Emergence of solitary and chimera states in adaptive pendulum networks under diverse learning rules

Abstract

We investigate the interplay between phase lag and adaptive learning rules in a network of identical pendulum oscillators, where the coupling strengths evolve dynamically in response to the oscillators' states. Specifically, we examine two biologically inspired adaptation mechanisms, Hebbian and spike-timing-dependent plasticity (STDP), and their influence on the emergence of collective dynamical patterns. Under Hebbian adaptation, the network exhibits a wide range of organized behaviors, including two-cluster, solitary, multi-antipodal, and chimera states. In contrast, STDP coupling induces splay, splay-cluster, and splay-chimera configurations. Importantly, we find that the solitary state arises spontaneously in this adaptive network without requiring delays, nonlocal coupling, or external perturbations; instead, it is induced purely by variations in the phase-lag parameter. To the best of our knowledge, such delay-free and symmetry-preserving emergence of solitary behavior has not been reported previously in adaptive oscillator systems. To systematically characterize the resulting dynamical transitions, we employ two complementary incoherence measures based on the local standard deviation of (i) time-averaged frequencies and (ii) instantaneous phases across spatial bins, enabling the construction of detailed two-parameter phase diagrams. Analytical stability analysis of the two-cluster state shows strong agreement with numerical simulations, revealing regions of pronounced multistability. These findings establish adaptive pendulum networks as a minimal yet powerful framework for studying self-organized synchronization, chimera formation, and multistable transitions driven by diverse adaptation mechanisms.

Figures (14)

• Figure 1: Evolution of coupling strengths under different adaptation rules characterized by the control parameter $\beta$. (a) Hebbian adaptation for $\beta = -\pi/2$, (b) STDP for $\beta = 0$, and (c) anti-Hebbian adaptation for $\beta = \pi/2$.
• Figure 2: (a) Coupling matrix $k_{ij}$ corresponding to the two cluster state. (b) Phase portrait of the instantaneous phases. (c) Snapshot of the instantaneous phases and (d) averaged frequency profile. (e) Space-time evolution of the TC state for $\alpha = 0.05\pi$. The other parameters are fixed as $\nu = 1.0$, $\epsilon = 0.01$, and $\beta = -0.5\pi$.
• Figure 3: (a) Coupling matrix $k_{ij}$ corresponding to the solitary state. (b) Phase portrait of the instantaneous phases. (c) Snapshot of the instantaneous phases and (d) averaged frequency profile. (e) Space-time evolution of the SS state for $\alpha = 0.12\pi$.
• Figure 4: (a) Coupling matrix $k_{ij}$ corresponding to the multi-antipodal cluster state. (b) Phase portrait of the instantaneous phases. (c) Snapshot of the instantaneous phases and (d) averaged frequency profile. (e) Space-time evolution of the MAC state for $\alpha = 0.25\pi$.
• Figure 5: (a) Coupling matrix $k_{ij}$ corresponding to the chimera state. (b) Phase portrait of the instantaneous phases. (c) Snapshot of the instantaneous phases and (d) averaged frequency profile. (e) Space-time evolution of the CHI state for $\alpha = 0.46\pi$.