in quantum simulations.This model may also find interesting applications in dynamics systems as a pure mathematical problem,which may be beyond the results in the Floquet theorem.
文摘基于开闭环控制的思想,设计了一类由外激力与线性误差反馈组成的开闭环控制器,研究了Mathieu-Duffing振子混沌轨道至任意目标周期轨道的控制问题;同时,利用Liapunov稳定性理论与二阶常微分方程初值问题的一个比较定理,证明了上述开闭环控制夹带盆(basin of entrainment)的全局性.最后,利用数值模拟,验证了理论结果的正确性.
基金Supported by the National Nature Science Foundation of China(51079097)Top Discipline Projects of Shanghai Municipal Education CommissionThe Innovation Projects of 2013 Shanghai Postgraduate Education(the second,20131129)
文摘为了研究大幅值参数激励M ath ieu方程,通过引入新的变换,将强参数激励系统转化为弱参数激励系统,利用改进的变形参数法求解无阻尼项的M ath ieu方程,分别得到具有π和2π方程的周期解和相应的过渡曲线,并利用过渡曲线得到方程的不稳定区域和稳定域.结果表明,该方法与文献[7]得到的过渡曲线基本吻合,通过数值法验证了用该方法得到稳定域比文献[7]的结果更为合理.
文摘本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应,得到了其非线性频率响应曲线;再利用Floquet理论对周期解进行稳定性分析,得到了两种类型的分岔及它们的位置.然后,基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的,另一个基频事先是未知的),推导了相应的两时间尺度IHB法,精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值,尤其在临界点附近处的准周期运动响应.得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合.最后,研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.
