Chirality-Driven Collective Dynamics in Oscillators with Attractive and Repulsive Couplings

Mechanics Engineering Physics
chirality collective dynamics coupled systems synchronization oscillation death

Background Introduction

Complex systems are ubiquitous in nature, appearing in various forms such as neural networks, social networks, and power grids. Understanding the dynamical transitions within these systems is often achieved through mathematical models, particularly coupled nonlinear oscillators, which exhibit a wide variety of collective behaviors. Chirality, which refers to the coexistence of clockwise and counterclockwise rotational dynamics, plays a critical role in shaping the behavior of coupled systems. However, the role of chirality in systems with competing attractive and repulsive couplings has not been fully explored. To address this, Sathiyadevi Kanagaraj, Premraj Durairaj, and Zhigang Zheng conducted this study to investigate the effects of chirality in globally coupled Stuart-Landau oscillators with attractive and repulsive interactions.

Source of the Paper

This paper was co-authored by Sathiyadevi Kanagaraj, Premraj Durairaj, and Zhigang Zheng, who are affiliated with the Institute of Systems Science and College of Information Science and Engineering, the College of Mechanical Engineering and Automation at Huaqiao University, and the Centre for Nonlinear Systems at Chennai Institute of Technology, respectively. The paper was accepted for publication on February 18, 2025, in the journal Nonlinear Dynamics, with the DOI 10.1007/s11071-025-11030-5.

Research Process and Results

1. Research Model and Experimental Design

The researchers used the Stuart-Landau (SL) oscillator as the research model and introduced global coupling with attractive and repulsive (AR) interactions. In the model, the oscillators’ frequencies were divided into clockwise and counterclockwise groups to simulate chiral effects. The specific model equation is:

$$ \dot{z}_j = (\lambda + i\omega_j - |z_j|^2)zj + \frac{1}{N} \sum{k=1,k \neq j}^N [\epsilon_1(z_k - z_j) - i\epsilon_2(z_k - z_j)] $$

Here, $z_j = x_j + iy_j$ represents the complex variables of the system, $\lambda$ is the control parameter, $\omega_j$ is the frequency, and $\epsilon_1$ and $\epsilon_2$ are the strengths of the attractive and repulsive couplings, respectively.

2. Study of Symmetric and Asymmetric Chirality Effects

The researchers first studied the system with identical frequencies and found that when the distribution of clockwise and counterclockwise frequencies was symmetric, the system transitioned from a mixed synchronization (MS) state to a mixed oscillation death (MOD) state. However, when the frequency distribution was asymmetric, the system exhibited symmetry-breaking clustering behavior, transitioning from a cluster oscillatory state (COS) to a cluster oscillation death state (COD).

3. Impact of Heterogeneous Frequencies

To further investigate the impact of heterogeneous frequencies, the researchers introduced a threshold parameter $\Delta$ to deviate the oscillators’ frequencies. The results showed that frequency heterogeneity induced transitions from a desynchronized state (DS) through a chimera-like state (CL) to a chiral wave state (CW), eventually reaching an oscillation death state (OD). As heterogeneity increased, the regions of desynchronization and chimera-like states expanded, while the region of chiral wave states decreased.

4. Bifurcation Analysis and Quantification

The researchers validated these dynamical transitions through bifurcation analysis. For instance, under symmetric frequency distribution, the system transitioned from oscillatory states to death states via saddle-node bifurcation (SN). Additionally, the researchers used a phase-reduced model and the strength of incoherence (SI) to quantify the observed dynamical states, further verifying the reliability of the results.

5. Validation in Other Systems

To demonstrate the robustness of chimera-like behavior, the researchers also validated these phenomena in globally coupled van der Pol (VDP) oscillators. The results showed that similar dynamical behaviors were reproduced in the VDP system.

Conclusions and Significance

This study reveals the critical role of chirality in oscillators with competing attractive and repulsive couplings. By analyzing the effects of symmetric and asymmetric chirality as well as heterogeneous frequencies, the researchers identified various dynamical transitions, such as from mixed synchronization to oscillation death and from cluster oscillatory states to cluster oscillation death. These findings not only deepen the understanding of chirality-driven dynamics but also provide important references for optimizing network dynamics and developing control strategies in complex systems.

Research Highlights

  1. Systematic Study of Chirality Effects: This is the first comprehensive exploration of the dynamical behaviors of chirality in oscillators with attractive and repulsive couplings.
  2. Impact of Heterogeneous Frequencies: The study reveals how frequency heterogeneity induces complex dynamical transitions from desynchronized states to oscillation death.
  3. Robustness of Chimera-like States: The universality of chimera-like behavior is validated in multiple systems, demonstrating its applicability in different coupled systems.
  4. Bifurcation Analysis and Quantification Methods: Through bifurcation analysis and strength of incoherence quantification, precise descriptions and validations of dynamical transitions are provided.

Research Value

This study offers new perspectives on understanding chiral phenomena in complex systems, with broad applications in biological systems, fluid dynamics, and engineering networks. The findings not only contribute to optimizing network design but also provide a theoretical foundation for developing new control strategies.