Stability Analysis of the Helmholtz Oscillator with Time Varying Mass

Stability of phase difference trajectories of networks of Kuramoto oscillators with time-varying couplings and intrinsic frequencies

We study dynamics of phase-differences (PDs) of coupled oscillators where both the intrinsic frequencies and the couplings vary in time. In case the coupling coefficients are all nonnegative, we prove that the PDs are asymptotically stable if there exists T>0 such that the aggregation of the time-varying graphs across any time interval of length $T$ has a spanning tree. We also consider the situation that the coupling coefficients may be negative and provide sufficient conditions for the asymptotic stability of the PD dynamics. Due to time-variations, the PDs are asymptotic to time-varying pat…

Weakly Coupled Oscillators in a Slowly Varying World

We extend the theory of weakly coupled oscillators to incorporate slowly varying inputs and parameters. We employ a combination of regular perturbation and an adiabatic approximation to derive equations for the phase-difference between a pair of oscillators. We apply this to the simple Hopf oscillator and then to a biophysical model. The latter represents the behavior of a neuron that is subject to slow modulation of a muscarinic current such as would occur during transient attention through cholinergic activation. Our method extends and simplifies the recent work of Kurebayashi (Physical Revi…

Entrainment to Subharmonic Trajectories in Oscillatory Discrete-Time Systems

A matrix $A$ is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix $A$ is called oscillatory if it is TN and some power of $A$ is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time $k$ is oscillatory. We analyze the properties of $n$-dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-…

Novel Synchronization of Pulse-Coupled Oscillators on Time-Varying Networks

In this article I investigate the novel synchronization behaviors of evolving pulse-coupled oscillator networks. Unlike previous models, the time-varying mechanism is inspired by neural network development, where seldom used links die out while heavily used ones get strengthened. Even small network with all-to-all connected topology can have exotic dynamics under this circumstance. The oscillators can coevolve with the network, and finally form a co-synchronized pair. Or if we only allow a fraction of oscillators to send pulses every time, the oscillators can synchronize with the network remai…

A stiction oscillator under slowly varying forcing: Uncovering small scale phenomena using blowup

In this paper, we consider a mass-spring-friction oscillator with the friction modelled by a regularized stiction model in the limit where the ratio of the natural spring frequency and the forcing frequency is on the same order of magnitude as the scale associated with the regularized stiction model. The motivation for studying this situation comes from \cite{bossolini2017b} which demonstrated new friction phenomena in this regime. The results of Bossolini et al 2017 led to some open problems, that we resolve in this paper. In particular, using GSPT and blowup we provide a simple geometric des…

时变高阶结构上的全局同步

Global synchronization on time-varying higher-order structures

本文研究时变高阶结构对全局同步的影响，通过扩展主稳定性形式体系，纳入高阶结构时间演化带来的额外贡献，并在Stuart-Landau振荡器和Lorenz混沌振荡器上验证了理论的有效性。