Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known. Let $\triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $\triangle(1, 2, 2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments. This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.

Many natural systems exhibit oscillations that show sizeable fluctuations in frequency and amplitude. This variability can arise from a wide variety of physical mechanisms. Phase descriptions that work for deterministic oscillators have a limited applicability for stochastic oscillators. In my talk I review attempts to generalize the phase concept to stochastic oscillations, specifically, the mean-return-time phase and the asymptotic phase. For stochastic systems described by Fokker-Planck and Kolmogorov-backward equations, I introduce a mapping of the system’s variables to a complex pointer (instead of a real-valued phase) that is based on the eigenfunction of the Kolmogorov equation. Under the new (complex-valued) description, the statistics of the oscillator’s spontaneous activity, of its response to external perturbations, and of the coordinated activity of (weakly) coupled oscillators, is brought into a universal and greatly simplified form. The theory is tested for three theoretical models of noisy oscillators arising from fundamentally different mechanisms: a damped harmonic oscillator with dynamical noise, a fluctuation-perturbed limit-cycle system, and an excitable system in which oscillations require noise to occur.

Spontaneous rhythmic oscillations are widely observed in real-world systems. Synchronized rhythmic oscillations often provide important functions for biological or engineered systems. One of the useful theoretical methods for analyzing rhythmic oscillations is the phase reduction theory for weakly perturbed limit-cycle oscillators, which systematically gives a low-dimensional description of the oscillatory dynamics using only the asymptotic phase of the oscillator. Recent advances in Koopman operator theory provide a new viewpoint on phase reduction, yielding an operator-theoretic definition of the classical notion of the asymptotic phase and, moreover, of the amplitudes, which characterize distances from the limit cycle. This led to the generalization of classical phase reduction to phase-amplitude reduction, which can characterize amplitude deviations of the oscillator from the unperturbed limit cycle in addition to the phase along the cycle in a systematic manner. In the talk, these theories are briefly reviewed and then applied to several examples of synchronizing rhythmic systems, including biological oscillators, networked dynamical systems, and rhythmic spatiotemporal patterns.

Spontaneous rhythmic oscillations are widely observed in real-world systems. Synchronized rhythmic oscillations often provide important functions for biological or engineered systems. One of the useful theoretical methods for analyzing rhythmic oscillations is the phase reduction theory for weakly perturbed limit-cycle oscillators, which systematically gives a low-dimensional description of the oscillatory dynamics using only the asymptotic phase of the oscillator. Recent advances in Koopman operator theory provide a new viewpoint on phase reduction, yielding an operator-theoretic definition of the classical notion of the asymptotic phase and, moreover, of the amplitudes, which characterize distances from the limit cycle. This led to the generalization of classical phase reduction to phase-amplitude reduction, which can characterize amplitude deviations of the oscillator from the unperturbed limit cycle in addition to the phase along the cycle in a systematic manner. In the talk, these theories are briefly reviewed and then applied to several examples of synchronizing rhythmic systems, including biological oscillators, networked dynamical systems, and rhythmic spatiotemporal patterns.