June  2017, 4(1&2): 71-118. doi: 10.3934/jcd.2017003

Rigorous continuation of bifurcation points in the diblock copolymer equation

1. 

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St W, Montreal, QC, H3A 0B9, Canada

2. 

Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030, USA

* Corresponding author: T. Wanner

Published  October 2017

We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through ${\mathbb{Z}}_2$ symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

Citation: Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

G. Arioli and H. Koch, Integration of dissipative partial differential equations: A case study, SIAM Journal on Applied Dynamical Systems, 9 (2010), 1119-1133.  doi: 10.1137/10078298X.  Google Scholar

[3]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.   Google Scholar

[4]

D. BlömkerB. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.  doi: 10.3934/dcds.2010.27.25.  Google Scholar

[5]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, in Mathematical Methods and Models in Phase Transitions (ed. A. Miranville), Nova Science Publishers, New York, 2005, 1–41.  Google Scholar

[6]

D. BlömkerE. Sander and T. Wanner, Degenerate nucleation in the Cahn-Hilliard-Cook model, SIAM Journal on Applied Dynamical Systems, 15 (2016), 459-494.  doi: 10.1137/15M1028844.  Google Scholar

[7]

M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[8]

R. ChoksiM. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.  doi: 10.1137/080728809.  Google Scholar

[9]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.  doi: 10.1023/A:1025722804873.  Google Scholar

[10]

P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems vol. 15 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. doi: 10.1142/4062.  Google Scholar

[11]

S. DayJ.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[12]

J. P. DesiH. EdreesJ. PriceE. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.  doi: 10.1137/100801378.  Google Scholar

[13]

M. Gameiro and J.-P. Lessard, Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation, Numer. Math., 117 (2011), 753-778.  doi: 10.1007/s00211-010-0350-3.  Google Scholar

[14]

M. GameiroJ.-P. Lessard and K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.  doi: 10.1016/j.matcom.2008.03.014.  Google Scholar

[15]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.  doi: 10.1017/S0308210500028079.  Google Scholar

[16]

A. HungriaJ.-P. Lessard and J. D. Mireles-James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[17]

I. JohnsonE. Sander and T. Wanner, Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.  doi: 10.3934/dcds.2013.33.3671.  Google Scholar

[18]

O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[19]

J. -P. Lessard, Continuation of solutions and studying delay differential equations via rigorous numerics, Proceedings of Symposia in Applied Mathematics. Google Scholar

[20]

J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006.  Google Scholar

[21]

J. -P. Lessard, E. Sander and T. Wanner, Matlab codes to perform the computer-assisted proofs, Available at http://archimede.mat.ulaval.ca/jplessard/rigbpcont/. Google Scholar

[22]

S. Maier-PaapeU. MillerK. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.  doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[23]

S. Maier-PaapeK. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.  doi: 10.1142/S0218127407017781.  Google Scholar

[24]

G. Moore and A. Spence, The calculation of turning points of nonlinear equations, SIAM Journal on Numerical Analysis, 17 (1980), 567-576.  doi: 10.1137/0717048.  Google Scholar

[25]

M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.  doi: 10.1081/NFA-100105107.  Google Scholar

[26]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[27]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[28]

M. Plum, An existence and inclusion method for two-point boundary value problems with turning points, Z. Angew. Math. Mech., 74 (1994), 615-623.  doi: 10.1002/zamm.19940741210.  Google Scholar

[29]

M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, J. Comput. Appl. Math., 60 (1995), 187-200.  doi: 10.1016/0377-0427(94)00091-E.  Google Scholar

[30]

S. Rump, INTLAB -INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104. Google Scholar

[31]

E. Sander and T. Wanner, Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.  doi: 10.1137/16M1061011.  Google Scholar

[32]

A. Spence and B. Werner, Nonsimple turning points and cusps, IMA Journal of Numerical Analysis, 2 (1982), 413-427.  doi: 10.1093/imanum/2.4.413.  Google Scholar

[33]

J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices of the American Mathematical Society,, 62 (2015), 1057-1061.  doi: 10.1090/noti1276.  Google Scholar

[34]

J. B. van den BergJ.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[35]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2d Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[36]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics Vol. 74. American Mathematical Society, to appear. Google Scholar

[37]

T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51. doi: 10.1007/978-4-431-56104-0_2.  Google Scholar

[38]

T. Wanner, Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.  doi: 10.3934/dcds.2017045.  Google Scholar

[39]

B. Werner and A. Spence, The computation of symmetry-breaking bifurcation points, SIAM Journal on Numerical Analysis, 21 (1984), 388-399.  doi: 10.1137/0721029.  Google Scholar

[40]

N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[41]

E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems, Springer-Verlag, New York – Berlin – Heidelberg, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

G. Arioli and H. Koch, Integration of dissipative partial differential equations: A case study, SIAM Journal on Applied Dynamical Systems, 9 (2010), 1119-1133.  doi: 10.1137/10078298X.  Google Scholar

[3]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.   Google Scholar

[4]

D. BlömkerB. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.  doi: 10.3934/dcds.2010.27.25.  Google Scholar

[5]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, in Mathematical Methods and Models in Phase Transitions (ed. A. Miranville), Nova Science Publishers, New York, 2005, 1–41.  Google Scholar

[6]

D. BlömkerE. Sander and T. Wanner, Degenerate nucleation in the Cahn-Hilliard-Cook model, SIAM Journal on Applied Dynamical Systems, 15 (2016), 459-494.  doi: 10.1137/15M1028844.  Google Scholar

[7]

M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[8]

R. ChoksiM. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.  doi: 10.1137/080728809.  Google Scholar

[9]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.  doi: 10.1023/A:1025722804873.  Google Scholar

[10]

P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems vol. 15 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. doi: 10.1142/4062.  Google Scholar

[11]

S. DayJ.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[12]

J. P. DesiH. EdreesJ. PriceE. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.  doi: 10.1137/100801378.  Google Scholar

[13]

M. Gameiro and J.-P. Lessard, Rigorous computation of smooth branches of equilibria for the three dimensional Cahn-Hilliard equation, Numer. Math., 117 (2011), 753-778.  doi: 10.1007/s00211-010-0350-3.  Google Scholar

[14]

M. GameiroJ.-P. Lessard and K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.  doi: 10.1016/j.matcom.2008.03.014.  Google Scholar

[15]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.  doi: 10.1017/S0308210500028079.  Google Scholar

[16]

A. HungriaJ.-P. Lessard and J. D. Mireles-James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[17]

I. JohnsonE. Sander and T. Wanner, Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.  doi: 10.3934/dcds.2013.33.3671.  Google Scholar

[18]

O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), 6 (1982), 427-434.  doi: 10.1090/S0273-0979-1982-15008-X.  Google Scholar

[19]

J. -P. Lessard, Continuation of solutions and studying delay differential equations via rigorous numerics, Proceedings of Symposia in Applied Mathematics. Google Scholar

[20]

J.-P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), 530-561.  doi: 10.1137/16M1056006.  Google Scholar

[21]

J. -P. Lessard, E. Sander and T. Wanner, Matlab codes to perform the computer-assisted proofs, Available at http://archimede.mat.ulaval.ca/jplessard/rigbpcont/. Google Scholar

[22]

S. Maier-PaapeU. MillerK. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.  doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[23]

S. Maier-PaapeK. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.  doi: 10.1142/S0218127407017781.  Google Scholar

[24]

G. Moore and A. Spence, The calculation of turning points of nonlinear equations, SIAM Journal on Numerical Analysis, 17 (1980), 567-576.  doi: 10.1137/0717048.  Google Scholar

[25]

M. T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim., 22 (2001), 321-356.  doi: 10.1081/NFA-100105107.  Google Scholar

[26]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[27]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[28]

M. Plum, An existence and inclusion method for two-point boundary value problems with turning points, Z. Angew. Math. Mech., 74 (1994), 615-623.  doi: 10.1002/zamm.19940741210.  Google Scholar

[29]

M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, J. Comput. Appl. Math., 60 (1995), 187-200.  doi: 10.1016/0377-0427(94)00091-E.  Google Scholar

[30]

S. Rump, INTLAB -INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104. Google Scholar

[31]

E. Sander and T. Wanner, Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.  doi: 10.1137/16M1061011.  Google Scholar

[32]

A. Spence and B. Werner, Nonsimple turning points and cusps, IMA Journal of Numerical Analysis, 2 (1982), 413-427.  doi: 10.1093/imanum/2.4.413.  Google Scholar

[33]

J. B. van den Berg and J.-P. Lessard, Rigorous numerics in dynamics, Notices of the American Mathematical Society,, 62 (2015), 1057-1061.  doi: 10.1090/noti1276.  Google Scholar

[34]

J. B. van den BergJ.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[35]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2d Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[36]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics Vol. 74. American Mathematical Society, to appear. Google Scholar

[37]

T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51. doi: 10.1007/978-4-431-56104-0_2.  Google Scholar

[38]

T. Wanner, Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.  doi: 10.3934/dcds.2017045.  Google Scholar

[39]

B. Werner and A. Spence, The computation of symmetry-breaking bifurcation points, SIAM Journal on Numerical Analysis, 21 (1984), 388-399.  doi: 10.1137/0721029.  Google Scholar

[40]

N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal., 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[41]

E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixed-Point Theorems, Springer-Verlag, New York – Berlin – Heidelberg, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

