September  2015, 20(7): 1933-1957. doi: 10.3934/dcdsb.2015.20.1933

Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period

1. 

Division of General Education, Kwangwoon University, Seoul, 139-701, South Korea

2. 

Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul, 130-701

3. 

Department of Mathematics and Taidar Institute of Mthematical Science, National Taiwan University, Taipei, 10617

Received  April 2014 Revised  March 2015 Published  July 2015

In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd periodic interval of period $2\lambda$. We fix the control parameter $\alpha \in (0,1)$ and study how the equation bifurcates to attractors as $\lambda$ varies. Using the center manifold analysis, we prove that the bifurcated attractors are homeomorphic to $S^1$ and consist of four or eight singular points and their connecting orbits. We verify the structure of the bifurcated attractors by investigating the stability of each singular point.
Citation: Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933
References:
[1]

H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency,, Phys. Rev. Lett., 58 (1987), 112.   Google Scholar

[2]

Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation,, J. Math. Anal. Appl., 421 (2015), 383.  doi: 10.1016/j.jmaa.2014.07.009.  Google Scholar

[3]

N. Ecrolani, D. McLaughlin and H. Roitner, Attractors and transients for a perturbed periodics KdV rquation: A nonlinear spectral analysis,, J. Nonlin. Sci., 3 (1993), 477.  doi: 10.1007/BF02429875.  Google Scholar

[4]

K. Elder, H. w. Xi, M. Deans and J. Gunton, Spatiotemporal chaos in the damped Kuramot-Sivashinsky equation,, AIP Conf. Proc., 342 (1995), 702.   Google Scholar

[5]

K. R. Edler, J. D. Gunton and N. Goldenfled, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinky equation,, Phys. Rev. E, 56 (1997), 1631.   Google Scholar

[6]

H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation,, Intern. J. Bifur. Chaos, 20 (2010), 619.  doi: 10.1142/S0218127410025922.  Google Scholar

[7]

H. Gomez and J. Paris, Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinky equation,, Phys. Rev E, 83 (2011).   Google Scholar

[8]

J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition,, Dis. Cont. Dyn. Sys. B, 17 (2012), 2431.  doi: 10.3934/dcdsb.2012.17.2431.  Google Scholar

[9]

J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation,, Bull. Korean Math. Soc., 49 (2012), 923.  doi: 10.4134/BKMS.2012.49.5.923.  Google Scholar

[10]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, (2005).  doi: 10.1142/9789812701152.  Google Scholar

[11]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer, (2014).  doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[12]

T. Ma and S. Wang, Rayleigh-Bénard convection: Dynamics and structure in the physical space,, Comm. Math. Sci., 5 (2007), 553.  doi: 10.4310/CMS.2007.v5.n3.a3.  Google Scholar

[13]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Dis. Cont. Dyn. Sys. B, 11 (2009), 741.  doi: 10.3934/dcdsb.2009.11.741.  Google Scholar

[14]

C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation,, Phys. Rev. E, 49 (1994), 166.  doi: 10.1103/PhysRevE.49.166.  Google Scholar

[15]

M. Paniconi and K. Edler, Stationary, dynamical, and chaotic states of the two-dimensional damped Kuramoto-Sivashinsky equation,, Phys. Rev. E, 56 (1997), 2713.  doi: 10.1103/PhysRevE.56.2713.  Google Scholar

[16]

L. A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equations,, Phys. D, 194 (2004), 95.  doi: 10.1016/j.physd.2004.01.043.  Google Scholar

[17]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation,, SIAM J. Appl. Dyn. Sys., 6 (2007), 208.  doi: 10.1137/050647232.  Google Scholar

[18]

S. Vogel and S. Linz, Contiuum modeling of sputter erosion under normal incidence: Interplay between nonlocality and nonlinearity,, Phys. Rev. B, 72 (2005).   Google Scholar

[19]

Q. Xiao and H. Gao, Bifurcation analysis of the Swift-Hohenberg equation with quitic nonlinearity,, Intern. J. Bifur. Chaos, 19 (2009), 2927.  doi: 10.1142/S0218127409024542.  Google Scholar

[20]

M. Yari, Attractor bifurcation and final patterns of the $N$-dimensional and generalized Swift-Hohenberg equations,, Dis. Cont. Dyn. Sys. B, 7 (2007), 441.  doi: 10.3934/dcdsb.2007.7.441.  Google Scholar

