September  2011, 10(5): 1331-1344. doi: 10.3934/cpaa.2011.10.1331

Existence of chaos in weakly quasilinear systems

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65203, United States

Received  February 2009 Revised  August 2010 Published  April 2011

The aim of this article is twofold: (1). develop a strategy to prove the existence of chaos in weakly quasilinear systems, (2). strengthen the existing results on chaos in partial differential equations. First, we study a sine-Gordon equation containing weakly quasilinear terms, and existence of chaos is proved. Then, we study a Ginzburg-Landau equation containing weakly quasilinear terms, and existence of chaos is proved under generic conditions. Finally, in the Appendix, we prove the existence of chaos in a reaction-diffusion equation.
Citation: Y. Charles Li. Existence of chaos in weakly quasilinear systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1331-1344. doi: 10.3934/cpaa.2011.10.1331
References:
[1]

K. Alligood, T. Sauer and J. Yorke, "Chaos,'', Springer, (1997).   Google Scholar

[2]

Y. Li, "Chaos in Partial Differential Equations,'', International Press, (2004).   Google Scholar

[3]

Y. Li, et al., Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation,, Comm. Pure Appl. Math., XLIX (1996), 1175.  doi: 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9.  Google Scholar

[4]

Y. Li, Chaos and shadowing lemma for autonomous systems of infinite dimensions,, J. Dynam. Diff. Eq., 15 (2003), 699.  doi: 10.1023/B:JODY.0000010062.09599.d8.  Google Scholar

[5]

Y. Li, Chaos and shadowing around a homoclinic tube,, Abstr. Appl. Anal., 16 (2003), 923.  doi: 10.1155/S1085337503304038.  Google Scholar

[6]

Y. Li, Homoclinic tubes and chaos in perturbed sine-Gordon equation,, Chaos, 20 (2004), 791.  doi: 10.1016/j.chaos.2003.08.013.  Google Scholar

[7]

Y. Li, Persistent homoclinic orbits for nonlinear Schrödinger equation under singular perturbation,, Dynamics of PDE, 1 (2004), 87.   Google Scholar

[8]

Y. Li, Existence of chaos for nonlinear Schrödinger equation under singular perturbation,, Dynamics of PDE, 1 (2004), 225.   Google Scholar

[9]

Y. Li, Chaos in Miles' equations,, Chaos, 22 (2004), 965.  doi: 10.1016/j.chaos.2004.03.018.  Google Scholar

[10]

Y. Li, Strange tori of the derivative nonlinear Schrödinger equation,, Letters in Mathematical Physics, 80 (2007), 83.  doi: 10.1007/s11005-007-0152-4.  Google Scholar

[11]

C. Sparrow, "The Lorenz Equations,'', Springer, (1982).   Google Scholar

[12]

H. Steinlein and H.-O. Walther, Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for $C^1$-maps in Banach spaces,, J. Dynam. Diff. Eq., 2 (1990), 325.  doi: 10.1007/BF01048949.  Google Scholar

[13]

W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53.   Google Scholar

show all references

References:
[1]

K. Alligood, T. Sauer and J. Yorke, "Chaos,'', Springer, (1997).   Google Scholar

[2]

Y. Li, "Chaos in Partial Differential Equations,'', International Press, (2004).   Google Scholar

[3]

Y. Li, et al., Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation,, Comm. Pure Appl. Math., XLIX (1996), 1175.  doi: 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9.  Google Scholar

[4]

Y. Li, Chaos and shadowing lemma for autonomous systems of infinite dimensions,, J. Dynam. Diff. Eq., 15 (2003), 699.  doi: 10.1023/B:JODY.0000010062.09599.d8.  Google Scholar

[5]

Y. Li, Chaos and shadowing around a homoclinic tube,, Abstr. Appl. Anal., 16 (2003), 923.  doi: 10.1155/S1085337503304038.  Google Scholar

[6]

Y. Li, Homoclinic tubes and chaos in perturbed sine-Gordon equation,, Chaos, 20 (2004), 791.  doi: 10.1016/j.chaos.2003.08.013.  Google Scholar

[7]

Y. Li, Persistent homoclinic orbits for nonlinear Schrödinger equation under singular perturbation,, Dynamics of PDE, 1 (2004), 87.   Google Scholar

[8]

Y. Li, Existence of chaos for nonlinear Schrödinger equation under singular perturbation,, Dynamics of PDE, 1 (2004), 225.   Google Scholar

[9]

Y. Li, Chaos in Miles' equations,, Chaos, 22 (2004), 965.  doi: 10.1016/j.chaos.2004.03.018.  Google Scholar

[10]

Y. Li, Strange tori of the derivative nonlinear Schrödinger equation,, Letters in Mathematical Physics, 80 (2007), 83.  doi: 10.1007/s11005-007-0152-4.  Google Scholar

[11]

C. Sparrow, "The Lorenz Equations,'', Springer, (1982).   Google Scholar

[12]

H. Steinlein and H.-O. Walther, Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for $C^1$-maps in Banach spaces,, J. Dynam. Diff. Eq., 2 (1990), 325.  doi: 10.1007/BF01048949.  Google Scholar

[13]

W. Tucker, A rigorous ODE solver and Smale's 14th problem,, Found. Comput. Math., 2 (2002), 53.   Google Scholar

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