# 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, 2016, 36 (8) : 4133-4177. doi: 10.3934/dcds.2016.36.4133
##### References:
 [1] M. Alfaro, F. Marcellán and M. L. Rezola, Estimates for Jacobi-Sobolev type orthogonal polynomials, Appl. Anal., 67 (1997), 157-174. doi: 10.1080/00036819708840602.  Google Scholar [2] H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control Optim., 16 (1978), 169-208. doi: 10.1137/0316013.  Google Scholar [3] H. T. Banks and F. Kappel, Spline approximations for functional differential equations, Journal of Differential Equations, 34 (1979), 496-522. doi: 10.1016/0022-0396(79)90033-0.  Google Scholar [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), 830-855. doi: 10.1137/0905059.  Google Scholar [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), 1687-1712. doi: 10.1175/1520-0469(1989)046<1687:IVIATA>2.0.CO;2.  Google Scholar [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), 1747-1773. doi: 10.1175/1520-0469(1982)039<1747:IVOAEB>2.0.CO;2.  Google Scholar [7] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [8] J. Burns, T. Herdman and H. Stech, Linear functional differential equations as semigroups on product spaces, SIAM Journal on Mathematical Analysis, 14 (1983), 98-116. doi: 10.1137/0514007.  Google Scholar [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), 1562-1577. doi: 10.1175/1520-0469(1990)047<1562:ASOSEO>2.0.CO;2.  Google Scholar [10] A. Casal and M. Freedman, A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations, Automatic Control, IEEE Transactions on, 25 (1980), 967-973. doi: 10.1109/TAC.1980.1102450.  Google Scholar [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, Oxford, 1998.  Google Scholar [12] M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Commun. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.  Google Scholar [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), 11766-11771. Google Scholar [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), 1684-1690. doi: 10.1073/pnas.1321816111.  Google Scholar [15] S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations, 26 (1977), 112-159. doi: 10.1016/0022-0396(77)90101-2.  Google Scholar [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.  Google Scholar [17] S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics, 30 (2002), 323-335. doi: 10.1023/A:1021220117746.  Google Scholar [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.  Google Scholar [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, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [20] H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large-Scale Ocean Circulation and El Niño, Springer, New York/Berlin, 2005. doi: 10.1007/978-94-015-9450-9.  Google Scholar [21] H. A. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43 (2005), RG3001. doi: 10.1029/2002RG000122.  Google Scholar [22] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617.  Google Scholar [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), 181-200. doi: 10.1006/jdeq.1995.1144.  Google Scholar [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), 2936-2950. doi: 10.1175/1520-0469(2000)057<2936:ESPLTT>2.0.CO;2.  Google Scholar [25] M. Ghil, Hilbert problems for the geosciences in the 21st century, Nonlin. Processes Geophys., 8 (2001), 211-211. doi: 10.5194/npg-8-211-2001.  Google Scholar [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), 1-41. doi: 10.1029/2000RG000092.  Google Scholar [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.  Google Scholar [28] M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Applied Mathematical Sciences, Springer-Verlag, New York/Berlin/London/Paris/ Tokyo, 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), Academic Press, San Diego, 70 (2001), 285-325. 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), 2493-2500. 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, Modeling and Economics of Extreme Events (eds. M. Chavez, M. Ghil and J. Urrutia-Fucugauchi), American Geophysical Union, in press, 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), 417-433. 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, New York, 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, Cambridge, 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), 737-759. 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), 101-112. 