skip to main content

Title: Traveling spiral wave chimeras in coupled oscillator systems: emergence, dynamics, and transitions

Systems of coupled nonlinear oscillators often exhibit states of partial synchrony in which some of the oscillators oscillate coherently while the rest remain incoherent. If such a state emerges spontaneously, in other words, if it cannot be associated with any heterogeneity in the system, it is generally referred to as a chimera state. In planar oscillator arrays, these chimera states can take the form of rotating spiral waves surrounding an incoherent core, resembling those observed in oscillatory or excitable media, and may display complex dynamical behavior. To understand this behavior we study stationary and moving chimera states in planar phase oscillator arrays using a combination of direct numerical simulations and numerical continuation of solutions of the corresponding continuum limit, focusing on the existence and properties of traveling spiral wave chimeras as a function of the system parameters. The oscillators are coupled nonlocally and their frequencies are drawn from a Lorentzian distribution. Two cases are discussed in detail, that of a top-hat coupling function and a two-parameter truncated Fourier approximation to this function in Cartesian coordinates. The latter allows semi-analytical progress, including determination of stability properties, leading to a classification of possible behaviors of both static and moving chimera states. The transition from stationary to moving chimeras is shown to be accompanied by the appearance of complex filamentary structures within the incoherent spiral wave core representing secondary coherence regions associated with temporal resonances. As the parameters are varied the number of such filaments may grow, a process reflected in a series of folds in the corresponding bifurcation diagram showing the drift speedsas a function of the phase-lag parameterα.

more » « less
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
IOP Publishing
Date Published:
Journal Name:
New Journal of Physics
Medium: X Size: Article No. 103023
["Article No. 103023"]
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Chimera states, or coherence–incoherence patterns in systems of symmetrically coupled identical oscillators, have been the subject of intensive study for the last two decades. In particular it is now known that the continuum limit of phase-coupled oscillators allows an elegant mathematical description of these states based on a nonlinear integro-differential equation known as the Ott–Antonsen equation. However, a systematic study of this equation usually requires a substantial computational effort. In this paper, we consider a special class of nonlocally coupled phase oscillator models where the above analytical approach simplifies significantly, leading to a semi-analytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a one-dimensional lattice, on a two-dimensional square lattice and on a three-dimensional cubic lattice, all three with periodic boundary conditions. For each of these systems we identify multiple symmetric coherence–incoherence patterns and compute their linear stability properties. In addition, we describe how chimera states in higher-dimensional models are inherited from lower-dimensional models and explain how they can be grouped according to their symmetry properties and global order parameter.

    more » « less
  2. In this work we present a systematic review of novel and interesting behaviour we have observed in a simplified model of a MEMS oscillator. The model is third order and nonlinear, and we expressit as a single ODE for a displacement variable. We find that a single oscillator exhibits limitcycles whose amplitude is well approximated by perturbation methods. Two coupled identicaloscillators have in-phase and out-of-phase modes as well as more complicated motions.Bothof the simple modes are stable in some regions of the parameter space while the bifurcationstructure is quite complex in other regions. This structure is symmetric; the symmetry is brokenby the introduction of detuning between the two oscillators. Numerical integration of the fullsystem is used to check all bifurcation computations. Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern. As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change, producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation. A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, is investigated using Lindstedt's perturbation method. Conditions are derived on the parameters of the model for a limit cycle to exist. The original model of the MEMS oscillator consists of two equations: a second order ODE which describes the physical motion of a microbeam, and a first order ODE which describes the heat conduction due to the laser. Starting with these equations, we derive a single governing ODE which is of third order and which leads to the definition of a linear operator called the MEMS operator. The addition of nonlinear terms in the model is shown to produce limit cycle behavior. The differential equations of motion of the system of two coupled oscillators are numerically integrated for varying values of the coupling parameter. It is shown that the in-phase mode loses stability as the coupling parameter is reduced below a certain value, and is replaced by two new periodic motions which are born in a pitchfork bifurcation. Then as this parameter is further reduced, the form of the bifurcating periodic motions grows more complex, with yet additional bifurcations occurring. This sequence of bifurcations leads to a situation in which the only periodic motion is a stable out-of-phase mode. The complexity of the resulting sequence of bifurcations is illustrated through a series of diagrams based on numerical integration. 
    more » « less
  3. The optical power spectrum is the prime observable to dissect, understand, and design the long- time behavior of small and large arrays of optically coupled semiconductor lasers. A long-standing issue has been identified within the literature of injection locking in photonic oscillators: first how the thickness of linewidth and the lineshape spectral envelope correlates with the deterministic evolution of the monochromatic injected laser oscillator and second how the presence of noise and the typically dense proximity in phase space of coexisting limit cycles of the coupled system are shaping and influencing the overall spectral behavior. In addition, we are critically interested in the regions where the basin of attraction has a fractal-like structure, still, the long-time orbits are P1 (period 1) and/or P3 (period 3) limit cycles. Numerically computed evidence shows that, when the coupled system lives in the regions of coexisting isolas and four-wave mixing (FWM) limit cycles, the overall optical power spectrum is deeply imprinted by a strong influence from the underlying noise sources. A particularly intriguing observation in this region of parameter space that we examine is that the isolas draw most of the trajectories on its phase space path.

