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In a chimera state, domains composed of synchronized oscillators coexist with ones composed of desynchronized oscillators. It is a common view that, in a chimera state, oscillators in coherent domains always share the same mean phase velocity. However, recent studies have suggested that oscillators in different coherent domains may have different mean phase velocities. In this work, we study a ring of nonlocally coupled Brusselators. We find a two-frequency chimera state with mixed phase regularities in which Brusselators in adjacent coherent domains oscillate at different velocities. Moreover, Brusselators in coherent domains with higher mean phase velocity are nearly in phase. In contrast, Brusselators in coherent domains with lower mean phase velocity are randomly partitioned into two groups in antiphase. We find that the local mean fields in these two types of coherent domains perform different dynamics. Based on local mean fields, we provide an explanation for the formation of this type of chimera state. Furthermore, the stability diagrams of the two-frequency chimera state with mixed phase regularities are investigated in different parameter planes

Twisted states, referred to traveling waves in an array of nonlocally coupled phase oscillators, have drawn some attention in recent years. In this work, we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution. We show that twisted standing waves and stationary twisted states appear successively with the increase of the coupling strength. In the continuum limit, we derive a low-dimensional reduced equation using the Ott–Antonsen ansatz, which verifies the twisted states in the simulations of finite networks of oscillators. We also theoretically investigate the stationary twisted states and their stabilities by using the reduced equation.

Chimera states refer to the dynamical states in which the inherent symmetry of the system is broken. The system composed of two interacting identical subpopulations of phase oscillators provides a platform to study chimera states. In this system, different types of chimera states have been identified and the transitions between them have been investigated. However, the parameter space is not fully explored in this system. In this work, we study a system comprised of two interacting subpopulations of nonidentical phase oscillators. Through numerical simulations and theoretical analyses, we find three symmetry-reserving states, including incoherent state, in-phase synchronous state, and antiphase synchronous state, and three types of symmetry-breaking states, including in-phase chimera states, antiphase chimera states, and weak chimera states. The stability diagrams of these dynamical states are explored on different parameter planes and transition scenarios amongst these states are investigated. We find that the weak chimera states act as the bridge between in-phase and antiphase chimera states. We also observe the existence of a period-two chimera state, chaotic chimera state, and drifting chimera states.

Spiral wave chimeras with spatially extended unsynchronized cores have been reported in nonlocally coupled phase oscillators and chaotic oscillators. Here, we investigate the two-dimensional nonlocally coupled FitzHugh–Nagumo oscillators with open boundary conditions. Rich dynamics of spiral wave chimeras is discovered numerically. Besides outwardly propagating and inwardly propagating single-spiral wave chimeras, we find the double-spiral wave chimera with two spiral chimeras rotating with each other and the breakup of spiral wave chimeras. We also find a transition between a normal spiral wave with a phase singularity at its center and spiral wave chimera.

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

The main purpose of the present paper is to develop the theory of center constructions on Hom–Hopf algebras. Let H be a Hom–Hopf algebra, we first introduce the notions of nth Yetter–Drinfeld modules and mth Drinfeld codouble for H. Also we prove that the category $${\mathcal {YD}}_H^H(n)$$ of nth Yetter–Drinfeld modules of H is a braided autonomous category. Finally, we show that $${\mathcal {YD}}_H^H(n)$$ and $$Corep^{i,j}(CD_m(H))$$ (i.e., the corepresentation category of the Drinfeld codouble of H) are braided isomorphic as the full subcategories of $$Corep^{i,j}(H)$$ .

The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$ being of maximal length, the simplicity of $\mathfrak{L}$ is also characterized in terms of connections of roots., Comment: 16pages. arXiv admin note: substantial text overlap with arXiv:1902.06260; text overlap with arXiv:1504.04236, arXiv:1508.02124 by other authors

We obtain necessary and sufficient conditions for the functor [Formula: see text] on the category of partial entwined modules that forgets the A-action to be separable. As an application, we prove a Maschke-type theorem for the category of partial entwined modules.

We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Second, we give the maps between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras give rise to functors between the category of Doi Hom-Hopf modules and study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf modules. Finally, as an application of our theory, we consider the braiding on the category of Hom-modules, the category of Hom-comodules and the category of Hom-Yetter-Drinfeld modules respectively.

In globally coupled phase oscillators, the distribution of natural frequency has strong effects on both synchronization transition and synchronous dynamics. In this work, we study a ring of nonlocally coupled phase oscillators with the frequency distribution made up of two Lorentzians with the same center frequency but with different half widths. Using the Ott-Antonsen ansatz, we derive a reduced model in the continuum limit. Based on the reduced model, we analyze the stability of the incoherent state and find the existence of multiple stability islands for the incoherent state depending on the parameters. Furthermore, we numerically simulate the reduced model and find a large number of twisted states resulting from the instabilities of the incoherent state with respect to different spatial modes. For some winding numbers, the stability region of the corresponding twisted state consists of two disjoint parameter regions, one for the intermediate coupling strength and the other for the strong coupling strength.

BiHom-Novikov agebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota-Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras., Comment: 19. arXiv admin note: substantial text overlap with arXiv:1501.00229, arXiv:1204.6373 by other authors

In this paper, we study symmetric pairs and pseudosymmetries in the Hom-Yetter–Drinfeld category [Formula: see text] over a Hom-Hopf algebra [Formula: see text]. We first show that the category [Formula: see text] over a (co)triangular Hom-Hopf algebra [Formula: see text] contains a rich symmetric subcategory. Also we prove that the (co)commutativity and trivial property of [Formula: see text] are determined by some symmetric pairs of objects in [Formula: see text]. Moreover, we find a sufficient and necessary condition for [Formula: see text] to be pseudosymmetric.

Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.