 Award ID(s):
 1634664
 Publication Date:
 NSFPAR ID:
 10309647
 Journal Name:
 International journal of engineering research and applications
 Volume:
 11
 Issue:
 10
 ISSN:
 22489622
 Sponsoring Org:
 National Science Foundation
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Abstract The passive conserved Swift–Hohenberg equation (or phasefieldcrystal [PFC] model) describes gradient dynamics of a singleorder parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of selfpropelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by paritybreaking drift bifurcations. Here, we briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC models with common gradient dynamics, (ii) two coupled passive PFC models where the coupling breaks the gradient dynamics structure and (iii) a passive PFC model coupled to an active PFC model.

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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 phasecoupled oscillators allows an elegant mathematical description of these states based on a nonlinear integrodifferential 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 semianalytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a onedimensional lattice, on a twodimensional square lattice and on a threedimensional 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 higherdimensional models are inherited from lowerdimensional models and explain how they can be grouped according to their symmetry properties and global order parameter.

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