# American Institute of Mathematical Sciences

April  2021, 8(2): 165-181. doi: 10.3934/jcd.2021008

## Chebyshev spectral methods for computing center manifolds

 Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

* Corresponding author: Kazuyuki Yagasaki

Present address: Nachi-Fujikoshi Corporation, 1-1-1 Fujikoshi-Honmachi, Toyama 930-8511, Japan

Received  August 2020 Revised  December 2020 Published  April 2021 Early access  March 2021

Fund Project: The second author was partially supported by JSPS Kakenhi Grant Number JP17H02859

We propose a numerical approach for computing center manifolds of equilibria in ordinary differential equations. Near the equilibria, the center manifolds are represented as graphs of functions satisfying certain partial differential equations (PDEs). We use a Chebyshev spectral method for solving the PDEs numerically to compute the center manifolds. We illustrate our approach for three examples: A two-dimensional system, the Hénon-Heiles system (a two-degree-of-freedom Hamiltonian system) and a three-degree-of-freedom Hamiltonian system which have one-, two- and four-dimensional center manifolds, respectively. The obtained results are compared with polynomial approximations and other numerical computations.

Citation: Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008
##### References:
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Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-53691-0.  Google Scholar [22] T. Sakajo and K. Yagasaki, Chaotic motion of the $N$-vortex problem on a sphere: I. Saddle-centers in two-degree-of-freedom Hamiltonians, J. Nonlinear Sci., 18 (2008), 485-525.  doi: 10.1007/s00332-008-9019-9.  Google Scholar [23] J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations: Theory and applications, SIAM J. Numer. Anal., 41 (2003), 325-363.  doi: 10.1137/S0036142901392742.  Google Scholar [24] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.  Google Scholar [25] A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in Dynamics Reported, Vol. 2 (eds. U. Kirchgraber and H. O. Walther), John Wiley and Sons, Chichester, (1989), 89–169.  Google Scholar [26] J. A. C. Weideman and S. C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software, 26 (2000), 465-519.  doi: 10.1145/365723.365727.  Google Scholar [27] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar [28] K. Yagasaki, Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers, Discrete Contin. Dyn. Syst. A, 29 (2011), 387-402.  doi: 10.3934/dcds.2011.29.387.  Google Scholar

