# American Institute of Mathematical Sciences

August  2016, 36(8): 4133-4177. doi: 10.3934/dcds.2016.36.4133

## Low-dimensional Galerkin approximations of nonlinear delay differential equations

 1 Department of Atmospheric & Oceanic Sciences, University of California, Los Angeles, CA 90095-1565, United States, United States 2 Geosciences Department and Laboratoire de Météorologie Dynamique (CNRS and IPSL), École Normale Supérieure, F-75231 Paris Cedex 05, France 3 Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  June 2015 Revised  August 2015 Published  March 2016

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.
Citation: Mickaël D. Chekroun, Michael Ghil, Honghu Liu, Shouhong Wang. Low-dimensional Galerkin approximations of nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133
##### References:
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##### References:
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Stepan, A collection on 'Climate dynamics: Multiple scales and memory effects', editorial,, R. Soc. Proc. A, 471 (2015). doi: 10.1098/rspa.2015.0097. Google Scholar [28] M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics,, Applied Mathematical Sciences, (1987). doi: 10.1007/978-1-4612-1052-8. Google Scholar [29] M. Ghil and A. W. Robertson, Solving problems with GCMs: General circulation models and their role in the climate modeling hierarchy,, in General Circulation Model Development: International Geophysics (ed. D. Randall), 70 (2001), 285. doi: 10.1016/S0074-6142(00)80058-3. Google Scholar [30] M. Ghil and A. W. Robertson, "Waves" vs."particles" in the atmosphere's phase space: A pathway to long-range forecasting?,, Proc. Natl. Acad. Sci. USA, 99 (2002), 2493. doi: 10.1073/pnas.012580899. Google Scholar [31] M. Ghil and I. Zaliapin, Understanding ENSO variability and its extrema: A delay differential equation approach,, in Observations, (2015). doi: 10.1002/9781119157052.ch6. Google Scholar [32] M. Ghil, I. Zaliapin and S. Thompson, A delay differential model of ENSO variability: Parametric instability and the distribution of extremes,, Nonlin. Processes Geophys., 15 (2008), 417. doi: 10.5194/npg-15-417-2008. Google Scholar [33] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [34] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, vol. 98 of Encyclopedia of Mathematics and its Applications,, Cambridge University Press, (2005). doi: 10.1017/CBO9781107325982. Google Scholar [35] K. Ito and R. Teglas, Legendre-tau approximations for functional-differential equations,, SIAM J. Control Optim., 24 (1986), 737. doi: 10.1137/0324046. Google Scholar [36] N. Jiang, D. Neelin and M. Ghil, Quasi-quadrennial and quasi-biennial variability in the equatorial Pacific,, Clim. Dyn., 12 (1995), 101. doi: 10.1007/BF00223723. Google Scholar [37] F. F. Jin, J. D. Neelin and M. Ghil, El Niño on the Devil's Staircase: Annual subharmonic steps to chaos,, Science, 264 (1994), 70. Google Scholar [38] F. Kappel, Semigroups and delay equations,, in Semigroups, (1984), 136. Google Scholar [39] F. Kappel and D. Salamon, Spline approximation for retarded systems and the Riccati equation,, SIAM J.Control Optim., 25 (1987), 1082. doi: 10.1137/0325060. Google Scholar [40] F. Kappel and W. Schappacher, Autonomous nonlinear functional differential equations and averaging approximations,, Nonlinear Analysis: Theory, 2 (1978), 391. doi: 10.1016/0362-546X(78)90048-2. Google Scholar [41] F. Kappel and K. 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