The aim of this paper is to obtain a high-order approximation of the canard explosion in the Brusselator equations. This classical chemical system has been extensively studied but, until now, only first-order approximation to the canard explosion has been provided. Here, with the help of the nonlinear time transformation method, we are able to obtain an approximation to any desired order. Our results strongly agree with those obtained by numerical continuation.

Keywords:

References

Algaba, A., Chung, K. W., Qin, B. W. & Rodríguez-Luis, A. J. [2019] “ A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens–Bogdanov normal form,” Nonlin. Dyn. 97, 979–990. Web of Science, Google Scholar
Algaba, A., Chung, K. W., Qin, B. W. & Rodríguez-Luis, A. J. [2020a] “ Computation of all the coefficients for the global connections in the $ℤ_{2}$ -symmetric Takens–Bogdanov normal forms,” Commun. Nonlin. Sci. Numer. Simulat. 81, 105012. Web of Science, Google Scholar
Algaba, A., Chung, K. W., Qin, B. W. & Rodríguez-Luis, A. J. [2020b] “ Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method,” Physica D 406, 132384. Web of Science, Google Scholar
Ambrosio, B., Aziz-Alaoui, M. A. & Yafia, R. [2018] “ Canard phenomenon in a slow–fast modified Leslie–Gower model,” Math. Biosci. 295, 48–54. Web of Science, Google Scholar
Atabaigi, A. & Barati, A. [2017] “ Relaxation oscillations and canard explosion in a predator–prey system of Holling and Leslie types,” Nonlin. Anal.: Real World Appl. 36, 139–153. Web of Science, Google Scholar
Baer, S. M. & Erneux, T. [1986] “ Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739. Web of Science, Google Scholar
Benoit, E., Callot, J. F., Diener, F. & Diener, M. [1981] “ Chasse au canard,” Collect. Math. 31–32, 37–119. Google Scholar
Bie, Q. [2011] “ Pattern formation in a general two-cell Brusselator model,” J. Math. Anal. Appl. 376, 551–564. Web of Science, Google Scholar
Brøns, M. & Bar-Eli, K. [1991] “ Canard explosion and excitation in a model of the Belousov–Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713. Web of Science, Google Scholar
Brøns, M. & Bar-Eli, K. [1994] “ Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. Roy. Soc. London A 445, 305–322. Web of Science, Google Scholar
Brøns, M. & Sturis, J. [2001] “ Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209. Web of Science, Google Scholar
Brøns, M. [2005] “ Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. Roy. Soc. A 461, 2289–2302. Google Scholar
Brown, K. J. & Davidson, F. A. [1995] “ Global bifurcation in the Brusselator system,” Nonlin. Anal. 24, 1713–1725. Web of Science, Google Scholar
Cao, Y. Y., Chung, K. W. & Xu, J. [2011] “ A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method,” Nonlin. Dyn. 64, 221–236. Web of Science, Google Scholar
Chan, H. S. Y., Chung, K. W. & Xu, Z. [1996] “ A perturbation-incremental method for strongly non-linear oscillators,” Int. J. Non-Lin. Mech. 31, 59–72. Web of Science, Google Scholar
Dhooge, A., Govaerts, W., Kuznetsov, Y. A., Meijer, H. G. E. & Sautois, B. [2008] “ New features of the software MatCont for bifurcation analysis of dynamical systems,” Math. Comp. Model. Dyn. 14, 147–175. Web of Science, Google Scholar
Doedel, E. J., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [2002] “ An analytical and numerical study of a modified van der Pol oscillator,” J. Sound Vibr. 256, 755–771. Web of Science, Google Scholar
Doedel, E. J., Champneys, A. R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X. & Zhang, C. [2012] “AUTO-07P: Continuation and bifurcation software for ordinary differential equations (with HomCont),” Technical report, Concordia University. Google Scholar
Dumortier, F. & Roussarie, R. [1996] “ Canard cycles and center manifolds,” Mem. Amer. Math. Soc. 121, 577. Web of Science, Google Scholar
Dumortier, F. & Roussarie, R. [2001] “ Multiple canard cycles in generalized Liénard equations,” J. Diff. Eqs. 174, 1–29. Web of Science, Google Scholar
Eckhaus, W. [1983] “ Relaxation oscillations including a standard chase on French ducks,” Asymptotic Analysis II, Lecture Notes in Mathematics, Vol. 985 (Springer, Berlin), pp. 449–494. Google Scholar
Epstein, I. R. & Pojman, J. A. [1998] An Introduction to Nonlinear Chemical Dynamics (Oxford University Press, NY). Google Scholar
Erneux, T. & Reiss, E. [1983] “ Brusselator isolas,” SIAM J. Appl. Math. 43, 1240–1246. Web of Science, Google Scholar
Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [1999] “ First-order approximation for canard periodic orbits in a van der Pol electronic oscillator,” Appl. Math. Lett. 12, 73–78. Web of Science, Google Scholar
Ginoux, J. M., Llibre, J. & Chua, L. O. [2013] “ Canards from Chua’s circuit,” Int. J. Bifurcation and Chaos 23, 1330010. Link, Web of Science, Google Scholar
Kolokolnikov, T., Erneux, T. & Wei, J. [2006] “ Mesa-type patterns in the one-dimensional Brusselator and their stability,” Physica D 214, 63–77. Web of Science, Google Scholar
Krupa, M. & Szmolyan, P. [2001] “ Relaxation oscillation and canard explosion,” J. Diff. Eqs. 174, 312–368. Web of Science, Google Scholar
Kuehn, C. [2010] “ From first Lyapunov coefficients to maximal canards,” Int. J. Bifurcation and Chaos 20, 1467–1475. Link, Web of Science, Google Scholar
Larter, R., Rabitz, H. & Kramer, M. A. [1984] “ Sensitivity analysis of limit cycles with application to the Brusselator,” J. Chem. Phys. 80, 4120–4128. Web of Science, Google Scholar
Li, C., Li, J., Ma, Z. & Zhu, H. [2014] “ Canard phenomenon for an SIS epidemic model with nonlinear incidence,” J. Math. Anal. Appl. 420, 987–1004. Web of Science, Google Scholar
Liao, M. & Wang, Q. R. [2016] “ Stability and bifurcation analysis in a diffusive Brusselator-type system,” Int. J. Bifurcation and Chaos 26, 1650119. Link, Web of Science, Google Scholar
Luo, A. C. J. & Guo, S. [2018] “ Period-1 evolutions to chaos in a periodically forced Brusselator,” Int. J. Bifurcation and Chaos 28, 1830046. Link, Web of Science, Google Scholar
Marius, G. [2008] “ Non-constant steady-state solutions for Brusselator type systems,” Nonlinearity 21, 2331–2345. Web of Science, Google Scholar
Matzinger, E. [2006] “ Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation,” J. Diff. Eqs. 220, 478–510. Web of Science, Google Scholar
Mishchenko, E. F., Kolesov, Y. S., Kolesov, A. Y. & Rozov, N. K. [1994] Asymptotic Methods in Singularly Perturbed Systems (Consultants Bureau, NY). Google Scholar
Moehlis, J. [2002] “ Canards in a surface oxidation reaction,” J. Nonlin. Sci. 12, 319–345. Web of Science, Google Scholar
Nicolis, G. & Prigogine, I. [1977] Self-Organization in Equilibrium Systems (Wiley, NY). Google Scholar
Osipov, V. V. & Ponizovskaya, E. V. [2000] “ Stochastic resonance in the Brusselator model,” Phys. Rev. E 61, 4603–4605. Web of Science, Google Scholar
Prigogine, I. & Lefever, R. [1968] “ Symmetry breaking instabilities in dissipative systems. II,” J. Chem. Phys. 48, 1695–1700. Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Fahsi, A. & Belhaq, M. [2016] “ On the heteroclinic connections in the 1:3 resonance problem,” Int. J. Bifurcation and Chaos 26, 1650143. Link, Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Rodríguez-Luis, A. J. & Belhaq, M. [2018] “ Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens–Bogdanov normal form with $D_{4}$ symmetry,” Chaos 28, 093107. Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Algaba, A. & Rodríguez-Luis, A. J. [2020a] “ High-order analysis of global bifurcations in a codimension-three Takens–Bogdanov singularity in reversible systems,” Int. J. Bifurcation and Chaos 30, 2050017. Link, Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Algaba, A. & Rodríguez-Luis, A. J. [2020b] “ Analytical approximation of cuspidal loops using a nonlinear time transformation method,” Appl. Math. Comput. 373, 125042. Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Algaba, A. & Rodríguez-Luis, A. J. [2020c] “ High-order study of the canard explosion in an aircraft ground dynamics model,” Nonlin. Dyn., https://doi.org/10.1007/s11071-020-05575-w (accepted). Web of Science, Google Scholar
Qin, B. W., Chung, K. W., Algaba, A. & Rodríguez-Luis, A. J. [2020d] “ Asymptotic expansions for a family of non-generic canards using parametric representation,” Appl. Math. Lett. 106, 106355. Web of Science, Google Scholar
Rankin, J., Desroches, M., Krauskopf, B. & Lowenberg, M. [2011] “ Canard cycles in aircraft ground dynamics,” Nonlin. Dyn. 66, 681–688. Web of Science, Google Scholar
Sun, W. & Zhang, Y. [2015] “ Control and synchronization of Julia sets in the forced Brusselator model,” Int. J. Bifurcation and Chaos 25, 1550113. Link, Web of Science, Google Scholar
Tyson, J. J. [1973] “ Some further studies of nonlinear oscillations in chemical systems,” J. Chem. Phys. 58, 3919–3930. Web of Science, Google Scholar
Vidal, A. & Françoise, J. P. [2012] “ Canard cycles in global dynamics,” Int. J. Bifurcation and Chaos 22, 1250026. Link, Web of Science, Google Scholar
Zuo, W. & Wei, J. [2012] “ Stability and bifurcation analysis in a diffusive Brusselator system with delayed feedback control,” Int. J. Bifurcation and Chaos 22, 1250037. Link, Web of Science, Google Scholar
Zvonkin, A. K. & Shubin, M. A. [1984] “ Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131. Web of Science, Google Scholar