Figure 1.  Equilibrium bifurcation diagrams of the Cahn-Hilliard model (left image) and the diblock copolymer model for $\sigma = 6$ (right image). In both figures, $\mu = 0$. Most of the shown branches actually correspond to two or more solution branches, since the $L^2$-norm of the associated solutions is used as the vertical diagram axis. The colors correspond to the indices of the solutions. Along the horizontal trivial solution branch they increase from zero (black) to five (cyan)
Figure 2.  Two degenerate symmetry-breaking bifurcations as described in Example 2.14
Figure 3.  Equilibrium solutions $u_0$ (in blue) of the diblock copolymer equation for $\sigma_0 = 6$, together with their associated kernel functions $\varphi _0$ (in orange). These two distinct stationary solutions are both saddle-node bifurcation points at the same parameter value $\lambda_0 \approx 262.9$ and the same $L^2$-norm close to $0.562$. In fact, the entire non-trivial portion of the bifurcation diagram is multiply covered. These equilibria are rigorously proved in Theorem 3.7
Figure 4.  Equilibrium solutions $u_0$ (in blue) and associated kernel functions $\varphi _0$ (in orange) of the diblock copolymer equation for $\sigma_0 = 6$, with $\lambda_0 \approx 142.1, 53.6,203.1$ for top left, top right, and bottom right, respectively. All four solutions are pitchfork bifurcation points. They are the first bifurcation points on the first four branches bifurcating from the trivial solution in the right image sof Figure 1. Note that as in Figure 3, the bifurcation diagram is a double cover; corresponding to each of these four solutions, there is another solution at the same point in the bifurcation diagram. See also Theorems 3.13, 3.14, and 3.15
Figure 5.  Piecewise linear curve approximation (in black) constructed using parameter continuation and existence of a global solution curve $\mathcal C$ of $f=0$ (in blue) nearby the approximations
Figure 6.  Equilibrium solutions $u_0$ (in blue) of the diblock copolymer equation for $\sigma_0 = 6$, together with their associated kernel functions (in red). On left $\lambda_0 \approx 681.4$, on right $\lambda_0 \approx 1343.3$. These two distinct stationary solutions are both saddle-node bifurcation points. The equilibrium on the left (respectively right) is rigorously proved in Theorem 3.8 (respectively Theorem 3.9)
Figure 7.  Three global $C^\infty$ branches of saddle-node bifurcation points of the diblock copolymer equation. The red (respectively green, blue) branch is proven in Theorem 3.10 (respectively Theorem 3.11, Theorem 3.12)
Figure 8.  (Left) Global $C^\infty$ branches of pitchfork bifurcations points of the nonlinear diblock-copolymer equation (1). The red (respectively green, blue) branch is proven in Theorem 3.16 (respectively Theorem 3.17, Theorem 3.18). (Right) Zoom-in of the branches
Figure 9.  The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.7. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$
Figure 10.  The cosine Fourier coefficients of the saddle-node bifurcation point from Theorem 3.8. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$
Figure 11.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.13. We show ${\bar a }_k$ for $k \ge 1$. Note that all even coefficients are $0$
Figure 12.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.14. We show ${\bar a }_k$ for $k \ge 2$. Note that all other coefficients are $0$
Figure 13.  The cosine Fourier coefficients of the pitchfork bifurcation point from Theorem 3.15. We show ${\bar a }_k$ for $k \ge 2$. Note that all other coefficients are $0$
Table 1.  Some of the partial derivatives of the bifurcation function $b(\lambda,\nu)$ at the point $(\lambda_0,0)$ up to order three, together with the required partial derivatives of $W$
$\begin{align*}D_{\lambda} b(\lambda_0,0)&= \psi_0^* D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] D_{\lambda\alpha} b(\lambda_0,0)&= \psi_0^* D_{\lambda u}F(\lambda_0,u_0)[\varphi _0] \\[1ex] & ~~~~ + \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{\lambda}W(\lambda_0,v_0)] \; , \\[1ex] D_{\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; , \\[1ex] D_{\alpha\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uuu}F(\lambda_0,u_0)[\varphi _0,\varphi _0,\varphi _0] \\[1ex] & ~~~~ + 3 \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{vv}W(\lambda_0,v_0)[\varphi _0,\varphi _0]] \; , \\[1ex] L D_{\lambda} W(\lambda_0,v_0)&= -(I-P) D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] L D_{vv} W(\lambda_0,v_0)[\varphi _0,\varphi _0]&= -(I-P) D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; . \end{align*} $
$\begin{align*}D_{\lambda} b(\lambda_0,0)&= \psi_0^* D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] D_{\lambda\alpha} b(\lambda_0,0)&= \psi_0^* D_{\lambda u}F(\lambda_0,u_0)[\varphi _0] \\[1ex] & ~~~~ + \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{\lambda}W(\lambda_0,v_0)] \; , \\[1ex] D_{\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; , \\[1ex] D_{\alpha\alpha\alpha} b(\lambda_0,0)&= \psi_0^* D_{uuu}F(\lambda_0,u_0)[\varphi _0,\varphi _0,\varphi _0] \\[1ex] & ~~~~ + 3 \psi_0^* D_{uu}F(\lambda_0,u_0)[\varphi _0,D_{vv}W(\lambda_0,v_0)[\varphi _0,\varphi _0]] \; , \\[1ex] L D_{\lambda} W(\lambda_0,v_0)&= -(I-P) D_{\lambda}F(\lambda_0,u_0) \; , \\[1ex] L D_{vv} W(\lambda_0,v_0)[\varphi _0,\varphi _0]&= -(I-P) D_{uu}F(\lambda_0,u_0)[\varphi _0,\varphi _0] \; . \end{align*} $
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