# American Institute of Mathematical Sciences

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:

show all references

##### References:
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)
Two degenerate symmetry-breaking bifurcations as described in Example 2.14
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
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
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
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)
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)
(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
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 $0$
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 $0$
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 $0$
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 $0$
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 $0$
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*}
 [1] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [2] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [3] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [4] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [5] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [6] Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464 [7] Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316 [8] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [9] Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 [10] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349 [11] Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459 [12] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [13] Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431 [14] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275 [15] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [16] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [17] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [18] Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347 [19] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [20] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

Impact Factor: