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Mathematical equations are bedrock in understanding the complex evolution of real-world phenomena. Dynamical systems based on ordinary differential equations are important tools to describe processes arising in biology, ecology, disease dynamics, economics, mechanics, medicine, etc. Time delay (time lag) is inevitable in these systems, and even a small-time delay can significantly alter a system’s dynamics and can manifest many intricate dynamical scenarios in a very simple-looking system. Ordinary differential equations cannot take into account of such time lags in models. Therefore, delay differential equation (DDE) framework is an important alternative in mathematical modeling.

Time delay in a dynamical system can arise in different forms, viz., constant delay, state-dependent delay, time-varying delay, etc. It is a common practice to study local stability and bifurcation in delayed differential equations w.r.t the time delay. Researchers often determine the first bifurcation threshold of the constant delay parameter from transition from a stable to an unstable state. However, the existence of multiple sequences of delay threshold and stability switching phenomenon are not explored in detail in many systems of interest. Further, modeling and applications of state-dependent and time-varying delays are limited due to its complicated mathematical analysis. Some DDE systems with multiple constant time delays have been investigated partially, but analytical results of varying all delays simultaneously are yet to be explored. In fact, there is a ample scope to investigate delayed partial differential equations models pertaining to pattern formations and interpretations. Impacts of small and large delays could be of interest in many real-world situations. One could also examine the benefits and dangers of delay in quantifying physical processes.

Another interesting theme of the delay differential equations is to apply them in neural progenitor fate decisions. The time series solutions also could produce quasi-periodicity, chaos, or hyperchaos which are still less explored in delay differential equations. Analytic methods are limited in fully solving practical models using delay differential equations in ecology, epidemiology, and engineering problems. One might be aware of the fact that delay differential equations are stiff in nature and appropriate mathematical methods, or discretization processes are essential to solve the systems numerically. Therefore, numerical methods are indispensable for understanding to comprehend the resulting data accurately. This enables the development of algorithms that provide correct (convergent) and high-quality information at a reasonable cost. Using real data and deriving appropriate time delay length from DDE models could be an interesting topic, especially for epidemiological systems.

We welcome original research and review articles on delay differential equations centered (but not exclusively) on the following themes. Topics of interest include, but are not limited to:
• Mathematical analysis of DDEs with multiple delays
• Large and small delay in population models
• Neural networks and cell dynamics with DDEs
• Numerical solutions of DDEs
• Complex dynamics and chaos in DDEs
• Synchronization in delayed complex networks
• Dynamics of stochastic DDEs
• DDEs in control systems and engineering
• Application of DDEs in opinion dynamics
• Connecting real data with DDE models

Keywords: Population models, Opinion dynamics, Multiple time delays, Stochastic DDEs, Synchronization and chaos, Numerical solutions of DDEs

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