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.
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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 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
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Numerically computed center manifold in (13): (a)
Numerically computed center manifold by our approach for
One-parameter families of periodic orbits around the saddle-center in (15): (a) Case (ⅰ); (b) case (ⅱ)
Numerically computed center manifold on the section
Polynomial approximations of the center manifold on the section
Differences between our numerical results or polynomial approximations and the one-parameter family of periodic orbits in Fig. 3 on the section
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
Numerically computed center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The data of
Polynomial approximations of the center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) case (ⅱ). The data of
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
Numerically computed center manifold for (17): (a) and (b) Case (ⅰ); (c) and (d) Case (ⅱ). The section