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), 70-72. Google Scholar [38] F. Kappel, Semigroups and delay equations, in Semigroups, Theory and Applications, Vol. II (Trieste, 1984), vol. 152 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986, 136-176.  Google Scholar [39] F. Kappel and D. Salamon, Spline approximation for retarded systems and the Riccati equation, SIAM J.Control Optim., 25 (1987), 1082-1117. doi: 10.1137/0325060.  Google Scholar [40] F. Kappel and W. Schappacher, Autonomous nonlinear functional differential equations and averaging approximations, Nonlinear Analysis: Theory, Methods & Applications, 2 (1978), 391-422. doi: 10.1016/0362-546X(78)90048-2.  Google Scholar [41] F. Kappel and K. Zhang, Equivalence of functional-differential equations of neutral type and abstract Cauchy problems, Monatshefte für Mathematik, 101 (1986), 115-133. doi: 10.1007/BF01298925.  Google Scholar [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), 461-477. doi: 10.1093/imamat/21.4.461.  Google Scholar [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), 205-214. doi: 10.4153/CMB-1984-030-7.  Google Scholar [44] B. Krauskopf and J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Niño southern oscillation, Proc. R. Soc. A, 470 (2014), 20140348, 18pp. doi: 10.1098/rspa.2014.0348.  Google Scholar [45] T. Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.  Google Scholar [46] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.  Google Scholar [47] K. Kunisch, Approximation schemes for the linear-quadratic optimal control problem associated with delay equations, SIAM J. Control and Optimization, 20 (1982), 506-540. doi: 10.1137/0320038.  Google Scholar [48] M. Lakshmanan and D. V. Senthilkumar, Dynamics of Nonlinear Time-delay Systems, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-14938-2.  Google Scholar [49] A. Lunardi, Linear and nonlinear diffusion problems, in Lecture Notes, 2004, URL http://www.math.unipr.it/~lunardi/LectureNotes/Cortona2004.pdf. Google Scholar [50] N. MacDonald, Biological Delay Systems: Linear Stability Theory, vol. 8 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1989.  Google Scholar [51] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.  Google Scholar [52] W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-based Approach, vol. 12 of Advances in Design and Control, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898718645.  Google Scholar [53] M. Münnich, M. A. Cane and S. E. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: Nonlinear cases, J. Atmos. Sci., 48 (1991), 1238-1248. Google Scholar [54] S.-I. Nakagiri and M. Yamamoto, Controllability and observability of linear retarded systems in banach spaces, International Journal of Control, 49 (1989), 1489-1504. doi: 10.1080/00207178908559721.  Google Scholar [55] A. H. Nayfeh, Order reduction of retarded nonlinear systems-the method of multiple scales versus center-manifold reduction, Nonlinear Dyn., 51 (2008), 483-500. doi: 10.1007/s11071-007-9237-y.  Google Scholar [56] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [57] L. Roques, M. D. Chekroun, M. Cristofol, S. Soubeyrand and M. Ghil, Parameter estimation for energy balance models with memory, Proc. R. Soc. A, 470 (2014), 20140349, 20pp. doi: 10.1098/rspa.2014.0349.  Google Scholar [58] J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, vol. 41 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar [59] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar [60] J. C. Sprott, A simple chaotic delay differential equation, Physics Letters A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083.  Google Scholar [61] G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Functions, vol. 210 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar [62] G. Szegő, Orthogonal Polynomials, 4th edition, American Mathematical Society, Providence, R.I., 1975. Google Scholar [63] E. Tziperman, L. Stone, M. A. Cane and H. Jarosh, El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264 (1994), 72-74. Google Scholar [64] V. B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Computat. Math. Math. Phys., 9 (1969), 1253-1262.  Google Scholar [65] C. P. Vyasarayani, Galerkin approximations for higher order delay differential equations, J. Comput. Nonlinear Dynamics, 7 (2012), 031004, 5pp. doi: 10.1115/1.4005931.  Google Scholar [66] P. Wahi and A. Chatterjee, Galerkin projections for delay differential equations, ASME. J. Dyn. Syst., Meas., Control, 127 (2005), 80-87. doi: 10.1115/DETC2003/VIB-48570.  