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##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley, Redwood City, CA, 1978. doi: 10.1090/chel/364.  Google Scholar [2] W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math., 80 (1998), 1-38.  doi: 10.1007/s002110050357.  Google Scholar [3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd. ed., Dover, Mineola, NY, 2001.  Google Scholar [4] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.  Google Scholar [5] A. R. Champneys and G. J. Lord, Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem, Phys. D, 102 (1997), 101-124. doi: 10.1016/S0167-2789(96)00206-0.  Google Scholar [6] E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, 2012. Available online from http://cmvl.cs.concordia.ca. Google Scholar [7] T. Eirola and J. von Pfaler, Numerical Taylor expansions for invariant manifolds, Numer. Math., 99 (2004), 25-46.  doi: 10.1007/s00211-004-0537-6.  Google Scholar [8] A. Farrés and À. Jorba, On the high order approximation of the centre manifold for ODEs, Discrete Contin. Dyn. Syst. B, 14 (2010), 977-1000.  doi: 10.3934/dcdsb.2010.14.977.  Google Scholar [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [10] J. Guckenheimer and A. Vladimirsky, A fast method for approximating invariant manifolds, SIAM J. Appl. Dynam. Sys., 3 (2004), 232-260.  doi: 10.1137/030600179.  Google Scholar [11] À. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations, Springer-Verlag, Cham, Switzerland, 2016. doi: 10.1007/978-3-319-29662-3.  Google Scholar [12] M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79.  doi: 10.1086/109234.  Google Scholar [13] À. Jorba, A methodology for the numerical computation of normal forms, centre manifolds and first integrals of Hamiltonian systems, Experiment. Math., 8 (1999), 155-195.  doi: 10.1080/10586458.1999.10504397.  Google Scholar [14] D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-90-481-2261-5.  Google Scholar [15] B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Springer-Verlag, Dordrecht, (2007), 117–154. doi: 10.1007/978-1-4020-6356-5_4.  Google Scholar [16] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.  doi: 10.1142/S0218127405012533.  Google Scholar [17] F. Ma and T. Küpper, A numerical method to calculate center manifolds of ODE's, Appl. Anal., 54 (1994), 1-15.  doi: 10.1080/00036819408840264.  Google Scholar [18] F. Ma and T. Küpper, Numerical calculation of invariant manifolds for maps, Numer. Linear Algebra Appl., 1 (1994), 141-150.  doi: 10.1002/nla.1680010205.  Google Scholar [19] MATLAB, Version 9.7.0 (R2019b), The MathWorks Inc., Natick, MA, 2019. Google Scholar [20] Mathematica, Version 12.0, Wolfram Research, Inc., Champaign, IL, 2019. Google Scholar [21] K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, 3$^{rd}$ edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-53691-0.  Google Scholar [22] T. Sakajo and K. Yagasaki, Chaotic motion of the $N$-vortex problem on a sphere: I. Saddle-centers in two-degree-of-freedom Hamiltonians, J. Nonlinear Sci., 18 (2008), 485-525.  doi: 10.1007/s00332-008-9019-9.  Google Scholar [23] J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations: Theory and applications, SIAM J. Numer. Anal., 41 (2003), 325-363.  doi: 10.1137/S0036142901392742.  Google Scholar [24] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598.  Google Scholar [25] A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in Dynamics Reported, Vol. 2 (eds. U. Kirchgraber and H. O. Walther), John Wiley and Sons, Chichester, (1989), 89–169.  Google Scholar [26] J. A. C. Weideman and S. C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software, 26 (2000), 465-519.  doi: 10.1145/365723.365727.  Google Scholar [27] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar [28] K. Yagasaki, Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers, Discrete Contin. Dyn. Syst. A, 29 (2011), 387-402.  doi: 10.3934/dcds.2011.29.387.  Google Scholar
Numerically computed center manifold in (13): (a) $\alpha = 1$; (b) $\alpha = -1$. Black lines represent numerical results obtained by our approach with $N_1 = 8$. For comparison, the 2nd-, 4th-, 16th- and 32nd-order polynomial approximations are plotted as purple, green, blue and red lines, respectively. Numerically computed orbits are also plotted as orange lines with arrows representing their directions
Numerically computed center manifold by our approach for $N_1 = 4$, $8$ and $16$ in (13): (a) $\alpha = 1$; (b) $\alpha = -1$. Red, black and blue lines, which agree almost completely in the computed ranges, represent the results for $N_1 = 4$, $8$ and $16$, respectively
One-parameter families of periodic orbits around the saddle-center in (15): (a) Case (ⅰ); (b) case (ⅱ)
