# Mathematical Modelling, Dynamical Behavior and Computational Techniques in Delay Differential Systems

Dynamical system theory is one of the essential modern theoretical developments for the qualitative theory of differential equations, which seeks to establish general properties of solutions from general principles without writing down any explicit solutions. For all dynamical systems, qualitative analysis and computational techniques are the primary requirements. Mathematical models developed in basic dynamical behavioral research have been used to predict and control the behavior system states in applied settings. Over the years, researchers who study basic behavioral processes have increasingly relied on mathematical models in their work.

Delayed interactions are a ubiquitous feature of many dynamical systems in science and engineering. Delays occur due to finite swiftness of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delays system can be described by delay differential equations and often include non-negligible nonlinear effects. Nonlinear properties play a principal role, causing complex dynamical behavior, which cannot be explained by just looking at the residents of a system. Therefore, delayed dynamical systems are crucial in applications and pose a fundamental challenge for theoretical and numerical studies. However, it is more applicable in mathematical modeling to understand the biological bases of a behavioral phenomenon. The existence of state processes with delays has represented a phenomenon that considerably affects their stability and dynamics. Studying the influence of uncertainties on system stability, dynamics, and control performance with optimal estimation poses a challenging mathematical exercise.

This Research Topic focuses on recent developments concerning various approaches to time-delay systems analysis and control design. Papers on the existence of solutions, system stability, and dynamic analysis are welcomed, including fractional, functional, neutral and stochastic systems for the study of exponential, asymptotic, strong, delay-dependent, delay-independent, L_2, L_p, H_2, H_∞, and other types of stability. Also, chaotic, bifurcation, stability switching, and eigenvalue analyses are welcome as well. The delays of nonlinear neutral and impulsive types and systems provided by algebraic-differential equations will give more attention; however, systems with nonlinear retarded delays are also acceptable. Modern control methods, both theoretical and numerical analysis and their applications also fall within the scope of this Research Topic, including networked control, switched systems, event-triggered control, Lyapunov–Razumikhin and Krasovskii-type approaches, etc.

We solicit high-quality original research papers in any aspect of dynamical systems, mathematical modeling, and new computational techniques with applications are especially encouraged. Relevant topics include but are not limited to:

• Models with delay in biology, economics, and engineering
• Existence and uniqueness of equilibrium points
• Numerical computational analysis
• Control in nonlinear functional and fractional differential equations
• Asymptotical analysis and synchronization of population dynamics
• Impulsive delay differential equations with uncertainty
• Infinite delay stochastic systems in abstract spaces
• Stability and stabilization on time-delay system
• Optimal control of time-delay system

Keywords: Delay differential equations, Stochastic control, Optimal control, Dynamical analysis, Existence theory, Fixed point techniques, Numerical solutions, Fractional control models, Population dynamics

Important Note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

Dynamical system theory is one of the essential modern theoretical developments for the qualitative theory of differential equations, which seeks to establish general properties of solutions from general principles without writing down any explicit solutions. For all dynamical systems, qualitative analysis and computational techniques are the primary requirements. Mathematical models developed in basic dynamical behavioral research have been used to predict and control the behavior system states in applied settings. Over the years, researchers who study basic behavioral processes have increasingly relied on mathematical models in their work.

Delayed interactions are a ubiquitous feature of many dynamical systems in science and engineering. Delays occur due to finite swiftness of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delays system can be described by delay differential equations and often include non-negligible nonlinear effects. Nonlinear properties play a principal role, causing complex dynamical behavior, which cannot be explained by just looking at the residents of a system. Therefore, delayed dynamical systems are crucial in applications and pose a fundamental challenge for theoretical and numerical studies. However, it is more applicable in mathematical modeling to understand the biological bases of a behavioral phenomenon. The existence of state processes with delays has represented a phenomenon that considerably affects their stability and dynamics. Studying the influence of uncertainties on system stability, dynamics, and control performance with optimal estimation poses a challenging mathematical exercise.

This Research Topic focuses on recent developments concerning various approaches to time-delay systems analysis and control design. Papers on the existence of solutions, system stability, and dynamic analysis are welcomed, including fractional, functional, neutral and stochastic systems for the study of exponential, asymptotic, strong, delay-dependent, delay-independent, L_2, L_p, H_2, H_∞, and other types of stability. Also, chaotic, bifurcation, stability switching, and eigenvalue analyses are welcome as well. The delays of nonlinear neutral and impulsive types and systems provided by algebraic-differential equations will give more attention; however, systems with nonlinear retarded delays are also acceptable. Modern control methods, both theoretical and numerical analysis and their applications also fall within the scope of this Research Topic, including networked control, switched systems, event-triggered control, Lyapunov–Razumikhin and Krasovskii-type approaches, etc.

We solicit high-quality original research papers in any aspect of dynamical systems, mathematical modeling, and new computational techniques with applications are especially encouraged. Relevant topics include but are not limited to:

• Models with delay in biology, economics, and engineering
• Existence and uniqueness of equilibrium points
• Numerical computational analysis
• Control in nonlinear functional and fractional differential equations
• Asymptotical analysis and synchronization of population dynamics
• Impulsive delay differential equations with uncertainty
• Infinite delay stochastic systems in abstract spaces
• Stability and stabilization on time-delay system
• Optimal control of time-delay system

Keywords: Delay differential equations, Stochastic control, Optimal control, Dynamical analysis, Existence theory, Fixed point techniques, Numerical solutions, Fractional control models, Population dynamics

Important Note: All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

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## Topic Editors

 16 December 2021 Manuscript

## Participating Journals

Manuscripts can be submitted to this Research Topic via the following journals: