Trend to equilibrium and diffusion limit for the inertial Kuramoto-Sakaguchi equation

In this paper, we study the inertial Kuramoto-Sakaguchi equation for interacting oscillatory systems. On the one hand, we prove the convergence toward corresponding phasehomogeneous stationary states in weighted Lebesgue norm sense when the coupling strength is small enough. In [10], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof. On the other hand, we investigate the diffusion limit when all oscillators are identical. In [19], authors studied the same problem using an energy estimate, renormlized solutions and a compactness method, through which error estimates could not be discussed. Here we provide error estimates for the diffusion limit with respect to the mass m $\ll$ 1 using a simple proof by imposing slightly more regularity on the solution

Collective behaviors of the Lohe hermitian sphere model with inertia

We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength

Emergent dynamics of various Cucker–Smale type models with a fractional derivative

In this paper, we demonstrate emergent dynamics of various Cucker–Smale type models, especially standard Cucker–Smale (CS), thermodynamic Cucker–Smale (TCS), and relativistic Cucker–Smale (RCS) with a fractional derivative in time variable. For this, we adopt the Caputo fractional derivative as a widely used standard fractional derivative. We first introduce basic concepts and previous properties based on fractional calculus to explain its unusual aspects compared to standard calculus. Thereafter, for each proposed fractional model, we provide several sufficient frameworks for the asymptotic flocking of the proposed systems. Unlike the flocking dynamics which occurs exponentially fast in the original models, we focus on the flocking dynamics that occur slowly at an algebraic rate in the fractional systems

Measure-valued death state and local sensitivity analysis for Winfree models with uncertain high-order couplings

We study the measure-valued death state and local sensitivity analysis of the Winfree model and its mean-field counterpart with uncertain high-order couplings. The Winfree model is the first mathematical model for synchronization, and it can cast as the effective approximation of the pulse-coupled model for synchronization, and it exhibits diverse asymptotic patterns depending on system parameters and initial data. For the proposed models, we present several frameworks leading to oscillator death in terms of system parameters and initial data, and the propagation of regularity in random space. We also present several numerical tests and compare them with analytical results

A Study on Korean Art Song Tradition with Emphasis on Vocal Works by Female Composers: Hanbyeol Kang, Wonju Lee, Soon-Ae Kim, and JaeEun Park

The Korean art song, Gagok (가곡), a genre rooted in the traditional poetic form Sijo and influenced by Western hymnals, has evolved significantly throughout the 20th and 21st centuries. English hymnals were introduced to late nineteenth-century Korea by the missionaries Horace G. Underwood (1859–1916) from a Northern Presbyterian mission and Henry G. Appenzeller (1858–1902) from a Northern Methodist mission. Throughout the period of Japanese rule over Korea from 1910 to 1945, Changga (창가) emerged as a form of secular song that aimed to promote the ideology of civilization development and independence while fostering a sense of patriotism. Public interest in art songs increased through 1960 with larger concerts, private recitals, recordings, and broadcasts. With the establishment of the Korean Composer Association (대한 음악가 협회) in the early 1960s, art song composition became more prevalent. Composers in the 20th and early 21st centuries started composing songs in their unique style.This study examines the works of four female composers– Hanbyeol Kang (b.1993), Wonju Lee (b.1979), Soon-Ae Kim (1920–2007), and JaeEun Park (b.1956)– to explore how they have used art songs to reflect and challenge the historical and social contexts of Korea. New beliefs, internal strife, and social upheavals in Korean culture influenced their cultural and artistic innovation in music. Through a feminist lens, this study analyzes how these composers’ works have contributed to a broader understanding of Korean women’s experiences and struggles. Drawing inspiration from female poets of various eras, these composers offered a unique perspective on the complexities of gender roles and societal expectations. Female composers in the 20th and 21st centuries have been influenced by Korean female poets, creating music that reflects historical events. Those poetic works are of female poets from the Joseon Dynasty, such as Nanseolheon Huh (1563–1589) and Maechang Yi (1573–1610), as well as modern poets, such as Namjo Kim (1927–2003) and Jeonghee Go (1948–1991). The four composers incorporated Korean cultural elements into their vocal works, inspired by the works of Korean female poets and traditional musical elements. Their poetry frequently delved into the complex issues of female social status in the Joseon Dynasty, and the broader social and political upheavals of the Korean War and the Gwangju Democratization Movement. Furthermore, this study provides an in-depth exploration of selected art songs and overview of Korean Art songs. It explores distinctive textual elements, traditional instrumentation, and cultural influences that shaped Korean art song as a genre. The study delves into the use of unique Korean textual elements, traditional instruments, and cultural influences. It explores distinctive textual elements, traditional instrumentation, and cultural influences that shaped Korean art song as a genre. The study delves into the use of unique Korean textual elements, traditional instruments, and cultural influences

Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia

Kang M, Rehmeier M. Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia. Communications in Mathematical Sciences . 2023;21(7):1875-1894.We study the emergence of phase-locking for Winfree oscillators under the effect of inertia. It is known that in a large coupling regime, oscillators governed by the deterministic secondorder Winfree model with inertia converge to a unique equilibrium. In contrast, in this paper we show the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study the effect of a new stochastically perturbed Winfree system with multiplicative noise and obtain lower estimates in probability for the pathwise emergence of such a synchronizing pattern, provided the noise is sufficiently small. We also provide numerical simulations which hint at the possibility of more general and stronger analytical results