Google Scholar [67] X. Wang, Approximating stationary statistical properties, Chinese Annals of Mathematics, Series B, 30 (2009), 831-844. doi: 10.1007/s11401-009-0178-2.  Google Scholar [68] G. F. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Diff. Equat., 20 (1976), 71-89. doi: 10.1016/0022-0396(76)90097-8.  Google Scholar [69] W. Wischert, A. Wunderlin, A. Pelster, M. Olivier and J. Groslambert, Delay-induced instabilities in nonlinear feedback systems, Physical Review E, 49 (1994), 203-219. doi: 10.1103/PhysRevE.49.203.  Google Scholar [70] I. Zaliapin and M. Ghil, A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema, Nonlin. Processes Geophys., 17 (2010), 123-135. doi: 10.5194/npg-17-123-2010.  Google Scholar [71] T. Živković and K. Rypdal, ENSO dynamics: Low-dimensional-chaotic or stochastic?, J. Geophys. Res. Atmos., 118 (2013), 2161-2168. Google Scholar

show all references

##### References:
 [1] M. Alfaro, F. Marcellán and M. L. Rezola, Estimates for Jacobi-Sobolev type orthogonal polynomials, Appl. Anal., 67 (1997), 157-174. doi: 10.1080/00036819708840602.  Google Scholar [2] H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control Optim., 16 (1978), 169-208. doi: 10.1137/0316013.  Google Scholar [3] H. T. Banks and F. Kappel, Spline approximations for functional differential equations, Journal of Differential Equations, 34 (1979), 496-522. doi: 10.1016/0022-0396(79)90033-0.  Google Scholar [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), 830-855. doi: 10.1137/0905059.  Google Scholar [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), 1687-1712. doi: 10.1175/1520-0469(1989)046<1687:IVIATA>2.0.CO;2.  Google Scholar [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), 1747-1773. doi: 10.1175/1520-0469(1982)039<1747:IVOAEB>2.0.CO;2.  Google Scholar [7] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.  Google Scholar [8] J. Burns, T. Herdman and H. Stech, Linear functional differential equations as semigroups on product spaces, SIAM Journal on Mathematical Analysis, 14 (1983), 98-116. doi: 10.1137/0514007.  Google Scholar [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), 1562-1577. doi: 10.1175/1520-0469(1990)047<1562:ASOSEO>2.0.CO;2.  Google Scholar [10] A. Casal and M. Freedman, A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations, Automatic Control, IEEE Transactions on, 25 (1980), 967-973. doi: 10.1109/TAC.1980.1102450.  Google Scholar [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, Oxford, 1998.  Google Scholar [12] M. D. Chekroun and N. E. Glatt-Holtz, Invariant measures for dissipative dynamical systems: Abstract results and applications, Commun. Math. Phys., 316 (2012), 723-761. doi: 10.1007/s00220-012-1515-y.  Google Scholar [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), 11766-11771. Google Scholar [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), 1684-1690. doi: 10.1073/pnas.1321816111.  Google Scholar [15] S. N. Chow and J. Mallet-Paret, Integral averaging and bifurcation, J. Differential Equations, 26 (1977), 112-159. doi: 10.1016/0022-0396(77)90101-2.  Google Scholar [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.  Google Scholar [17] S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics, 30 (2002), 323-335. doi: 10.1023/A:1021220117746.  Google Scholar [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.  Google Scholar [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, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar [20] H. A. Dijkstra, Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large-Scale Ocean Circulation and El Niño, Springer, New York/Berlin, 2005. doi: 10.1007/978-94-015-9450-9.  Google Scholar [21] H. A. Dijkstra and M. Ghil, Low-frequency variability of the large-scale ocean circulation: A dynamical systems approach, Rev. Geophys., 43 (2005), RG3001. doi: 10.1029/2002RG000122.  Google Scholar [22] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617.  Google Scholar [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), 181-200. doi: 10.1006/jdeq.1995.1144.  Google Scholar [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), 2936-2950. doi: 10.1175/1520-0469(2000)057<2936:ESPLTT>2.0.CO;2.  Google Scholar [25] M. Ghil, Hilbert problems for the geosciences in the 21st century, Nonlin. Processes Geophys., 8 (2001), 211-211. doi: 10.5194/npg-8-211-2001.  Google Scholar [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), 1-41. doi: 10.1029/2000RG000092.  