Existence of chaos in weakly quasilinear systems

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

Existence of chaos in weakly quasilinear systems

Y. Charles Li ^1,

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

Received February 2009 Revised August 2010 Published April 2011

Abstract
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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.

Keywords: Ginzburg-Landau equation., Quasilinear system, homoclinic orbit, sine-Gordon equation, chaos.

Mathematics Subject Classification: Primary 35, 37; Secondary 34, 4.

Citation: Y. Charles Li. Existence of chaos in weakly quasilinear systems. Communications on Pure and 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.
[2]	Y. Li, "Chaos in Partial Differential Equations,'' International Press, Somerville, MA, USA, 2004.
[3]	Y. Li, et al., Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation, Comm. Pure Appl. Math., XLIX (1996), 1175-1255. doi: 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9.
[4]	Y. Li, Chaos and shadowing lemma for autonomous systems of infinite dimensions, J. Dynam. Diff. Eq., 15 (2003), 699-730. doi: 10.1023/B:JODY.0000010062.09599.d8.
[5]	Y. Li, Chaos and shadowing around a homoclinic tube, Abstr. Appl. Anal., 16 (2003), 923-931. doi: 10.1155/S1085337503304038.
[6]	Y. Li, Homoclinic tubes and chaos in perturbed sine-Gordon equation, Chaos, Solitons and Fractals, 20 (2004), 791-798. doi: 10.1016/j.chaos.2003.08.013.
[7]	Y. Li, Persistent homoclinic orbits for nonlinear Schrödinger equation under singular perturbation, Dynamics of PDE, 1 (2004), 87-123.
[8]	Y. Li, Existence of chaos for nonlinear Schrödinger equation under singular perturbation, Dynamics of PDE, 1 (2004), 225-237.
[9]	Y. Li, Chaos in Miles' equations, Chaos, Solitons and Fractals, 22 (2004), 965-974. doi: 10.1016/j.chaos.2004.03.018.
[10]	Y. Li, Strange tori of the derivative nonlinear Schrödinger equation, Letters in Mathematical Physics, 80 (2007), 83-99. doi: 10.1007/s11005-007-0152-4.
[11]	C. Sparrow, "The Lorenz Equations,'' Springer, 1982.
[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), 325-365. doi: 10.1007/BF01048949.
[13]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.

show all references

References:

[1]	K. Alligood, T. Sauer and J. Yorke, "Chaos,'' Springer, 1997.
[2]	Y. Li, "Chaos in Partial Differential Equations,'' International Press, Somerville, MA, USA, 2004.
[3]	Y. Li, et al., Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation, Comm. Pure Appl. Math., XLIX (1996), 1175-1255. doi: 10.1002/(SICI)1097-0312(199611)49:11<1175::AID-CPA2>3.0.CO;2-9.
[4]	Y. Li, Chaos and shadowing lemma for autonomous systems of infinite dimensions, J. Dynam. Diff. Eq., 15 (2003), 699-730. doi: 10.1023/B:JODY.0000010062.09599.d8.
[5]	Y. Li, Chaos and shadowing around a homoclinic tube, Abstr. Appl. Anal., 16 (2003), 923-931. doi: 10.1155/S1085337503304038.
[6]	Y. Li, Homoclinic tubes and chaos in perturbed sine-Gordon equation, Chaos, Solitons and Fractals, 20 (2004), 791-798. doi: 10.1016/j.chaos.2003.08.013.
[7]	Y. Li, Persistent homoclinic orbits for nonlinear Schrödinger equation under singular perturbation, Dynamics of PDE, 1 (2004), 87-123.
[8]	Y. Li, Existence of chaos for nonlinear Schrödinger equation under singular perturbation, Dynamics of PDE, 1 (2004), 225-237.
[9]	Y. Li, Chaos in Miles' equations, Chaos, Solitons and Fractals, 22 (2004), 965-974. doi: 10.1016/j.chaos.2004.03.018.
[10]	Y. Li, Strange tori of the derivative nonlinear Schrödinger equation, Letters in Mathematical Physics, 80 (2007), 83-99. doi: 10.1007/s11005-007-0152-4.
[11]	C. Sparrow, "The Lorenz Equations,'' Springer, 1982.
[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), 325-365. doi: 10.1007/BF01048949.
[13]	W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math., 2 (2002), 53-117.

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