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# Chebyshev spectral methods for computing center manifolds

• * Corresponding author: Kazuyuki Yagasaki

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

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.

Mathematics Subject Classification: Primary: 37M21, 65P99; Secondary: 34C45.

 Citation:

• Figure 1.  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

Figure 2.  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

Figure 3.  One-parameter families of periodic orbits around the saddle-center in (15): (a) Case (ⅰ); (b) case (ⅱ)

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

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

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 $16$

Figure 7.  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))

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

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

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 $8$, $j = 1$-$4$

Figure 11.  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

•  [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2$^{nd}$ edition, Addison-Wesley, Redwood City, CA, 1978. doi: 10.1090/chel/364. [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. [3] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd. ed., Dover, Mineola, NY, 2001. [4] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] MATLAB, Version 9.7.0 (R2019b), The MathWorks Inc., Natick, MA, 2019. [20] Mathematica, Version 12.0, Wolfram Research, Inc., Champaign, IL, 2019. [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. [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. [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. [24] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. [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. [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. [27] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2$^{nd}$ edition, Springer-Verlag, New York, 2003. [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.

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