Google Scholar [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.  Google Scholar [28] M. Ghil and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Applied Mathematical Sciences, Springer-Verlag, New York/Berlin/London/Paris/ Tokyo, 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), Academic Press, San Diego, 70 (2001), 285-325. 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), 2493-2500. 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, Modeling and Economics of Extreme Events (eds. M. Chavez, M. Ghil and J. Urrutia-Fucugauchi), American Geophysical Union, in press, 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), 417-433. 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, New York, 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, Cambridge, 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), 737-759. 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), 101-112. 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), 70-72. Google Scholar [38] F. Kappel, Semigroups and delay equations, in Semigroups, Theory and Applications, Vol. II (Trieste, 1984), vol. 152 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1986, 136-176.  Google Scholar [39] F. Kappel and D. Salamon, Spline approximation for retarded systems and the Riccati equation, SIAM J.Control Optim., 25 (1987), 1082-1117. doi: 10.1137/0325060.  Google Scholar [40] F. Kappel and W. Schappacher, Autonomous nonlinear functional differential equations and averaging approximations, Nonlinear Analysis: Theory, Methods & Applications, 2 (1978), 391-422. doi: 10.1016/0362-546X(78)90048-2.  Google Scholar [41] F. Kappel and K. Zhang, Equivalence of functional-differential equations of neutral type and abstract Cauchy problems, Monatshefte für Mathematik, 101 (1986), 115-133. doi: 10.1007/BF01298925.  Google Scholar [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), 461-477. doi: 10.1093/imamat/21.4.461.  Google Scholar [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), 205-214. doi: 10.4153/CMB-1984-030-7.  Google Scholar [44] B. Krauskopf and J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Niño southern oscillation, Proc. R. Soc. A, 470 (2014), 20140348, 18pp. doi: 10.1098/rspa.2014.0348.  Google Scholar [45] T. Krisztin, Global dynamics of delay differential equations, Periodica Mathematica Hungarica, 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.  Google Scholar [46] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993.  Google Scholar [47] K. Kunisch, Approximation schemes for the linear-quadratic optimal control problem associated with delay equations, SIAM J. Control and Optimization, 20 (1982), 506-540. doi: 10.1137/0320038.  Google Scholar [48] M. Lakshmanan and D. V. Senthilkumar, Dynamics of Nonlinear Time-delay Systems, Springer, Heidelberg, 2010. doi: 10.1007/978-3-642-14938-2.  Google Scholar [49] A. Lunardi, Linear and nonlinear diffusion problems, in Lecture Notes, 2004, URL http://www.math.unipr.it/~lunardi/LectureNotes/Cortona2004.pdf. Google Scholar [50] N. MacDonald, Biological Delay Systems: Linear Stability Theory, vol. 8 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1989.  Google Scholar [51] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.  Google Scholar [52] W. Michiels and S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-based Approach, vol. 12 of Advances in Design and Control, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898718645.  Google Scholar [53] M. Münnich, M. A. Cane and S. E. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: Nonlinear cases, J. Atmos. Sci., 48 (1991), 1238-1248. Google Scholar [54] S.-I. Nakagiri and M. Yamamoto, Controllability and observability of linear retarded systems in banach spaces, International Journal of Control, 49 (1989), 1489-1504. doi: 10.1080/00207178908559721.  Google Scholar [55] A. H. Nayfeh, Order reduction of retarded nonlinear systems-the method of multiple scales versus center-manifold reduction, Nonlinear Dyn., 51 (2008), 483-500. doi: 10.1007/s11071-007-9237-y.  Google Scholar [56] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [57] L. Roques, M. D. Chekroun, M. Cristofol, S. Soubeyrand and M. Ghil, Parameter estimation for energy balance models with memory, Proc. R. Soc. A, 470 (2014), 20140349, 20pp. doi: 10.1098/rspa.2014.0349.  