[21]

M. Zhen, Numerical Bifurcation Analysis for Reaction-Diffusion Equations,, Springer Series in Computational Mathematics, (2000).  doi: 10.1007/978-3-662-04177-2.  Google Scholar

show all references

References:
[1]

H. Chaté and P. Manneville, Transition to turbulence via spatiotemporal intermittency,, Phys. Rev. Lett., 58 (1987), 112.   Google Scholar

[2]

Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation,, J. Math. Anal. Appl., 421 (2015), 383.  doi: 10.1016/j.jmaa.2014.07.009.  Google Scholar

[3]

N. Ecrolani, D. McLaughlin and H. Roitner, Attractors and transients for a perturbed periodics KdV rquation: A nonlinear spectral analysis,, J. Nonlin. Sci., 3 (1993), 477.  doi: 10.1007/BF02429875.  Google Scholar

[4]

K. Elder, H. w. Xi, M. Deans and J. Gunton, Spatiotemporal chaos in the damped Kuramot-Sivashinsky equation,, AIP Conf. Proc., 342 (1995), 702.   Google Scholar

[5]

K. R. Edler, J. D. Gunton and N. Goldenfled, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinky equation,, Phys. Rev. E, 56 (1997), 1631.   Google Scholar

[6]

H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation,, Intern. J. Bifur. Chaos, 20 (2010), 619.  doi: 10.1142/S0218127410025922.  Google Scholar

[7]

H. Gomez and J. Paris, Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinky equation,, Phys. Rev E, 83 (2011).   Google Scholar

[8]

J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition,, Dis. Cont. Dyn. Sys. B, 17 (2012), 2431.  doi: 10.3934/dcdsb.2012.17.2431.  Google Scholar

[9]

J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation,, Bull. Korean Math. Soc., 49 (2012), 923.  doi: 10.4134/BKMS.2012.49.5.923.  Google Scholar

[10]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific, (2005).  doi: 10.1142/9789812701152.  Google Scholar

[11]

T. Ma and S. Wang, Phase Transition Dynamics,, Springer, (2014).  doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[12]

T. Ma and S. Wang, Rayleigh-Bénard convection: Dynamics and structure in the physical space,, Comm. Math. Sci., 5 (2007), 553.  doi: 10.4310/CMS.2007.v5.n3.a3.  Google Scholar

[13]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Dis. Cont. Dyn. Sys. B, 11 (2009), 741.  doi: 10.3934/dcdsb.2009.11.741.  Google Scholar

[14]

C. Misbah and A. Valance, Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation,, Phys. Rev. E, 49 (1994), 166.  doi: 10.1103/PhysRevE.49.166.  Google Scholar

[15]

M. Paniconi and K. Edler, Stationary, dynamical, and chaotic states of the two-dimensional damped Kuramoto-Sivashinsky equation,, Phys. Rev. E, 56 (1997), 2713.  doi: 10.1103/PhysRevE.56.2713.  Google Scholar

[16]

L. A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equations,, Phys. D, 194 (2004), 95.  doi: 10.1016/j.physd.2004.01.043.  Google Scholar

[17]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation,, SIAM J. Appl. Dyn. Sys., 6 (2007), 208.  doi: 10.1137/050647232.  Google Scholar

[18]

S. Vogel and S. Linz, Contiuum modeling of sputter erosion under normal incidence: Interplay between nonlocality and nonlinearity,, Phys. Rev. B, 72 (2005).   Google Scholar

[19]

Q. Xiao and H. Gao, Bifurcation analysis of the Swift-Hohenberg equation with quitic nonlinearity,, Intern. J. Bifur. Chaos, 19 (2009), 2927.  doi: 10.1142/S0218127409024542.  Google Scholar

[20]

M. Yari, Attractor bifurcation and final patterns of the $N$-dimensional and generalized Swift-Hohenberg equations,, Dis. Cont. Dyn. Sys. B, 7 (2007), 441.  doi: 10.3934/dcdsb.2007.7.441.  Google Scholar

[21]

M. Zhen, Numerical Bifurcation Analysis for Reaction-Diffusion Equations,, Springer Series in Computational Mathematics, (2000).  doi: 10.1007/978-3-662-04177-2.  Google Scholar

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