Nonlinear Dynamics in PDEs: Stability, Bifurcation, and Pattern Formation
Keywords:
Bifurcation, PDE, Nonlinear dynamics, InstabilityAbstract
In physics, biology, engineering, and other scientific fields, partial differential equations (PDEs) are fundamental tools for modelling dynamic systems. Stability and bifurcation are two basic ideas that frequently control how their solutions behave, particularly in different situations. The conditions under which solutions to nonlinear PDEs either stay stable or experience qualitative changes as a result of parameter variation are examined in this research paper. To investigate the dynamics of representative systems like reaction-diffusion equations and the Navier-Stokes equations, numerical simulations are used in conjunction with analytical methods such as Lyapunov stability theory, linearisation techniques, and bifurcation theory, including pitchfork and Hopf bifurcations. In-depth analysis is done on how boundary conditions, eigenvalue spectra, and nonlinear feedback affect the behaviour of the solution. In addition to offering useful insights into the critical transitions seen in real-world systems like fluid flow, chemical reactions, and biological pattern formation, our goal is to advance the theoretical understanding of solution trajectories in PDEs.
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S. K. Ghosh, S. Gupta, and A. S. K. Tiwari, “Bifurcation and stability of periodic solutions for a reaction-diffusion system with spatial heterogeneity,” J. Math. Anal. Appl., vol. 530, no. 1, pp. 1–20, 2024.
M. N. Kasyanov, M. I. Shapiro, and I. V. Taranenko, “Stability and bifurcation analysis of traveling wave solutions for a reaction-diffusion equation with delay,” Appl. Math. Lett., vol. 152, pp. 108203, 2024.
T. I. Aizawa, Y. Takeda, and H. Yagisawa, “Local bifurcation of periodic solutions in a nonlinear wave equation,” Nonlinear Dyn., vol. 111, pp. 839–854, 2024.
R. M. I. Goecker, H. M. D. Kloss, and J. S. Y. Tan, “Stability analysis of equilibrium solutions in a degenerate parabolic system,” J. Differ. Equ., vol. 366, pp. 202–226, 2024.
A. S. Hwang and M. K. Kasyanov, “Bifurcation of solitary waves in a two-component reaction-diffusion system,” Physica D: Nonlinear Phenomena, vol. 455, pp. 133032, 2023.
C. T. W. Cheng, J. P. H. de Jong, and M. N. J. West, “Stability of traveling wave solutions for a generalized Korteweg-de Vries equation,” J. Math. Phys., vol. 64, no. 2, 2023.
V. G. Lychagin, S. A. Komarov, and D. O. Shcherbakov, “Bifurcation phenomena in nonlocal reaction-diffusion equations,” Discrete Contin. Dyn. Syst., vol. 43, pp. 1985–2003, 2023.
J. M. G. Vargas, A. L. A. Oliveira, and B. F. M. Neves, “Stability analysis of a discrete reaction-diffusion model with variable coefficients,” Math. Models Methods Appl. Sci., vol. 33, pp. 1154–1172, 2023.
Y. T. Li, Z. Q. Wang, and Q. L. Zheng, “Dynamics of a predator-prey model with nonlinear functional response and diffusion,” Appl. Math. Model., vol. 117, pp. 232–248, 2023.
L. J. K. Zhao, S. M. Li, and R. W. M. Lee, “Stability and bifurcation of traveling wave solutions for a nonlinear parabolic equation,” SIAM J. Appl. Dyn. Syst., vol. 22, pp. 98–117, 2023.
J. Hale and H. Koçak, Dynamics and Bifurcations, Springer, 1991.
H. Poincaré, Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1892.
A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, 1992 (translated from 1892 Russian edition).
M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” J. Funct. Anal., vol. 8, no. 2, pp. 321–340, 1971.
S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., CRC Press, 2018.
J. Carr, Applications of Center Manifold Theory, Springer, 1981.
A. M. Turing, “The chemical basis of morphogenesis,” Philos. Trans. R. Soc. Lond. B, vol. 237, no. 641, pp. 37–72, 1952.
J. D. Murray, Mathematical Biology, 3rd ed., Springer, 2003.
P. Grindrod, Patterns and Waves: The Theory and Applications of Reaction–Diffusion Equations, Oxford University Press, 1991.
D. D. Joseph and Y. Renardy, Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications, Springer, 1992.
P. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd ed., Cambridge University Press, 2012.
G. Dangelmayr and J. Guckenheimer, “On the symmetry-related bifurcations of periodic solutions of ordinary differential equations,” Physica D, vol. 23, no. 1–3, pp. 127–153, 1986.
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer, 1995.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, 2009.
E. Doedel et al., “AUTO-07P: Continuation and bifurcation software for ordinary differential equations,” Concordia University, Montreal, Canada, 2007.
K. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer, 2004..
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