Non-Hermitian Global Synchronization

Abstract

Synchronization of coupled nonlinear oscillators is a prevalent phenomenon in natural systems and can play important roles in various fields of modern science, such as laser arrays and electric networks. However, achieving robust global synchronization has always been a significant challenge due to its extreme susceptibility to initial conditions and structural perturbations. Here, a novel approach is presented to achieve robust global synchronization by manipulating the interplay between non-Hermitian physics and nonlinear dynamics. Remarkably, the initial-state-independent non-Hermitian skin and topological global synchronization are proposed, exhibiting diverse anomalous effects such as the enlarged-size triggered non-Hermitian global synchronization and nonlinear skin states-dominated global synchronization. To validate the findings, nonlinear topoelectrical circuits for experimental observation of non-Hermitian global synchronization are designed and fabricated. The work opens up a promising avenue for establishing resilient global synchronization with potential applications in constructing high-radiance laser arrays and topologically synchronized networks.

Keywords: nonlinear synchronization; non‐Hermitian skin effects; topoelectrical circuits; topological states.

Figure 1

Theoretical results of non‐Hermitian skin global synchronization. a). The scheme of the tight‐binding lattice model for realizing non‐Hermitian skin global synchronization. The Stuart‐Landau oscillator is added at each lattice site, and the non‐reciprocal coupling is used to couple nearest neighbored lattice sites. The right chart displays the wave evolution of a single Stuart‐Landau oscillator. (b1) and (b2). Numerical results of wave dynamics for the anti‐phase and in‐phase synchronized states. Black and red lines plot the real part of wave amplitudes at odd and even sites. Two insets present the locally enlarged views. It is noted that there are indeed 8 red lines, where two pairs of red lines are nearly overlapping. (c1). Numerical results of the linear eigen‐spectrum accomplished by IPR of each eigenstate. (c2). Spatial profiles of linear eigenstates for the non‐reciprocal chain. Black and red lines correspond to linear eigenstates with minimum IPR. (d1)‐(d2) and (e1)‐(e2) Numerical results of frequency spectra and spatial profiles of anti‐phase and in‐phase synchronized states (at the time marked by blue lines in (b1)‐b(2)). f). The variation of the order‐parameter R_o as a function of the non‐reciprocal coupling strength J ₋/J ₊ with N  =  15. The red block marks the region sustaining non‐Hermitian linear skin synchronization. g). The numerical result of R_o as a function of the lattice size N with a fixed non‐reciprocal strength (J ₊ =  1.5,   J ₋ =  1), where red and green blocks correspond to the regions of non‐Hermitian linear skin‐state synchronization and nonlinear skin‐state synchronization, respectively. Here, the parameters related Stuart‐Landau oscillator are set as ω₀ =  0.1, α  =  5e ⁻³, β  =  5e ⁻⁴.

Figure 2

The influence of disorder on non‐Hermitian skin synchronization. a–f). The probabilities for the appearance of non‐Hermitian global synchronization as a function of the system size with the disorder strength being W  =  0, 0.01ω₀, 0.1ω₀, 0.4ω₀, 0.7ω₀ and ω₀. Other parameters are identical to that used in Figure 1g.

Figure 3

Theoretical results of non‐Hermitian topological global synchronization. a). The lattice model for the realization of non‐Hermitian topological global synchronization. The reciprocal and non‐reciprocal couplings are used for intercell and intracell couplings. The Stuart‐Landau oscillators are added at each “A” and “B” sublattices. b–e) Numerical results of the spatial profile of each linear eigenstate with J = 0.2, 0.35, 0.5, and 1.0. (f1)‐(i1). Numerical results of wave dynamics of all oscillators with the intercell coupling being J  =  0.2, 0.35, 0.5, and 1.0, respectively. The blue vertical dashed lines mark the time that we used to plot the spatial profiles in Figures (f3)‐(i3). (f2)‐(i2) and (f3)‐(i3) present the corresponding frequency spectra of all oscillators and steady‐state spatial distributions of the system, respectively. j). Numerical results for the variation of order parameter R_o as a function of the intercell coupling strength J with the lattice length being N  =  25. (k). Numerical results for the variation of order parameter R_o as a function of the lattice length N with the intercell coupling strength being J  =  0.4. Here, other parameters are set as J ₊ =  0.56, J ₋ =  0.1, ω₀ =  0.1, α  =  5e⁻⁴, and β  =  5e⁻⁵. The appearances of the non‐Hermitian linear skin‐state synchronization, nonlinear skin‐state synchronization and topological global synchronization are highlighted in purple, green and red blocks, respectively.

Figure 4

Experimental results of non‐Hermitian linear skin‐state synchronization in electric circuits. a). The schematic diagram of the designed electric circuit to simulate the non‐Hermitian linear skin‐state synchronization. An effective lattice site is formed by four circuit nodes connected by nonreciprocal resistances ± R_w , with each node grounded by a nonlinear Chua diode to act as the Stuart‐Landau oscillator. Non‐reciprocal coupling is achieved by crossly connecting adjacent circuit nodes using non‐reciprocal resistance R ₂ ± R ₁ to realize the non‐reciprocal coupling. b). A photograph image of two‐coupled sites in the fabricated circuit sample for simulating non‐Hermitian skin synchronization is shown, along with a bottom chart plotting the photo of a nonlinear Chua diode. c–e). Measured voltage signals, the FT frequency spectra, and the steady‐state voltage profile of the circuit sample with N = 5. The multi‐frequency dynamical evolution is observed. f–h). Experimental results of the voltage evolution, the FT frequency spectra, and the steady‐state voltage profile in the circuit sample with N = 9. The non‐Hermitian linear skin‐state synchronization is realized. Other circuit parameters are set as R_w =  20 kΩ, R ₁ =  0.8 kΩ, R ₂ =  4 kΩ, C  =  100 nF, R ₃ =  20 kΩ, R ₄ =  10 kΩ, and R ₅ =  90 kΩ.

Figure 5

Experimental results of non‐Hermitian topological global synchronization in electric circuits. a). The schematic diagram illustrates the implementation of a single unit cell in an electric circuit, featuring two sublattices, non‐reciprocal intercell coupling, and reciprocal intracell coupling. Two bottom insets present the grounding of two types of sublattices. This setup is designed to simulate non‐Hermitian topological global synchronization. b). The photograph image of a single unit cell in the fabricated circuit, corresponding to the circuit diagram in (a). c) and f). Measured voltage signals of all nodes in two circuit samples with N  =  9 and 15. d) and e). The experimental results of FT frequency spectra and distribution of steady‐state voltage profiles in the short‐length circuit with N  =  9. g) and h) The measured FT frequency spectra and steady‐state voltage profile in the long‐length circuit with N  =  15. It is observed that non‐Hermitian topological global synchronization exists in the circuit sample with a larger size. Here, other parameters are set as R_w =  20 kΩ, R ₁ =  3.03 kΩ, R ₂ =  4.35 kΩ, R  =  2 kΩ, C  =  1 nF, R ₃ =  200 kΩ, R ₄ =  70 kΩ, and R ₅ =  630 kΩ.