D. V. Senthilkumar

From the firing of neurons to the formation of social consensus, the interplay between conformist and contrarian behaviors regulates collective dynamics. We unravel how adaptation timescales induce distinct synchronization behaviors in mixed populations of conformists and contrarians. While slow adaptation leaves contrarians in a persistent state of incoherence, fast adaptation traps them into a scaffold formed by conformists, stabilized by rapid feedback from the evolving network structure, leading to the emergence of synchronized clusters. Notably, contrarians are entrained only within an intermediate coupling window and require a conformist majority. We analytically reduce a low-dimensional description using a collective coordinate approach and demonstrate that a conformist majority and a threshold adaptation rate are necessary conditions for contrarians to synchronize in adaptive networks, which agrees well with numerical simulations. Our results highlight the critical role of adaptation timescales in reshaping collective behaviors in mixed populations, offering insights into consensus formation and polarization in adaptive social networks.

We investigate a three-dimensional Kuramoto model with nonlinear mean-field coupling, where the interaction strength depends on the macroscopic coherence through nonlinear feedback from the global order parameter. In the thermodynamic limit, a self-consistency analysis reveals multiple solution branches, including an additional branch between partially coherent and completely synchronized states, leading to bistability, controlled by the feedback parameters. This is a distinctive feature of the three-dimensional Kuramoto model with nonlinear mean-field coupling, which is absent in the classical three-dimensional Kuramoto model. Using the reduced dynamics based on the Ott-Antonsen ansatz and direct numerical simulation, we show that the system exhibits a two-step synchronization transition. While the transition from incoherence to partial coherence remains discontinuous at zero coupling for all parameter values, the transition from partial coherence to completely synchronized states can be continuous or discontinuous depending upon the feedback parameters. In the discontinuous regime, the coexistence of steady states leads to hysteresis. The observed hysteresis remains robust over the range of system sizes investigated. We further show that the hysteresis begins only above a threshold value of the feedback strength. These results demonstrate that nonlinear adaptation of the coupling provides a mechanism for controlling synchronization transitions and bistability in higher-dimensional systems.

We investigate a variant of coupled Kuramoto oscillators in which the effective coupling is modulated by the global coherence in a nonlinear fashion. In the thermodynamic limit, a self-consistency analysis reveals the manifestation of coexisting incoherent and coherent states in a finite range of the coupling strength, resulting in the bistability, the width of which is controlled by the feedback strength and the degree of the nonlinearity in the order parameter. For finite populations, intrinsic fluctuations fundamentally reshape this scenario: the synchronization transition manifests even at coupling strengths below the thermodynamic threshold. Strong feedback in combination with weak nonlinearity accelerates the onset of coherence at smaller coupling strengths, whereas strong feedback together with pronounced nonlinearity significantly enlarges the bistable region. Notably, this trend contrasts with the thermodynamic limit, where increasing nonlinearity suppresses the bistability. As the system size increases, the critical coupling strength approaches the thermodynamic-limiting value. Stronger nonlinear feedback facilitates the onset of the coherence closer to the thermodynamic critical coupling even for relatively smaller system sizes, thereby steering both the nature of the transition and its finite-size scaling. In addition, the theoretical estimates of the order parameter using the Ott-Antonsen ansatz and the finite-size scaling also corroborate the observed finite-size effects.

The classical Kuramoto oscillator typically involves a single scalar state variable representing the phase of each oscillator. In contrast, swarmalators, an extension of the framework of Kuramoto oscillators, incorporate two coupled variables per unit: one describing spatial position and the other representing phase or orientation. Depending on the application, these state variables may differ in their nature and interpretation. However, many natural systems involve interactions among three or more coupled state variables, raising the question of how to effectively describe their collective dynamics. In this work, we introduce an oscillator model that accommodates at least three mutually coupled state variables, each of which evolves on a circle. Consequently, the dynamics of the system are constrained to a 3-torus manifold (T^{3}=S^{1}×S^{1}×S^{1}) that allows analytical treatment of the distinct collective states. This framework reveals a variety of collective behaviors and offers broad applicability to systems in sociology, collective animal motion, neuroscience, and beyond.

We examine a nontrivial generalization of the coupled Kuramoto model with both symmetry-breaking interaction and time delay coupling. We show that this model exhibits three unique dynamical behaviors: incoherent state, stationary synchronized state, and standing wave state. In contrast to the symmetry-preserving coupling, time delay in the symmetry-breaking interaction results in three multistability regions. We elucidate that the latter two bistable regions emerge only in the presence of time delay in the symmetry-breaking coupling. The stability of the stationary synchronized state remains unaffected by the time delay interaction in the system. We deduce the low-dimensional equations for the complex order parameters using the Ott-Antonsen ansatz. We also deduce the Hopf, pitchfork, and saddle-node bifurcation curves from the evolution equations for the complex order parameters mediating the dynamical transitions. Simulation results of the original discrete set of equations for the generalized Kuramoto model agree well with the analytical bifurcation curves.