# Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations

@article{Chekroun2015LowDimensionalGA, title={Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations}, author={Micka{\"e}l D. Chekroun and Michael Ghil and Honghu Liu and Shouhong Wang}, journal={arXiv: Chaotic Dynamics}, year={2015} }

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive…

## 20 Citations

4. Galerkin Approximations For The Optimal Control Of Nonlinear Delay Differential Equations

- MathematicsHamilton-Jacobi-Bellman Equations
- 2018

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error…

Galerkin approximations for the optimal control of nonlinear delay differential equations

- Mathematics
- 2017

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error…

Approximating strange attractors and Lyapunov exponents of delay differential equations using Galerkin projections

- Computer Science, Mathematics
- 2018

Examples demonstrate that the strange attractors and Lyapunov exponents of chaotic DDE solutions can be reliably approximated by a smaller number of ODEs using the proposed approach compared to the standard method-of-lines approach, leading to faster convergence and improved computational efficiency.

Capture the past to portray the future - Numerical bifurcation analysis of delay equations, with a focus on population dynamics

- Mathematics
- 2018

Many mathematical models of population dynamics are formulated as Volterra integral equations coupled with integro-differential equations. They can be interpreted as delay equations, where the value…

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

- Mathematics
- 2017

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The…

Perturbative-Iterative Computation of Inertial Manifolds of Systems of Delay-Differential Equations with Small Delays

- MathematicsAlgorithms
- 2020

The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein.

Homogeneous Herz spaces with variable exponents and regularity results

- Mathematics
- 2018

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubtedly of major concern. As many other…

Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models.

- MathematicsChaos
- 2020

By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of…

Reduced-Order Models for Coupled Dynamical Systems: Koopman Operator and Data-driven Methods.

- Mathematics
- 2020

These findings support the physical basis and robustness of the EMR methodology and illustrate the practical relevance of the perturbative expansion used for deriving the parametrizations.

Reduced-order models for coupled dynamical systems: Data-driven methods and the Koopman operator.

- MathematicsChaos
- 2021

These findings support the physical basis and robustness of the EMR methodology and illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.

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