June  2016, 21(4): 1225-1236. doi: 10.3934/dcdsb.2016.21.1225

Dynamic transitions of generalized Kuramoto-Sivashinsky equation

1. 

Department of Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  January 2015 Revised  September 2015 Published  March 2016

In this article, we study the dynamic transition for the one dimensional generalized Kuramoto-Sivashinsky equation with periodic condition. It is shown that if the value of the dispersive parameter $\nu$ is strictly greater than $\nu^{\ast}$, then the transition is Type-I (continuous) and the bifurcated periodic orbit is an attractor as the control parameter $\lambda$ crosses the critical value $\lambda_0$. In the case where $\nu$ is strictly less than $\nu^{\ast}$, then the transition is Type-II (jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter $\lambda$ crosses the critical value $\lambda_0$. The value of $\nu^{\ast}$ is also calculated in this paper.
Citation: Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
References:
[1]

B. Barker, M. A. Johnson, P. Noble, L. M. Rodrigues and K. Zumbrun, Nonlinear modulational stability of periodic traveling-wave solutions of the generalized kuramoto-sivashinsky equation,, Physica D, 258 (2013), 11. doi: 10.1016/j.physd.2013.04.011.

[2]

H. Dijkstra, T. Sengul and S. Wang, Dynamic transitions of surface tension driven convection,, Physica D, 247 (2013), 7. doi: 10.1016/j.physd.2012.12.008.

[3]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).

[4]

A. P. Hooper and R. Grimshaw, Nonlinear instabilitity at the interface between two viscous fluids,, Phys. Fluids, 28 (1985), 37. doi: 10.1063/1.865160.

[5]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theo. Phys., 55 (1976), 356. doi: 10.1143/PTP.55.356.

[6]

T. Ma and S. Wang, Stability and Bifurcation of Nonlinear Evolutions Equations,, Science Press, (2007).

[7]

T. Ma and S. Wang, Cahn-hilliard equations and phase transition dynamics for binary system,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741.

[8]

T. Ma and S. Wang, Phase separation of binary systems,, Physica A, 388 (2009), 4811. doi: 10.1016/j.physa.2009.07.044.

[9]

T. Ma and S. Wang, Dynamic model and phase transitions for liquid helium,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2957943.

[10]

T. Ma and S. Wang, Dynamic bifurcation and stability in the rayleigh-benard convection,, Commun. Math. Sci., 2 (2004), 159. doi: 10.4310/CMS.2004.v2.n2.a2.

[11]

T. Ma and S. Wang, Phase transitions for belousov-zhabotinsky reactions,, Math. Methods Appl. Sci., 34 (2011), 1381. doi: 10.1002/mma.1446.

[12]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific Series on Nonlinear Science, (2005). doi: 10.1142/9789812701152.

[13]

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

[14]

S. Wang and P. Yang, Remarks on the rayleigh-benard convection on spherical shells,, J. Math. Fluid Mech., 15 (2013), 537. doi: 10.1007/s00021-012-0128-8.

[15]

G. I. Sivashinsky, On flame propagation under conditions of stoichiometry,, SIAM J. Appl. Math, 39 (1980), 67. doi: 10.1137/0139007.

[16]

G. I. Sivashinsky, Instabilities, pattern-formation and turbulence in flames,, Annu. Rev. Fluid Mech., 15 (1983), 179. doi: 10.1146/annurev.fl.15.010183.001143.

show all references

References:
[1]

B. Barker, M. A. Johnson, P. Noble, L. M. Rodrigues and K. Zumbrun, Nonlinear modulational stability of periodic traveling-wave solutions of the generalized kuramoto-sivashinsky equation,, Physica D, 258 (2013), 11. doi: 10.1016/j.physd.2013.04.011.

[2]

H. Dijkstra, T. Sengul and S. Wang, Dynamic transitions of surface tension driven convection,, Physica D, 247 (2013), 7. doi: 10.1016/j.physd.2012.12.008.

