# American Institute of Mathematical Sciences

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

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##### 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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