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Low-dimensional Galerkin approximations of nonlinear delay differential equations. (English) Zbl 1336.34088

Summary: 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 rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE’s strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

MSC:

34K07 Theoretical approximation of solutions to functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
41A10 Approximation by polynomials
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
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