Abstract
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 ofthe linear operator associated with the underlying DDE. Thesepolynomials are then used to design a general Galerkin scheme for whichwe derive rigorous convergence results and show that it can benumerically 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 articleprovides a good approximation by low-dimensional ODE systemsof the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.
Highlights
Background and motivationWe introduce the functional framework that will be adopted in Section 4.2 for the derivation of Galerkin approximations of a given nonlinear system of delay differential equations (DDEs)
We propose to avoid these technical difficulties in approximating DDEs as systems of ordinary differential equations (ODEs) by using an alternative polynomial basis: the elements of this basis belong naturally to the domain of the underlying linear operator, but they have not been used in the DDE literature so far
This interpretation relies on a formulation of the evolution in time of the initial state {x(θ) : θ ∈ [−τ, 0]} as the solution of a partial differential equation (PDE); see Remark 4.1
Summary
Systems of delay differential equations (DDEs) are widely used in many fields such as the biosciences, climate dynamics, chemistry, control theory, economics, and engineering [BGV82,DHL14,DvGVLW95, GCS15,GC87,GZT08,HVL93,KS14,Kua93,LS10,Mac89,MN07,RCC+14,Smi11,Ste89]. Finite-time uniform convergence results are derived for the proposed Galerkin approximations of nonlinear systems of DDEs, subject to simple and checkable conditions on the nonlinear term These conditions are identified in Section 4.2; see Corollaries 4.2 and 4.3. We outline here a useful interpretation of our proposed scheme regarding the finite-dimensional approximation of the linear part A of general systems of DDEs, when considered in the framework of Hilbert spaces, cf (4.24) This interpretation relies on a formulation of the evolution in time of the initial state {x(θ) : θ ∈ [−τ, 0]} as the solution of a partial differential equation (PDE); see Remark 4.1. It is shown that our Galerkin scheme provides a good approximation by low-dimensional ODE systems of the DDE’s strange attractor, as well as the statistical features that characterize the associated nonlinear dynamics
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