American Institute of Mathematical Sciences

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

Chebyshev spectral methods for computing center manifolds

Takeshi Saito †, and  Kazuyuki Yagasaki ,

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.

Keywords: Center manifold, dynamical system, numerical computation, Chebyshev spectral method, Hamiltonian system.
Mathematics Subject Classification: Primary: 37M21, 65P99; Secondary: 34C45.
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:
[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.

show all references

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.

[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.

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[2]

Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365
[3]

A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97
[4]

Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080
[5]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[6]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104
[7]

Jingwei Hu, Jie Shen, Yingwei Wang. A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions. Kinetic and Related Models, 2020, 13 (4) : 677-702. doi: 10.3934/krm.2020023
[8]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419
[9]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
[10]

Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485
[11]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402
[12]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[13]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[14]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692
[15]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[16]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure and Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161
[17]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[18]

Chui-Jie Wu, Hongliang Zhao. Generalized HWD-POD method and coupling low-dimensional dynamical system of turbulence. Conference Publications, 2001, 2001 (Special) : 371-379. doi: 10.3934/proc.2001.2001.371
[19]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks and Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[20]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51

 Impact Factor: 

Tools

Metrics

  • PDF downloads (209)
  • HTML views (353)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy