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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 365436, 11 pages
Attractor Bifurcation for Extended Fisher-Kolmogorov Equation
1Sino-European Institute of Aviation Engineering, Civil Aviation University of China, Tianjin 300300, China
2School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Received 7 July 2013; Accepted 7 August 2013
Academic Editor: Feliz Minhós
Copyright © 2013 Honglian You et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- G. T. Dee and W. van Saarloos, “Bistable systems with propagating fronts leading to pattern formation,” Physical Review Letters, vol. 60, pp. 2641–2644, 1988.
- N. N. Akhmediev, A. V. Buryak, and M. Karlsson, “Radiationless optical solitons with oscillating tails,” Optics Communications, vol. 110, no. 5-6, pp. 540–544, 1994.
- B. Buffoni, A. R. Champneys, and J. F. Toland, “Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system,” Journal of Dynamics and Differential Equations, vol. 8, no. 2, pp. 221–279, 1996.
- P. Coullet, C. Elphick, and D. Repaux, “Nature of spatial chaos,” Physical Review Letters, vol. 58, no. 5, pp. 431–434, 1987.
- J. Swift and P. C. Hohenberg, “Hydrodynamic fluctuations at the convective instability,” Physical Review A, vol. 15, pp. 319–328, 1977.
- R. A. Fisher, “The advance of advantageous genes,” Annals of Eugenics, vol. 7, pp. 355–369, 1937.
- A. Kolmogorov, I. Petrovsky, and N. Piskunov, “Etude de l'équation de la diffusion avec croissance de la quantité de matière etson application à un problème biologique,” Bulletin Université d'Etat à Moscou, Série Internationale A, vol. 1, pp. 1–25, 1937.
- W. Zimmermann, “Propagating fronts near a Lifshitz point,” Physical Review Letters, vol. 66, no. 11, p. 1546, 1991.
- J. B. van den Berg, “Uniqueness of solutions for the extended Fisher-Kolmogorov equation,” Comptes Rendus de l'Académie des Sciences, vol. 326, no. 4, pp. 447–452, 1998.
- J. V. Chaparova, L. A. Peletier, and S. A. Tersian, “Existence and nonexistence of nontrivial solutions of semilinear fourth- and sixth-order differential equations,” Advances in Differential Equations, vol. 8, no. 10, pp. 1237–1258, 2003.
- P. Danumjaya and A. K. Pani, “Numerical methods for the extended Fisher-Kolmogorov (EFK) equation,” International Journal of Numerical Analysis and Modeling, vol. 3, no. 2, pp. 186–210, 2006.
- M. A. Peletier, “Non-existence and uniqueness results for fourth-order Hamiltonian systems,” Nonlinearity, vol. 12, no. 6, pp. 1555–1570, 1999.
- L. A. Peletier and W. C. Troy, “Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions,” SIAM Journal on Mathematical Analysis, vol. 28, no. 6, pp. 1317–1353, 1997.
- J. Kwapisz, “Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation,” Journal of Differential Equations, vol. 165, no. 1, pp. 235–253, 2000.
- L. A. Peletier and W. C. Troy, “Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: kinks,” Differential and Integral Equations, vol. 8, no. 6, pp. 1279–1304, 1995.
- L. A. Peletier and W. C. Troy, “A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation,” Topological Methods in Nonlinear Analysis, vol. 6, no. 2, pp. 331–355, 1995.
- M. V. Bartuccelli, “On the asymptotic positivity of solutions for the extended Fisher-Kolmogorov equation with nonlinear diffusion,” Mathematical Methods in the Applied Sciences, vol. 25, no. 8, pp. 701–708, 2002.
- D. Hilhorst, L. A. Peletier, and R. Schätzle, “-limit for the extended Fisher-Kolmogorov equation,” Proceedings of the Royal Society of Edinburgh A, vol. 132, no. 1, pp. 141–162, 2002.
- J. K. Hale, Asymptotic Behavior of Dissipative Systems, vol. 25 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988.
- L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, vol. 45 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 2001.
- L. A. Peletier, W. C. Troy, and R. C. A. M. van der Vorst, “Stationary solutions of a fourth-order nonlinear diffusion equation,” Differentsial'nye Uravneniya, vol. 31, pp. 327–337, 1995 (Russian).
- W. D. Kalies, J. Kwapisz, and R. C. A. M. van der Vorst, “Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria,” Communications in Mathematical Physics, vol. 193, no. 2, pp. 337–371, 1998.
- J. B. van den Berg, Dynamics and equilibria of fourth order dfferential equations [Ph.D. thesis], Leiden University, Leiden, Netherlands, 2000.
- W. D. Kalies, J. Kwapisz, J. B. van den Berg, and R. C. A. M. van der Vorst, “Homotopy classes for stable periodic and chaotic patterns in fourth-order Hamiltonian systems,” Communications in Mathematical Physics, vol. 214, no. 3, pp. 573–592, 2000.
- W. D. Kalies and R. C. A. M. van der Vorst, “Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation,” Journal of Differential Equations, vol. 131, no. 2, pp. 209–228, 1996.
- L. A. Peletier and W. C. Troy, “Chaotic spatial patterns described by the extended Fisher-Kolmogorov equation,” Journal of Differential Equations, vol. 129, no. 2, pp. 458–508, 1996.
- J. B. van den Berg and R. C. A. M. van der Vorst, “Stable patterns for fourth-order parabolic equations,” Duke Mathematical Journal, vol. 115, no. 3, pp. 513–558, 2002.
- T. Ma and S. Wang, Bifurcation Theory and Applications, vol. 53, World Scientific, Hackensack, NJ, USA, 2005.
- T. Ma and S. Wang, “Dynamic bifurcation of nonlinear evolution equations,” Chinese Annals of Mathematics B, vol. 26, no. 2, pp. 185–206, 2005.
- T. Ma and S. Wang, “Dynamic bifurcation and stability in the Rayleigh-Bénard convection,” Communications in Mathematical Sciences, vol. 2, no. 2, pp. 159–183, 2004.
- T. Ma and S. Wang, “Attractor bifurcation theory and its applications to Rayleigh-Bénard convection,” Communications on Pure and Applied Analysis, vol. 2, no. 4, pp. 591–599, 2003.
- M. Yari, “Attractor bifurcation and final patterns of the -dimensional and generalized Swift-Hohenberg equations,” Discrete and Continuous Dynamical Systems B, vol. 7, no. 2, pp. 441–456, 2007.
- Q. Huang and J. Tang, “Bifurcation of a limit cycle in the AC-driven complex Ginzburg-Landau equation,” Discrete and Continuous Dynamical Systems B, vol. 14, no. 1, pp. 129–141, 2010.
- T. Ma, J. Park, and S. Wang, “Dynamic bifurcation of the Ginzburg-Landau equation,” SIAM Journal on Applied Dynamical Systems, vol. 3, no. 4, pp. 620–635, 2004.
- Y. Zhang, L. Song, and W. Axia, “Dynamical bifurcation for the Kuramoto-Sivashinsky equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 4, pp. 1155–1163, 2011.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
- D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.