    more » « less
  4. Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of rotators with higher-order Fourier modes in the coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct, coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops’ eye. As their mythological counterparts, the cyclops states are giants that dominate the system’s phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling’s repulsion. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks. 
    more » « less
  5. Abstract

    This study derives simple analytical expressions for the theoretical height profiles of particle number concentrations (Nt) and mean volume diameters (Dm) during the steady-state balance of vapor growth and collision–coalescence with sedimentation. These equations are general for both rain and snow gamma size distributions with size-dependent power-law functions that dictate particle fall speeds and masses. For collision–coalescence only,Nt(Dm) decreases (increases) as an exponential function of the radar reflectivity difference between two height layers. For vapor deposition only,Dmincreases as a generalized power law of this reflectivity difference. Simultaneous vapor deposition and collision–coalescence under steady-state conditions with conservation of number, mass, and reflectivity fluxes lead to a coupled set of first-order, nonlinear ordinary differential equations forNtandDm. The solutions to these coupled equations are generalized power-law functions of heightzforDm(z) andNt(z) whereby each variable is related to one another with an exponent that is independent of collision–coalescence efficiency. Compared to observed profiles derived from descending in situ aircraft Lagrangian spiral profiles from the CRYSTAL-FACE field campaign, these analytical solutions can on average capture the height profiles ofNtandDmwithin 8% and 4% of observations, respectively. Steady-state model projections of radar retrievals aloft are shown to produce the correct rapid enhancement of surface snowfall compared to the lowest-available radar retrievals from 500 m MSL. Future studies can utilize these equations alongside radar measurements to estimateNtandDmbelow radar tilt elevations and to estimate uncertain microphysical parameters such as collision–coalescence efficiencies.

    Significance Statement

    While complex numerical models are often used to describe weather phenomenon, sometimes simple equations can instead provide equally good or comparable results. Thus, these simple equations can be used in place of more complicated models in certain situations and this replacement can allow for computationally efficient and elegant solutions. This study derives such simple equations in terms of exponential and power-law mathematical functions that describe how the average size and total number of snow or rain particles change at different atmospheric height levels due to growth from the vapor phase and aggregation (the sticking together) of these particles balanced with their fallout from clouds. We catalog these mathematical equations for different assumptions of particle characteristics and we then test these equations using spirally descending aircraft observations and ground-based measurements. Overall, we show that these mathematical equations, despite their simplicity, are capable of accurately describing the magnitude and shape of observed height and time series profiles of particle sizes and numbers. These equations can be used by researchers and forecasters along with radar measurements to improve the understanding of precipitation and the estimation of its properties.

    more » « less