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On the influence of cross-diffusion in pattern formation
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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