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In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation u (x , t) = ϕ (ξ) , ξ = k x + v t α , we reduce the PDE to an ODE which depends on the fractional order α , then the analysis depends on the order α. Moreover, as α = 1 , the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order α. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative. [ABSTRACT FROM AUTHOR]

According to the master stability function (MSF) framework, synchronized regions play an extremely important role in network synchronization. On these grounds, this paper casts sight on network synchronous state stability via studying the bifurcation (or transition) problem of network synchronized regions with varying nodal dynamics, and the effects of time delay on the bifurcation of synchronized regions. Theoretical and numerical investigations show that in complex networks with coupling delay, there exist rich bifurcation behaviors of synchronized regions. The coupling delay can not only enlarge or narrow synchronized regions, but also change bifurcation points. More importantly, a very small delay can result in the conversion of an unbounded or empty synchronized region into a bounded one, implying that coupling delay can enhance or suppress synchronization in complex dynamical networks. These results will further strengthen our understanding for synchronous state stability in complex dynamical networks. [ABSTRACT FROM AUTHOR]