[3]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).

[4]

A. P. Hooper and R. Grimshaw, Nonlinear instabilitity at the interface between two viscous fluids,, Phys. Fluids, 28 (1985), 37. doi: 10.1063/1.865160.

[5]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium,, Prog. Theo. Phys., 55 (1976), 356. doi: 10.1143/PTP.55.356.

[6]

T. Ma and S. Wang, Stability and Bifurcation of Nonlinear Evolutions Equations,, Science Press, (2007).

[7]

T. Ma and S. Wang, Cahn-hilliard equations and phase transition dynamics for binary system,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741.

[8]

T. Ma and S. Wang, Phase separation of binary systems,, Physica A, 388 (2009), 4811. doi: 10.1016/j.physa.2009.07.044.

[9]

T. Ma and S. Wang, Dynamic model and phase transitions for liquid helium,, J. Math. Phys., 49 (2008). doi: 10.1063/1.2957943.

[10]

T. Ma and S. Wang, Dynamic bifurcation and stability in the rayleigh-benard convection,, Commun. Math. Sci., 2 (2004), 159. doi: 10.4310/CMS.2004.v2.n2.a2.

[11]

T. Ma and S. Wang, Phase transitions for belousov-zhabotinsky reactions,, Math. Methods Appl. Sci., 34 (2011), 1381. doi: 10.1002/mma.1446.

[12]

T. Ma and S. Wang, Bifurcation Theory and Applications,, World Scientific Series on Nonlinear Science, (2005). doi: 10.1142/9789812701152.

[13]

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

[14]

S. Wang and P. Yang, Remarks on the rayleigh-benard convection on spherical shells,, J. Math. Fluid Mech., 15 (2013), 537. doi: 10.1007/s00021-012-0128-8.

[15]

G. I. Sivashinsky, On flame propagation under conditions of stoichiometry,, SIAM J. Appl. Math, 39 (1980), 67. doi: 10.1137/0139007.

[16]

G. I. Sivashinsky, Instabilities, pattern-formation and turbulence in flames,, Annu. Rev. Fluid Mech., 15 (1983), 179. doi: 10.1146/annurev.fl.15.010183.001143.

[1]

Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729

[2]

Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701

[3]

Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91

[4]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557

[5]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95

[6]

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

[7]

L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555

[8]

Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247

[9]

Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems & Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036

[10]

Aslihan Demirkaya. The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D. Conference Publications, 2009, 2009 (Special) : 198-207. doi: 10.3934/proc.2009.2009.198

[11]

Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281

[12]

Fred C. Pinto. Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 117-136. doi: 10.3934/dcds.1999.5.117

[13]

Raimund Bürger, Gerardo Chowell, Pep Mulet, Luis M. Villada. Modelling the spatial-temporal progression of the 2009 A/H1N1 influenza pandemic in Chile. Mathematical Biosciences & Engineering, 2016, 13 (1) : 43-65. doi: 10.3934/mbe.2016.13.43

[14]

Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers, Peter D. Lee. 4D-CT reconstruction with unified spatial-temporal patch-based regularization. Inverse Problems & Imaging, 2015, 9 (2) : 447-467. doi: 10.3934/ipi.2015.9.447

[15]

Yoon-Sik Cho, Aram Galstyan, P. Jeffrey Brantingham, George Tita. Latent self-exciting point process model for spatial-temporal networks. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1335-1354. doi: 10.3934/dcdsb.2014.19.1335

[16]

A. V. Babin. Preservation of spatial patterns by a hyperbolic equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 1-19. doi: 10.3934/dcds.2004.10.1

[17]

Kousuke Kuto, Tohru Tsujikawa. Stationary patterns for an adsorbate-induced phase transition model I: Existence. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1105-1117. doi: 10.3934/dcdsb.2010.14.1105

[18]

Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787

[19]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157

[20]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]