Google Scholar [58] J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, vol. 41 of Springer Series in Computational Mathematics, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar [59] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar [60] J. C. Sprott, A simple chaotic delay differential equation, Physics Letters A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083.  Google Scholar [61] G. Stepan, Retarded Dynamical Systems: Stability and Characteristic Functions, vol. 210 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar [62] G. Szegő, Orthogonal Polynomials, 4th edition, American Mathematical Society, Providence, R.I., 1975. Google Scholar [63] E. Tziperman, L. Stone, M. A. Cane and H. Jarosh, El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science, 264 (1994), 72-74. Google Scholar [64] V. B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Computat. Math. Math. Phys., 9 (1969), 1253-1262.  Google Scholar [65] C. P. Vyasarayani, Galerkin approximations for higher order delay differential equations, J. Comput. Nonlinear Dynamics, 7 (2012), 031004, 5pp. doi: 10.1115/1.4005931.  Google Scholar [66] P. Wahi and A. Chatterjee, Galerkin projections for delay differential equations, ASME. J. Dyn. Syst., Meas., Control, 127 (2005), 80-87. doi: 10.1115/DETC2003/VIB-48570.  Google Scholar [67] X. Wang, Approximating stationary statistical properties, Chinese Annals of Mathematics, Series B, 30 (2009), 831-844. doi: 10.1007/s11401-009-0178-2.  Google Scholar [68] G. F. Webb, Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Diff. Equat., 20 (1976), 71-89. doi: 10.1016/0022-0396(76)90097-8.  Google Scholar [69] W. Wischert, A. Wunderlin, A. Pelster, M. Olivier and J. Groslambert, Delay-induced instabilities in nonlinear feedback systems, Physical Review E, 49 (1994), 203-219. doi: 10.1103/PhysRevE.49.203.  Google Scholar [70] I. Zaliapin and M. Ghil, A delay differential model of ENSO variability, Part 2: Phase locking, multiple solutions, and dynamics of extrema, Nonlin. Processes Geophys., 17 (2010), 123-135. doi: 10.5194/npg-17-123-2010.  Google Scholar [71] T. Živković and K. Rypdal, ENSO dynamics: Low-dimensional-chaotic or stochastic?, J. Geophys. Res. Atmos., 118 (2013), 2161-2168. Google Scholar
 [1] Rodrigo Donizete Euzébio, Jaume Llibre. Periodic solutions of El Niño model through the Vallis differential system. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3455-3469. doi: 10.3934/dcds.2014.34.3455 [2] Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801 [3] Matthieu Arfeux, Jan Kiwi. Topological cubic polynomials with one periodic ramification point. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1799-1811. doi: 10.3934/dcds.2020094 [4] Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005 [5] He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021 [6] Cunsheng Ding, Chunming Tang. Infinite families of $3$-designs from o-polynomials. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020082 [7] Darren C. Ong. Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Conference Publications, 2013, 2013 (special) : 605-609. doi: 10.3934/proc.2013.2013.605 [8] Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021121 [9] Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357 [10] Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002 [11] Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006 [12] Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465 [13] Elisavet Konstantinou, Aristides Kontogeorgis. Some remarks on the construction of class polynomials. Advances in Mathematics of Communications, 2011, 5 (1) : 109-118. doi: 10.3934/amc.2011.5.109 [14] Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004 [15] Bin Han. Some multivariate polynomials for doubled permutations. Electronic Research Archive, 2021, 29 (2) : 1925-1944. doi: 10.3934/era.2020098 [16] Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 529-544. doi: 10.3934/dcdsb.2019252 [17] Cecilia Cavaterra, Maurizio Grasselli. Asymptotic behavior of population dynamics models with nonlocal distributed delays. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 861-883. doi: 10.3934/dcds.2008.22.861 [18] Janos Kollar. Polynomials with integral coefficients, equivalent to a given polynomial. Electronic Research Announcements, 1997, 3: 17-27. [19] Nur Fadhilah Ibrahim. An algorithm for the largest eigenvalue of nonhomogeneous nonnegative polynomials. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 75-91. doi: 10.3934/naco.2014.4.75 [20] Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

2019 Impact Factor: 1.338