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Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

In this study, we principally investigate a fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. Taking advantage of transformation of the variable, we obtain an isovalent version of the considered fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. The isovalent version includes fractional‐order and integer‐order equations. Utilizing the stability criterion and bifurcation theory of fractional‐order differential equation, a novel delay‐independent bifurcation condition to ensure the appearance of Hopf bifurcation for the fractional‐order stage‐structured predator–prey system is set up. The impact of time delay on the stability and bifurcation is clearly revealed. Numerical simulation figures are presented to sustain the rationality of the derived key conclusions. [ABSTRACT FROM AUTHOR]

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

• Stability properties of DEs with a general distributed delay are investigated. • Saddle-node and Hopf bifurcation curves are determined in a general framework. • Stability region of the equilibrium is fully characterized in the parameter plane. • Criticality of Hopf bifurcation is analyzed by the multiple times scales method. • Theoretical results are showcased in the framework of a simple neural model. For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model. [ABSTRACT FROM AUTHOR]