Chaos of Multi-dimensional Weakly Hyperbolic Equations with General Nonlinear Boundary Conditions

• Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Amer. Math. Monthly 99, 332–334 (1992)

  Article  MathSciNet  Google Scholar 

• Bartnik, R., Isenberg, J.: The constraint equations. The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity. Birkhäuser Basel, 1–38 (2004)

• Battelli, F., Fečkan, M.: Chaos in the beam equation. J. Differ. Equ. 209, 172–227 (2005)

  Article  MathSciNet  Google Scholar 

• Benzoni-Gavage, S., Serre, D.: Multi-Dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications. Clarendon Press, Oxford University Press, Oxford (2007)

• Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75(2), 400–422 (1988)

  Article  MathSciNet  Google Scholar 

• Chen, G., Hsu, S.B., Zhou, J.: Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis. Trans. Amer. Math. Soc. 350, 4265–4311 (1998)

  Article  MathSciNet  Google Scholar 

• Chen, G., Hsu, S.B., Zhou, J.: Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part II: Energy injection, period doubling and homoclinic orbits. Internat. J. Bifur. Chaos 8(3), 423–445 (1998)

  Article  MathSciNet  Google Scholar 

• Chen, Z.J., Huang, T.W., Huang, Y.: Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions. J. Math. Phys. 59, 022704 (2018)

  Article  MathSciNet  Google Scholar 

• Cicognani, M., Colombini, F.: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem. J. Differ. Equ. 221(1), 143–157 (2006)

  Article  MathSciNet  Google Scholar 

• Colombini, F., Nishitani, T.: On finitely degenerate hyperbolic operators of second order. Osaka J. Math. 41, 933–947 (2004)

  MathSciNet  Google Scholar 

• Colombini, F.: Energy estimates at infinity for hyperbolic equations with oscillating coefficients. J. Differ. Equ. 231(2), 598–610 (2006)

  Article  MathSciNet  Google Scholar 

• Columbini, F., Jannelli, E., Spagnolo, S.: Well-posedness in the Gevrey classes of the Cauchy problem for non-strictly hyperbolic equation with coefficients depending on time. Ann. Scuold Norm. Sup. Pisa Cl. Sci. 2, 291–312 (1983)

• Columbini, F., Spagnolo, S.: An example of a weakly hyperbolic Cauchy problem not well posed in \(C^{\infty }\). Acta Math. 148(1), 243–253 (1982)

  Article  MathSciNet  Google Scholar 

• Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discrete Contin. Dyn. Syst. 35(2), 653–668 (2015)

  Article  MathSciNet  Google Scholar 

• Crabb, R., Mackey, M.C., Rey, A.D.: Propagating fronts, chaos and multistability in a cell replication model. Chaos 6(3), 477–492 (1996)

  Article  Google Scholar 

• Dai, X.P., Huang, T.W., Huang, Y., Chen, G.: Chaotic oscillations of solutions of first order hyperbolic systems in 1D with nonlinear boundary conditions. Internat. J. Bifur. Chaos 24, 1450072 (2014)

  Article  MathSciNet  Google Scholar 

• Devaney, R.L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley Publishing Company, Boston (1989)

  Google Scholar 

• Granero-Belinchón, R., Hunter, J.K.: On a nonlocal analog of the Kuramoto–Sivashinsky equation. Nonlinearity 28, 1103 (2015)

  Article  MathSciNet  Google Scholar 

• Garetto, C., Ruzhansky, M.: Weakly hyperbolic equations with non-analytic coefficients and lower order terms. Math. Ann. 357, 401–440 (2013)

  Article  MathSciNet  Google Scholar 

• Huang, J.Z.: Strange dynamic behavior induced by time delay for a Rijke tube system with periodic excitation. J. Nonlinear Sci. 30, 767–791 (2019)

  Article  MathSciNet  Google Scholar 

• Huang, Y.: Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations. Internat. J. Bifur. Chaos 13, 1183–1195 (2003)

  Article  MathSciNet  Google Scholar 

• Huang, Y., Zou, X.: Co-existence of chaos and stable periodic orbits in a simple discrete neural network. J. Nonlinear Sci. 15, 291–303 (2005)

  Article  MathSciNet  Google Scholar 

• Jannelli E. The hyperbolic symmetrizer: theory and applications. Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini’s 60th Birthday, 113–139 (2009)

• Khan, M., Shah, T., Gondal, M.A.: An efficient technique for the construction of substitution box with chaotic partial differential equation. Nonlinear Dyn. 73(3), 1795–1801 (2013)