are plotted as black dashed lines">Figure 4.  Numerically computed center manifold on the section $x_2 = 0$ for (15): (a) Case (ⅰ); (b) case (ⅱ). Green, blue and red lines represent numerical results obtained by the approach with $N_1 = N_2 = 4$, $8$ and $16$, respectively. For comparison, the one-parameter families of periodic orbits in Fig. 3 are plotted as black dashed lines
are plotted as black dashed lines">Figure 5.  Polynomial approximations of the center manifold on the section $x_2 = 0$ for (15): (a) Case (ⅰ); (b) case (ⅱ). Purple, green, blue and red lines represent the 2nd-, 4th-, 8th- and 16th-order polynomial approximations. For comparison, the one-parameter family of periodic orbits in Fig. 3 are plotted as black dashed lines
on the section $x_2 = 0$ for (15): (a) Case (ⅰ); (b) case (ⅱ). Green and blue lines, respectively, represent the 8th- and 16th-order polynomial approximations, and red and black lines, respectively, our numerical results with $N_1 = N_2 = 8$ and $16$">Figure 6.  Differences between our numerical results or polynomial approximations and the one-parameter family of periodic orbits in Fig. 3 on the section $x_2 = 0$ for (15): (a) Case (ⅰ); (b) case (ⅱ). Green and blue lines, respectively, represent the 8th- and 16th-order polynomial approximations, and red and black lines, respectively, our numerical results with $N_1 = N_2 = 8$ and $16$
One-parameter families of periodic orbits around the saddle-center in (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The one-parameter families of periodic orbits which are tangent to the $(x_1,x_3)$- plane (resp. the $(x_2,x_4)$-plane) at the origin are plotted in Figs. (a) and (c) (resp. Figs. (b) and (d))
and (b) (resp. in Figs. 7(c) and (d)) are, respectively, used in Figs. (a) and (b) (resp. Figs. (c) and (d)). Blue and red lines represent numerical results obtained by the approach with $N_j = 4$ and $8$, $j = 1$-$4$, respectively. For comparison, the one-parameter families of periodic orbits in Fig. 7 are plotted as black dashed lines">Figure 8.  Numerically computed center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The data of $x_j$, $j = 1$-$4$, for the one-parameter families of periodic orbits displayed in Figs. 7(a) and (b) (resp. in Figs. 7(c) and (d)) are, respectively, used in Figs. (a) and (b) (resp. Figs. (c) and (d)). Blue and red lines represent numerical results obtained by the approach with $N_j = 4$ and $8$, $j = 1$-$4$, respectively. For comparison, the one-parameter families of periodic orbits in Fig. 7 are plotted as black dashed lines
and (b) (resp. in Figs. 7(c) and (d)) are, respectively, used in Figs. (a) and (b) (resp. Figs. (c) and (d)). Green, blue and red lines represent the 2nd-, 4th- and 8th-order polynomial approximations. For comparison, the one-parameter families of periodic orbits in Fig. 7 are plotted as black dashed lines">Figure 9.  Polynomial approximations of the center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The data of $x_j$, $j = 1$-$4$, for the one-parameter families of periodic orbits displayed in Figs. 7(a) and (b) (resp. in Figs. 7(c) and (d)) are, respectively, used in Figs. (a) and (b) (resp. Figs. (c) and (d)). Green, blue and red lines represent the 2nd-, 4th- and 8th-order polynomial approximations. For comparison, the one-parameter families of periodic orbits in Fig. 7 are plotted as black dashed lines
for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The section $x_2 = x_3 = x_4 = 0$ is taken in Figs. (a) and (c) while the section $x_1 = x_3 = x_4 = 0$ is taken in Figs. (b) and (d). Green and blue lines, respectively, represent the 4th- and 8th-order polynomial approximations, and red and black lines, respectively, our numerical results with $N_j = 4$ and $8$, $j = 1$-$4$">Figure 10.  Differences between our numerical results or polynomial approximations and the one-parameter family of periodic orbits in Fig. 7 for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The section $x_2 = x_3 = x_4 = 0$ is taken in Figs. (a) and (c) while the section $x_1 = x_3 = x_4 = 0$ is taken in Figs. (b) and (d). Green and blue lines, respectively, represent the 4th- and 8th-order polynomial approximations, and red and black lines, respectively, our numerical results with $N_j = 4$ and $8$, $j = 1$-$4$
Numerically computed center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) Case (ⅱ). The section $x_2 = x_3 = x_4 = 0$ is taken in Figs. (a) and (c) while the section $x_1 = x_3 = x_4 = 0$ is taken in Figs. (b) and (d). Red and blue lines, respectively, represent numerical results obtained by the approach with $N_j = 8$, $j = 1$-$4$, and the 8th-order polynomial approximations
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