Traveling waves in fully coupled networks of linear oscillators (Q2116201)

The authors study the existence and stability of travelling-wave-type solutions in fully coupled networks of nonlinear oscillators. To that end, they formulate an auxiliary system of delay equations, and they state conditions for the occurrence of stable periodic cycles therein, thus reducing the question of stability to the location of multipliers of the corresponding linearisation. In particular, the authors show that it suffices to consider the existence and stability of a family of canonical travelling waves in order to determine analogous properties for all travelling-type solutions induced by those. In sum, they hence develop a general methodology which generalises the approach proposed previously for unidirectionally coupled oscillators [\textit{S.D. Glyzin}, et al., Theor. Math. Phys. 175, No. 1, 499--517; translation from Teor. Mat. Fiz. 175, No. 1, 62--83 (2013; Zbl 1364.34058)]. To showcase their methodology, the authors consider a general system of weakly coupled oscillators which allows for the asymptotic verification of their stability conditions, by perturbation off cycles of the corresponding uncoupled nonlinear oscillator. To verify that these conditions can, in fact, be fulfilled, they present the specific example of a planar system of weakly coupled oscillators which can exhibit coexistence of stable cycles, or ``buffering''. Finally, the authors study numerically the structure of attractors in fully coupled networks of quasi-harmonic oscillators in parameter regimes where all travelling-wave-type solutions are unstable. They consider an example that admits explicit travelling wave solutions, and they show that either a chaotic stationary regime or an invariant torus may arise in such regimes.