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We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract…
In this paper, we propose a general approach for approximate simulation and analysis of delay differential equations (DDEs) with distributed time delays based on methods for ordinary differential equations (ODEs). The key innovation is that…
We analyze a structure-preserving model order reduction technique for delay and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring error bounds based on Hankel…
Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote…
We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classic differential equation models with…
Data-driven methodologies are nowadays ubiquitous. Their rapid development and spread have led to applications even beyond the traditional fields of science. As far as dynamical systems and differential equations are concerned, neural…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
In this paper, we consider the dynamics of a delayed reaction-diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary…
Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order…
Complex systems manifest a small number of instabilities and bifurcations that are canonical in nature, resulting in universal pattern forming characteristics as a function of some parametric dependence. Such parametric instabilities are…
Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states,…
We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic…
Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^\alpha x(t)=\delta…
Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature.…
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the…
We establish a discrete operator--theoretic framework for the analysis of implicit Euler and Lie--Trotter splitting schemes for delay differential equations (DDEs). Both schemes are formulated in terms of discrete resolvent operators acting…
This manuscript deals with the stability and bifurcation analysis of the equation $D^{2\alpha}x(t)+c D^{\alpha}x(t)=a x(t)+b x(t-\tau)$, where $0<\alpha<1$ and $\tau>0$. We sketch the boundaries of various stability regions in the parameter…
In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. FDEs bring many advantages to model the physical systems in the nature or man-made systems in the industry. Because this…
In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a…
The intersection of machine learning and dynamical systems has generated considerable interest recently. Neural Ordinary Differential Equations (NODEs) represent a rich overlap between these fields. In this paper, we develop a continuous…