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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H. T. Banks and F. Kappel, Spline approximations for functional differential equations,, Journal of Differential Equations, 34 (1979), 496. doi: 10.1016/0022-0396(79)90033-0.

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H. T. Banks, I. G. Rosen and K. Ito, A spline based technique for computing Riccati operators and feedback controls in regulator problems for delay equations,, SIAM J. Sci. Stat. Comput., 5 (1984), 830. doi: 10.1137/0905059.

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show all references

References:
[1]

M. Alfaro, F. Marcellán and M. L. Rezola, Estimates for Jacobi-Sobolev type orthogonal polynomials,, Appl. Anal., 67 (1997), 157. doi: 10.1080/00036819708840602.

[2]

H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations,, SIAM J. Control Optim., 16 (1978), 169. doi: 10.1137/0316013.

[3]

H. T. Banks and F. Kappel, Spline approximations for functional differential equations,, Journal of Differential Equations, 34 (1979), 496. doi: 10.1016/0022-0396(79)90033-0.

[4]

H. T. Banks, I. G. Rosen and K. Ito, A spline based technique for computing Riccati operators and feedback controls in regulator problems for delay equations,, SIAM J. Sci. Stat. Comput., 5 (1984), 830. doi: 10.1137/0905059.

[5]

D. S. Battisti and A. C. Hirst, Interannual variability in a tropical atmosphere-ocean model: Influence of the basic state, ocean geometry and nonlinearity,, J. Atmos. Sci., 46 (1989), 1687. doi: 10.1175/1520-0469(1989)046<1687:IVIATA>2.0.CO;2.

[6]

K. Bhattacharya, M. Ghil and I. Vulis, Internal variability of an energy-balance model with delayed albedo effects,, Journal of the Atmospheric Sciences, 39 (1982), 1747. doi: 10.1175/1520-0469(1982)039<1747:IVOAEB>2.0.CO;2.

[7]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011).

[8]

J. Burns, T. Herdman and H. Stech, Linear functional differential equations as semigroups on product spaces,, SIAM Journal on Mathematical Analysis, 14 (1983), 98. doi: 10.1137/0514007.

[9]

M. A. Cane, M. Münnich and S. E. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part I: Linear analysis,, J. Atmos. Sci., 47 (1990), 1562. doi: 10.1175/1520-0469(1990)047<1562:ASOSEO>2.0.CO;2.

[10]

A. Casal and M. Freedman, A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations,, Automatic Control, 25 (1980), 967. doi: 10.1109/TAC.1980.1102450.

[11]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications,, The Clarendon Press, (1998).

[12]

M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications,, Commun. Math. Phys., 316 (2012), 723. doi: 10.1007/s00220-012-1515-y.

[13]

M. D. Chekroun, D. Kondrashov and M. Ghil, Predicting stochastic systems by noise sampling, and application to the El Niño-Southern Oscillation,, Proc. Natl. Acad. Sci. USA, 108 (2011), 11766.

[14]

M. D. Chekroun, J. D. Neelin, D. Kondrashov, J. C. McWilliams and M. Ghil, Rough parameter dependence in climate models: The role of Ruelle-Pollicott resonances,, PNAS, 111 (2014), 1684. doi: 10.1073/pnas.1321816111.

[15]

S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation,, J. Differential Equations, 26 (1977), 112. doi: 10.1016/0022-0396(77)90101-2.

[16]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21,, Springer, (1995). doi: 10.1007/978-1-4612-4224-6.

[17]

S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations,, Nonlinear Dynamics, 30 (2002), 323. doi: 10.1023/A:1021220117746.

[18]

J. I. Diaz, A. Hidalgo and T. L., Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean,, Proc. R. Soc. A, 470 (2014). doi: 10.1098/rspa.2014.0376.

[19]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis, vol. 110 of Applied Mathematical Sciences,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2.

[20]

H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large-Scale Ocean Circulation and El Niño,, Springer, (2005). doi: 10.1007/978-94-015-9450-9.

[21]

H. A. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach,, Rev. Geophys., 43 (2005). doi: 10.1029/2002RG000122.

[22]

J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Modern Phys., 57 (1985), 617. doi: 10.1103/RevModPhys.57.617.

[23]

T. Faria and L. T. Magalhães, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation,, J. differential equations, 122 (1995), 181. doi: 10.1006/jdeq.1995.1144.

[24]

E. Galanti and E. Tziperman, ENSO's phase locking to the seasonal cycle in the fast-SST, fast-wave, and mixed-mode regimes,, J. Atmos. Sci., 57 (2000), 2936. doi: 10.1175/1520-0469(2000)057<2936:ESPLTT>2.0.CO;2.

[25]

M. Ghil, Hilbert problems for the geosciences in the 21st century,, Nonlin. Processes Geophys., 8 (2001), 211. doi: 10.5194/npg-8-211-2001.

[26]

M. Ghil, M. R. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. W. Robertson, A. Saunders, Y. Tian, F. Varadi and P. Yiou, Advanced spectral methods for climatic time series,, Rev. Geophys., 40 (2002), 1. doi: 10.1029/2000RG000092.

[27]

M. Ghil, M. D. Chekroun and G. Stepan, A collection on 'Climate dynamics: Multiple scales and memory effects', editorial,, R. Soc. Proc. A, 471 (2015). doi: 10.1098/rspa.2015.0097.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

K. Ito and R. Teglas, Legendre-tau approximations for functional-differential equations,, SIAM J. Control Optim., 24 (1986), 737. doi: 10.1137/0324046.

[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.

[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.

[38]

F. Kappel, Semigroups and delay equations,, in Semigroups, (1984), 136.

[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.

[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.

[41]

F. Kappel and K. Zhang, Equivalence of functional-differential equations of neutral type and abstract Cauchy problems,, Monatshefte für Mathematik, 101 (1986), 115. doi: 10.1007/BF01298925.

[42]

N. D. Kazarinoff, Y.-H. Wan and P. Van den Driessche, Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations,, IMA J. Appl. Math., 21 (1978), 461. doi: 10.1093/imamat/21.4.461.

[43]

T. H. Koornwinder, Orthogonal polynomials with weight function $(1-x)^\alpha(1+ x) ^\beta + M \delta (x+ 1)+ N \delta (x-1)$,, Canad. Math. Bull., 27 (1984), 205. doi: 10.4153/CMB-1984-030-7.

[44]

B. Krauskopf and J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Niño southern oscillation,, Proc. R. Soc. A, 470 (2014). doi: 10.1098/rspa.2014.0348.

[45]

T. Krisztin, Global dynamics of delay differential equations,, Periodica Mathematica Hungarica, 56 (2008), 83. doi: 10.1007/s10998-008-5083-x.

[46]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).

[47]

K. Kunisch, Approximation schemes for the linear-quadratic optimal control problem associated with delay equations,, SIAM J. Control and Optimization, 20 (1982), 506. doi: 10.1137/0320038.

[48]

M. Lakshmanan and D. V. Senthilkumar, Dynamics of Nonlinear Time-delay Systems,, Springer, (2010). doi: 10.1007/978-3-642-14938-2.

[49]

A. Lunardi, Linear and nonlinear diffusion problems,, in Lecture Notes, (2004).

[50]

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