Functional Differential Equations and Dynamic Systems

Functional differential equations encompass differential systems in which derivatives at any given time depend not only on the present state but also on past or distributed states, parameters or other functionals. These equations generalise classical ordinary differential equations by incorporating delays, memory effects or dependence on integral terms, thereby capturing richer dynamics in physical, biological and engineering contexts. Dynamic systems theory provides the conceptual and analytical framework for exploring the time evolution of such equations, identifying stability properties, bifurcation phenomena and long-term behaviour. Recent advances have deepened our understanding of existence and uniqueness of solutions in infinite-dimensional spaces, extended analytical techniques to measure-driven and discontinuous settings, and developed robust numerical and computational methods. Applications range from modelling neural networks and population dynamics to describing viscoelastic materials and fluid stratification, highlighting global significance across disciplines.

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Contemporary work on Stieltjes differential systems has broadened the concept of differentiation to include derivators that may change sign or exhibit controlled variation. This approach has led to the definition of g-exponential maps and Peano-type existence theorems, with demonstrations on fluid stratification in buoyant jets and plumes. In parallel, the theory of measure differential inclusions has matured, offering general existence results for trajectories governed by multivalued mappings with respect to varying measures. Selection principles and bounded variation hypotheses underpin minimisation problems and the continuous dependence of solutions, enriching optimal control and variational analysis. On the computational front, novel numerical schemes based on quadrature formulae for Lebesgue–Stieltjes integrals have proven consistent, convergent and stable. Such techniques enable the approximation of Stieltjes differential equations with no closed-form solutions, validated against explicit solutions and applied to realistic population models and multi-derivator systems.

Functional Differential Equations and Dynamic Systems publication trend

The graph below shows the total number of articles in functional differential equations and dynamic systems across all publications each year (not limited to Nature Index journals).

Technical terms

Functional differential equation: A differential equation in which the derivative at a point depends on the state of the system over an interval or through functional expressions beyond the instantaneous value.

Delay differential equation: A subclass of functional differential equations where the dependence involves fixed or variable time lags, modelling memory or propagation effects.

Differential inclusion: A generalisation of a differential equation in which the derivative belongs to a multivalued mapping, accommodating set-valued dynamics and discontinuities.

Stieltjes derivative: A derivative defined with respect to a nondecreasing function (the derivator), unifying discrete and continuous calculus under a single operator.

Derivator: The nondecreasing function with respect to which the Stieltjes derivative is taken, which may encode impulses or nonmonotonic behaviour.

Dynamic system: A mathematical model describing the evolution of variables over time according to specified rules, often expressed via differential or difference equations.

References

1. Stieltjes differential systems with nonmonotonic derivators. Boundary Value Problems (2020).
2. Measure Differential Inclusions: Existence Results and Minimum Problems. Set-Valued and Variational Analysis (2020).
3. Continuous dependence results for set-valued measure differential problems. Electronic Journal of Qualitative Theory of Differential Equations (2015).
4. Numerical Solution of Stieltjes Differential Equations. Mathematics (2020).