  Article  MathSciNet  Google Scholar 

• Li, L.L., Chen, Y.L., Huang, Y.: Nonisotropic spatiotemporal chaotic vibrations of the one-dimensional wave equation with a mixing transport term and general nonlinear boundary condition. J. Math. Phys. 51, 102703 (2010)

  Article  MathSciNet  Google Scholar 

• Li, L.L., Tian, J., Chen, G.: Chaotic vibration of a two-dimensional non-strictly hyperbolic equation. Canad. Math. Bull. 61, 768–786 (2018)

  Article  MathSciNet  Google Scholar 

• Liu, C.J., Xie, F., Yang, T.: Justification of Prandtl ansatz for MHD boundary layer. SIAM J. Math. Anal. 51(3), 2748–2791 (2019)

  Article  MathSciNet  Google Scholar 

• Li, T.Y., Yorke, J.A.: Period three implies chaos. Amer. Math. Monthly 8(10), 985–992 (1975)

  Article  MathSciNet  Google Scholar 

• Li, Y.: Chaos in Partial Differential Equations. International Press, Somerville (2004)

  Google Scholar 

• Li, Z., Shi, Y., Liang, W.: Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces. Nonlinear Anal. 72, 757–770 (2010)

  Article  MathSciNet  Google Scholar 

• Lim, T.S., Lu, Y., Nolen, J.H.: Quantitative propagation of chaos in a bimolecular chemical reaction-diffusion model. SIAM J. Math. Anal. 52(2), 2098–2133 (2020)

  Article  MathSciNet  Google Scholar 

• Liu, J., Huang, Y., Sun, H., Xiao, M.: Numerical methods for weak solution of wave equation with van der pol type nonlinear boundary conditions. Numer. Meth. Part. Differ. Equ. 32, 373–398 (2016)

  Article  MathSciNet  Google Scholar 

• Marotto, F.R.: On redefining a snap-back repeller. Chaos Solit. Fract. 25, 25–28 (2005)

  Article  MathSciNet  Google Scholar 

• Moore, D.R., Toomre, J., Knobloch, E., Weiss, N.O.: Period doubling and chaos in partial differential equations for thermosolutal convection. Nature 303(5919), 663–667 (1983)

  Article  Google Scholar 

• Núñez, C., Obaya, R.: Li-Yorke chaos in nonautonomous Hopf bifurcation patterns-I. Nonlinearity 32, 3940 (2019)

  Article  MathSciNet  Google Scholar 

• Shi, Y.M., Yu, P.: On chaos of the logistic maps. Dyn. Contin. Discrete Impuls. Syst. Ser. B. 14, 175–195 (2007)

• Shi, Y., Yu, P.: Chaos induced by regular snap-back repellers. J. Math. Anal. Appl. 337, 1480–1494 (2008)

  Article  MathSciNet  Google Scholar 

• Wilczak, D., Zgliczynski, P.: A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto–Sivashinsky PDE on the line. J. Differ. Equ. 269, 8509–8548 (2020)

  Article  MathSciNet  Google Scholar 

• Xiang, Q.M., Wu, Z.H., Park, J.H., Guo, B.Z.: Observability and observers for a class of 2-D hyperbolic PDE chaotic systems. SIAM J. Control. Optim. 61(4), 2282–2304 (2023)

  Article  MathSciNet  Google Scholar 

• Xiang, Q.M., Yang, Q.G.: Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition. J. Math. Anal. Appl. 462, 730–7746 (2018)

  Article  MathSciNet  Google Scholar 

• Xiang, Q.M., Yang, Q.G.: Nonisotropic chaotic vibrations of a 2D hyperbolic PDE. Chaos 30, 023127 (2020)

  Article  MathSciNet  Google Scholar 

• Xiang, Q.M., Zhu, P.X., Yang, Q.G., Park, J.H.: Nonisotropic chaos induced by snap-back repellers and heteroclinic cycles of 3-D hyperbolic PDEs. Nonlinear Dyn. 108, 4399–4413 (2022)

  Article  Google Scholar 

• Yang, Q.G., Xiang, Q.M.: Chaotic vibrations of 3D linear hyperbolic PDEs with linear perturbations of superlinear boundary conditions. J. Math. Anal. Appl. 507, 125743 (2022)

  Article  MathSciNet  Google Scholar 

• Zhu, P.X., Yang, Q.G.: Chaos of multi-dimensional linear hyperbolic PDEs. Proc. Amer. Math. Soc. 151(4), 1593–1607 (2023)

  MathSciNet  Google Scholar 

• Zhu, P.X., Yang, Q.G.: Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition. J. Math. Phys. 63, 072702 (2022)

  Article  MathSciNet  Google Scholar 

Download references