Related papers: Augmented Phase Reduction for Periodic Orbits Near…
We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic…
The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The…
Networks of coupled nonlinear oscillators can display a wide range of emergent behaviours under variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by…
The goal of this paper is to show how to produce a piece of rigorous bifurcation diagram of periodic orbits for an ODE. We study the Rossler system, one of the textbook examples of ODEs generating nontrivial dynamics, for the parameter…
We have found a way for penetrating the space of the dynamical systems towards systems of arbitrary dimension exhibiting the nonlinear mixing of a large number of oscillation modes through which extraordinarily complex time evolutions…
Biological oscillators coordinate individual cellular components so that they function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch is unavoidable in biological implementations. We…
There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology,…
Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the…
We introduce a simple model system to study synchronization theoretically in quantum oscillators that are not just in limit-cycle states, but rather display a more complex bistable dynamics. Our oscillator model is purely dissipative, with…
The dynamics of a one-degree of freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of…
We show that a ring of phase oscillators coupled with transmission delays can be used as a pattern recognition system. The introduced model encodes patterns as stable periodic orbits. We present a detailed analysis of the underlying…
Controlling rhythmic systems, typically modeled as limit-cycle oscillators, is an important subject in real-world problems. Phase reduction theory, which simplifies the multidimensional oscillator state under weak input to a single phase…
We formulate a phase-reduction method for a general class of noisy limit cycle oscillators and find that the phase equation is parametrized by the ratio between time scales of the noise correlation and amplitude relaxation of the limit…
In this paper, the dynamical heteroclinic orbit and attractor have been employed to make the late-time behaviors of the model insensitive to the initial condition and thus alleviates the fine tuning problem in cosmological dynamical system…
We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of…
We study the evolution of an oscillator interacting via the most general bilinear coupling (with time-independent coefficients) with an ``environment'' consisting of a set of other harmonic oscillators. We are mainly interested in a…
We present a phase-amplitude reduction framework for analyzing collective oscillations in networked dynamical systems. The framework, which builds on the phase reduction method, takes into account not only the collective dynamics on the…
We present two calculations for a class of robust homoclinic cycles with symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal. Firstly, we compute optimal conditions…
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but…
Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator…