For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li–Yorke chaos. For this end, we consider a system $(I, f_{\infty})$ induced by a sequence of continuous maps and its functional envelope $(L^{1} (I, I), H_{\infty})$ , and show that, under some considerable condition, $H_{\infty}$ is transitive if and only if $f_{\infty}$ is weakly mixing of order $3$ ; $H_{\infty}$ is Li–Yorke chaotic and sensitive if $f_{\infty}$ is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.

Keywords:

References

Akin, E. [2004] “ Lectures on Cantor and Mycielski sets for dynamical systems,” Chapel Hill Ergodic Theory Workshops, Contemp. Math., Vol. 356 (Amer. Math. Soc., Providence, RI), pp. 21–79. Crossref, Google Scholar
Arcila, M. M. & Peris, A. [2013] “ Mixing properties for nonautonomous linear dynamics and invariant sets,” Appl. Math. Lett. 26, 215–218. Crossref, Web of Science, Google Scholar
Auslander, J., Kolyada, S. & Snoha, L. [2007] “ Functional envelope of a dynamical system,” Nonlinearity 20, 2245–2269. Crossref, Web of Science, Google Scholar
Balibrea, F. & Oprocha, P. [2012] “ Weak mixing and chaos in nonautonomous discrete systems,” Appl. Math. Lett. 25, 1135–1141. Crossref, Web of Science, Google Scholar
Blanchard, F., Glasner, E., Kolyada, S. & Maass, A. [2002] “ On Li–Yorke pairs,” J. Reine Angew. Math. 547, 51–68. Crossref, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Zhou, J. [1998a] “ 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. Crossref, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Zhou, J. [1998b] “ Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part II, energy injection, period doubling and homoclinic orbits,” Int. J. Bifurcation and Chaos 8, 423–445. Link, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Zhou, J. [1998c] “ Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part III, natural hysteresis memory effects,” Int. J. Bifurcation and Chaos 8, 447–470. Link, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Zhou, J. [1998d] “ Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the spane,” J. Math. Phys. 39, 6459–6489. Crossref, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Zhou, J. [2002a] “ Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition,” Int. J. Bifurcation and Chaos 12, 535–559. Link, Web of Science, Google Scholar
Chen, G., Hsu, S. B. & Huang, T. [2002b] “ Analyzing displacement terms memory effect in a van der Pol type boundary condition to prove chaotic vibration of the wave equation,” Int. J. Bifurcation and Chaos 12, 965–981. Link, Web of Science, Google Scholar
Chen, G., Huang, T. & Huang, Y. [2004] “ Chaotic behavior of interval maps and total variations of iterates,” Int. J. Bifurcation and Chaos 14, 2161–2186. Link, Web of Science, Google Scholar
Chen, G. & Huang, Y. [2011] “ Chaotic maps: Dynamics, fractals and rapid fluctuations, synthesis,” Lectures on Mathematics and Statistics, ed. Krantz, S. G. (Morgan & Claypool Publishers, Williston, VT). Crossref, Google Scholar
Chen, Z. & Huang, Y. [2017] “ Functional envelopes relative to the point-open topology on a subset,” Discr. Contin. Dyn. Syst. 37, 1227–1246. Crossref, Web of Science, Google Scholar
Chen, Z., Huang, T., Huang, Y. & Liu, X. [2018] “ Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions,” J. Math. Phys. 59, 022704. Crossref, Web of Science, Google Scholar
Dai, X. [2015] “ Chaotic dynamics of continuous-time topological semiflow on Polish spaces,” J. Diff. Eqs. 258, 2794–2805. Crossref, Web of Science, Google Scholar
Grosse-Erdmann, K.-G. & Peris, A. [2011] Linear Chaos, Universitext (Springer-Verlag, London). Crossref, Google Scholar
Huang, Y. [2003] “ Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations,” Int. J. Bifurcation and Chaos 13, 1183–1195. Link, Web of Science, Google Scholar
Huang, Y., Luo, J. & Zhou, Z. [2005] “ Rapid fluctuations of snapshots of one dimensional linear wave equation with a van der Pol nonlinear boundary condition,” Int. J. Bifurcation and Chaos 15, 567–580. Link, Web of Science, Google Scholar
Iwanik, A. [1989] “Independent sets of transitive points,” Dynamical Systems and Ergodic Theory, Warsaw, 1986, in: Banach Center Publ., Vol. 23 (PWN, Warsaw), pp. 277–282. Google Scholar
Li, T. & Yorke, J. A. [1975] “ Period three implies chaos,” Amer. Math. Monthly 82, 985–992. Crossref, Web of Science, Google Scholar
Li, L. & Huang, Y. [2010] “ Growth rates of total variations of snapshots of 1D linear wave equations with nonlinear right-end boundary conditions,” J. Math. Anal. Appl. 361, 69–85. Crossref, Web of Science, Google Scholar
Li, L., Huang, Y., Chen, G. & Huang, T. [2015] “ Chaotic oscillations of second order linear hyperbolic equations with nonlinear boundary conditions: A factorizable but noncommutative case,” Int. J. Bifurcation and Chaos 25, 1530032-1–20. Link, Web of Science, Google Scholar
Munkres, J. [2000] Topology, 2nd edition, Pearson. Google Scholar
Ruette, S. [2017] Chaos on the Interval, University Lecture Series, Vol. 67 (American Mathematical Society, Providence, RI). Crossref, Google Scholar