Hsien-Ching Kao

Twisted chimera states and multicore spiral chimera states on a two-dimensional torus.

Chimera states consisting of domains of coherently and incoherently oscillating oscillators in a two-dimensional periodic array of nonlocally coupled phase oscillators are studied. In addition to the one-dimensional chimera states familiar from one spatial dimension, two-dimensional structures termed twisted chimera states and spiral wave chimera states are identified in simulations. The properties of many of these states, including stability, are determined using an evolution equation for a complex order parameter and are found to be in agreement with the simulations.

Localized rotating convection with no-slip boundary conditions

Localized patches of stationary convection embedded in a background conduction state are called convectons. Multiple states of this type have recently been found in two-dimensional Boussinesq convection in a horizontal fluid layer with stress-free boundary conditions at top and bottom, and rotating about the vertical. The convectons differ in their lengths and in the strength of the self-generated shear within which they are embedded, and exhibit slanted snaking. We use homotopic continuation of the boundary conditions to show that similar structures exist in the presence of no-slip boundary conditions at the top and bottom of the layer and show that such structures exhibit standard snaking. The homotopic continuation allows us to study the transformation from slanted snaking characteristic of systems with a conserved quantity, here the zonal momentum, to standard snaking characteristic of systems with no conserved quantity.

Chimera states in systems of nonlocal nonidentical phase-coupled oscillators.

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multicluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.

Exact solutions of the cubic–quintic Swift–Hohenberg equation and their bifurcations

Many systems of physical interest may be modelled by the bistable Swift–Hohenberg equation with cubic–quintic nonlinearity. We construct a two-parameter family of exact meromorphic solutions of the time-independent equation and use these to construct a one-parameter family of exact periodic solutions on the real line. These are of two types, differing in their symmetry properties, and are connected via an exact heteroclinic solution. We use these exact solutions as initial points for numerical continuation and show that some of these lie on secondary branches while others fall on isolas. The approach substantially enhances our understanding of the solution space of this equation.

Front propagation in weakly subcritical pattern-forming systems

The speed and stability of fronts near a weakly subcritical steady-state bifurcation are studied, focusing on the transition between pushed and pulled fronts in the bistable Ginzburg-Landau equation. Exact nonlinear front solutions are constructed and their stability properties investigated. In some cases, the exact solutions are stable but are not selected from arbitrary small amplitude initial conditions. In other cases, the exact solution is unstable to modulational instabilities which select a distinct front. Chaotic front dynamics may result and is studied using numerical techniques.