BIFURCATION FROM TWO EQUILIBRIA OF STEADY STATE SOLUTIONS FOR NON-REVERSIBLE AMPLITUDE EQUATIONS

Yancong Xu; Rui Xu; Yu Yang

doi:10.11948/2017088

2017 Volume 7 Issue 4

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Yancong Xu, Rui Xu, Yu Yang. BIFURCATION FROM TWO EQUILIBRIA OF STEADY STATE SOLUTIONS FOR NON-REVERSIBLE AMPLITUDE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1448-1462. doi: 10.11948/2017088

Citation:

Yancong Xu, Rui Xu, Yu Yang. BIFURCATION FROM TWO EQUILIBRIA OF STEADY STATE SOLUTIONS FOR NON-REVERSIBLE AMPLITUDE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(4): 1448-1462. doi: 10.11948/2017088

BIFURCATION FROM TWO EQUILIBRIA OF STEADY STATE SOLUTIONS FOR NON-REVERSIBLE AMPLITUDE EQUATIONS

1 Department of Mathematics, Hangzhou Normal University, No. 16, Xuelin Street, Hangzhou, China;
2 School of Science and Technology, Zhejiang International Studies University, No. 140, Wensan Road, Hangzhou, China

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Abstract

Abstract

In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.
- Bifurcation /
- stability /
- perturbed amplitude equations /
- eigenvalue problems
MSC: 37G25;37G20;37C29

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References

[1]	I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys, 2002, 74(1), 99-143. Google Scholar
[2]	D. J. Benny, Long waves in liquid film, J. Math. Phys, 1966, 45, 150-155. Google Scholar
[3]	D. Blömker and W. W. Mohammed, Amplitude equations for SPDEs with cubic nonlinearities, Stochastics, Int. J. Probab. Stoch. Process, 2013, 85(2), 181-215. Google Scholar
[4]	J. Burke, S. M. Houghton and E. Knobloch, Swift-Hohenberg equation with broken reflection symmetry, Phys. Rev. E, 2009, 80, 036202. Google Scholar
[5]	P. Coullet and G. Iooss, Instabilities of one-dimensional cellular patterns, Phys. Rev. Lett, 1990, 64(8), 866-869. Google Scholar
[6]	S. M. Cox and P. C. Matthews, New instabilities of two-dimensional rotating convection and magnetoconvection, Physica D, 2001, 149(3), 210-229. Google Scholar
[7]	J. H. P. Dawes, Localized pattern formation with a large-scale mode:slanted snaking, SIAM J. Appl. Dyn. Syst., 2008, 7(1), 186-206. Google Scholar
[8]	A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 2001, 50(1), 421-425. Google Scholar
[9]	A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 2002, 155, 737. Google Scholar
[10]	A. Doelman, G. Hek and N. Valkhoff, Stabilization by slow diffusion in a real Ginzburg-Landau system, J. Nonlin. Sci., 2004, 14(3), 237-278. Google Scholar
[11]	S. Fauve, Pattern forming instabilities, Hydrodynamics and Nonlinear Instabilities, eds. Godréche, C. and Manneville, P. (Cambridge University Press), 1998, 387-491. Google Scholar
[12]	G. Fibich and D. Shpigelman, Positive and necklace solitary waves on bounded domains, Physica D, 2016, 315, 13=-32. Google Scholar
[13]	M. Ghergu, Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Transactions of the American Mathematical Society, 2009, 361(8), 3953-3976. Google Scholar
[14]	K. Klepel, K. Blömker and W. W. Mohammed, Amplitude equation for the generalized Swift Hohenberg equation with noise, ZAMP-Zeitschrift fur angewandte Mathematik und Physik, 2014, 65, 1107-1126. Google Scholar
[15]	C. P. Li and G. Chen, Bifurcation from an equilibrium of the steady state Kuramoto-Sivashinsky equation in two spatial dimensions, Int. J. Bifurcation and Chaos, 2011, 12(12), 103-114. Google Scholar
[16]	Y. Li, Hopf bifurcation in general systems of Brusselator type, Nonlinear Anal. R. W. A., 2016, 28, 32-47. Google Scholar
[17]	P. C. Matthews and S. M. Cox, One dimensional pattern formation with Galilean invariance near a stationary bifurcation, Phys. Rev. E, 2002, 62, 1-4. Google Scholar
[18]	P. C. Matthews and S. M. Cox, Pattern formation with a conservation law, Nonlinearity, 2000, 13(4), 1293-1320. Google Scholar
[19]	A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, Handbook of Dynamical Systems 2(North-Holland, Amsterdam), 2002, pp. 759-834. Google Scholar
[20]	J. Norbury, M. Winter and J. Wei, Existence and stability of singular patterns in a Ginzburg-Landau equation coupled with a mean field, Nonlinearity, 2002, 15(6), 2077-2096. Google Scholar
[21]	L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the SwiftHohenberg equation, Physica D, 2004, 194(1-2), 95-126. Google Scholar
[22]	L. A. Peletier and J. F. Williams, Some canonical bifurcations in the SwiftHohenberg equation, SIAM J. Appl. Dyn. Syst., 2007, 6(1), 208-235. Google Scholar
[23]	J. B. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 1977, 15(1), 319-328. Google Scholar
[24]	L. Shi and H. J. Gao, Bifurcation analysis of an amplitude equation, Int. J. Bifurcation and Chaos., 2013, 23(5), 1350081. Google Scholar
[25]	B. Sandstede and Y. C. Xu, Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 2012, 27(3), 317-329. Google Scholar
[26]	A. Zippelius and E. D. Siggia, Stability of the finite-amplitude convection, Phys. Fluids, 1983, 26(10), 2905-2915. Google Scholar