Classical open systems with balanced gain and loss, i.e., parity-time ($\mathcal{PT}$) symmetric systems, have attracted tremendous attention over the past decade. Their exotic properties arise from exceptional point degeneracies of non-Hermitian Hamiltonians that govern their dynamics. In recent years, increasingly sophisticated models of $\mathcal{PT}$ symmetric systems with time-periodic (Floquet) driving, time-periodic gain and loss, and time-delayed coupling have been investigated, and such systems have been realized across numerous platforms comprising optics, acoustics, mechanical oscillators, optomechanics, and electrical circuits. Here, we introduce a $\mathcal{PT}$ symmetric (balanced gain and loss) system with memory and investigate its dynamics analytically and numerically. Our model consists of two coupled $LC$ oscillators with positive and negative resistance, respectively. We introduce memory by replacing either the resistor with a memristor, or the coupling inductor with a meminductor, and investigate the circuit energy dynamics as characterized by $\mathcal{PT}$ symmetric or $\mathcal{PT}$ symmetry broken phases. Due to the resulting nonlinearity, we find that energy dynamics depend on the sign and strength of initial voltages and currents, as well as the distribution of initial circuit energy across its different components. Surprisingly, at strong inputs, the system exhibits self-organized Floquet dynamics, including a $\mathcal{PT}$ symmetry broken phase at vanishingly small dissipation strength. Our results indicate that $\mathcal{PT}$ symmetric systems with memory show a rich landscape.

中文翻译：

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带存储器的奇偶时间对称系统

具有平衡损益的经典开放系统，即奇偶时间（$\mathcal{PT}$）对称系统在过去十年中引起了极大的关注。它们的奇异特性源于控制它们动力学的非赫米特哈密顿量的异常点退化。近年来，越来越复杂的模型$\mathcal{PT}$已经研究了具有时间周期（Floquet）驱动，时间周期增益和损耗以及时间延迟耦合的对称系统，并且已经在包括光学，声学，机械振荡器，光机和电路的众多平台上实现了这种系统。在这里，我们介绍一个$\mathcal{PT}$具有内存的对称（平衡增益和损耗）系统，并通过分析和数值研究其动力学。我们的模型包括两个耦合$\mathrm{大号}C$分别具有正电阻和负电阻的振荡器。我们通过用忆阻器代替电阻器或用忆阻器代替耦合电感器来引入存储器，并研究电路能量动态，其特征在于$\mathcal{PT}$ 对称或 $\mathcal{PT}$对称断相。由于产生的非线性，我们发现能量动力学取决于初始电压和电流的符号和强度，以及初始电路能量在其不同组件上的分布。出乎意料的是，在强大的输入下，该系统展现出自组织的Floquet动力学，包括$\mathcal{PT}$对称的破碎相，消散强度几乎消失了。我们的结果表明$\mathcal{PT}$ 具有内存的对称系统显示出丰富的景观。