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This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

The local dynamics of systems of two equations with delay is considered. The main assumption is that the delay parameter is large enough. Critical cases in the problem of the stability of the equilibrium state are identified, and it is shown that they are of infinite dimension. Methods of infinite-dimensional normalization are used and further developed. The main result is the construction of special nonlinear boundary value problems that play the role of normal forms. Their nonlocal dynamics determine the behavior of all solutions of the original system in a neighborhood of the equilibrium state. [ABSTRACT FROM AUTHOR]

This paper is concerned with the limiting dynamics of stochastic retarded 3D non-autonomous Navier-Stokes-Voight (NSV) equations driven by Laplace-multiplier noise. We first prove the existence, uniqueness, forward compactness and forward longtime stability of pullback random attractors (PRAs). We then establish the upper semicontinuity of PRAs from non-autonomy to autonomy. Finally, we study the upper semicontinuity of PRAs under an analogue of Hausdorff semi-distance as the memory time tends to zero. Because of the solution has no higher regularity, the forward pullback asymptotic compactness of solutions in the state space is proved by the spectrum decomposition technique. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the regular dynamics for the 3D delay Brinkman–Forchheimer (BF) equations. We first prove the existence, uniqueness and time-dependent property of regular tempered pullback attractors as well as the existence of invariant measures for the 3D BF equations with non-autonomous abstract delay. We then study the asymptotic autonomy of regular pullback attractors for the 3D BF equations with autonomous abstract delay. Finally, we discuss the upper semicontinuity of regular pullback attractors as the delay time approaches to zero for the 3D BF equations with variable delay and distributed delay. [ABSTRACT FROM AUTHOR]

We are motivated by the idea that certain properties of delay differential and difference equations with constant coefficients arise as a consequence of their one‐periodic nature. We apply the recently introduced definition of periodicity for arbitrary isolated time scales to linear delay dynamic equations and a class of nonlinear delay dynamic equations. Utilizing a derived identity of higher order delta derivatives and delay terms, we rewrite the considered linear and nonlinear delayed dynamic equations with one‐periodic coefficients as a linear autonomous dynamic system with constant matrix. As the simplification of a constant matrix is only obtained for one‐periodic coefficients, dynamic equations with one‐periodic coefficients are the simplest form compared to the commonly used constant coefficients. [ABSTRACT FROM AUTHOR]

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space H s with s ≥ 2 - 2 α and α ∈ (1 2 , 1) . First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax–Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin–Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay. [ABSTRACT FROM AUTHOR]