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   https://doi.org/10.14232/ejqtde.2025.1.37  

   Gedeon, Tomas; Humphries, Antony; Mackey, Michael; Walther, Hans-Otto; Wang, Zhao (January 2025, Electronic Journal of Qualitative Theory of Differential Equations)

   We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable. 

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2. Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators

   https://doi.org/10.1007/s10915-022-01891-y  

   Banks, J. W.; Buckner, B. Brett; Hagstrom, T. (August 2022, Journal of Scientific Computing)
3. Elimination of unknowns for systems of algebraic differential-difference equations

   https://doi.org/10.1090/tran/8219  

   Li, Wei; Ovchinnikov, Alexey; Pogudin, Gleb; Scanlon, Thomas (January 2021, Transactions of the American Mathematical Society)

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4. Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.

   Levin, A. (June 2018, Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018)

   Multivariate dimension polynomials associated with finitely generated differential and difference field extensions arise as natural generalizations of the univariate differential and difference dimension polynomials. It turns out, however, that they carry more information about the corresponding extensions than their univariate counterparts. We extend the known results on multivariate dimension polynomials to the case of difference-differential field extensions with arbitrary partitions of sets of basic operators. We also describe some properties of multivariate dimension polynomials and their invariants. 

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5. Algebraic groups as difference Galois groups of linear differential equations

   https://doi.org/10.1016/j.jpaa.2021.106854  

   Bachmayr, Annette; Wibmer, Michael (February 2022, Journal of Pure and Applied Algebra)