Hidden Dynamics in One-Dimensional Wave Equation

• Xu Zhang ^[1] ; Liang Guo ^[1] ; Iram Hussan ^[1]
  1. [1] Shandong University

     Shandong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • There are two kinds of chaotic attractors in ordinary differential equations, self-excited and hidden. However, there is no corresponding hidden attractors (or dynamics) in partial differential equations. In this article, a kind of one-dimensional wave equations with a certain kind of boundary conditions is provided, which can be regarded as the first partial differential equation with hidden dynamics. The coexistence of regular and chaotic dynamics (i.e., the dynamical behavior is by turns regular and complicated, or the oscillation is partly hidden and partly visible) can be observed, where the existence of chaotic dynamics is verified by the snap-back repeller theory. Further, three examples with numerical experiments are given to illustrate the theoretical results.

• Referencias bibliográficas
  • 1. Aamo, O.M., Krstic, M.: Global stabilization of a nonlinear Ginzburg-Landau model of vortex shedding. Eur. J. Control. 10, 105–116 (2004)
  • 2. Anfinsen, H., Aamo, O.M.: Disturbance rejection in the interior domain of linear 2 × 2 hyperbolic systems. IEEE Trans. Autom. Control 60,...
  • 3. Banks, J.J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Amer. Math. Monthly 99, 332–334 (1992)
  • 4. Bao, B.C., Li, H.Z., Zhu, L., Zhang, X., Chen, M.: Initial-switched boosting bifurcations in 2d hyperchaotic map. Chaos 30, 033107 (2020)
  • 5. Caballero, R., Marín-Rubio, P., Valero, J.: Existence and characterization of attractors for a nonlocal reaction-diffusion equation with...
  • 6. Cerpa, E., Coron, J.M.: Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition. IEEE Trans. Autom....
  • 7. Chen, G., Hsu, S., Zhou, J.: Chaotic vibrations of the one-dimensional wave equation due to a selfexcitation boundary condition. II. Energy...
  • 8. Chen, G., Hsu, S., Zhou, J.: Chaotic vibrations of the one-dimensional wave equation due to a selfexcitation boundary condition. III. Natural...
  • 9. Chen, G., Hsu, S., Zhou, J., Chen, G., Crosta, G.: Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary...
  • 10. Chen, Y., Huang, Y., Li, L.: The persistence of snap-back repeller under small c1 perturbations in Banach spaces. Int. J. Bifurcation...
  • 11. Chen, Z., Yang, D.: Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations....
  • 12. Bonotto, E.D.M., Uzal, J.M.: Global attractors for a class of discrete dynamical systems. J. Dyn. Diff. Equat. (2024). https://doi.org/10.1007/s10884-024-10356-9
  • 13. Deutscher, J.: Finite-time output regulation for linear 2 × 2 hyperbolic systems using backstepping. Automatica 75, 54–62 (2017)
  • 14. Dudkowski, D., Jafari, S., Kapitaniak, T., Kuznetsov, N.V., Leonov, G.A., Prasad, A.: Hidden attractors in dynamical systems. Phys. Reports...
  • 15. Gardini, L., Tramontana, F.: Snap-back repellers and chaotic attractors. Phys. Rev. E 81, 046202 (2010)
  • 16. Hasan, A., Aamo, O.M., Foss, B.: Boundary control for a class of pseudo-parabolic differential equations. Syst. Control Lett. 62, 63–69...
  • 17. Huang, Y.: Growth rates of total variations of snapshots of the 1d linear wave equation with composite nonlinear boundary reflection relations....
  • 18. Huang, Y.: A new characterization of non- isotropic chaotic vibrations of the one-dimensional linear wave equation with a van der Pol...
  • 19. Jadachowski, L., Meurer, T., Kugi, A.: Backstepping observers for linear pdes on higher-dimensional spatial domains. Automatica 51, 85–97...
  • 20. Jafari, S., Pham, V.-T., Golpayegani, S.M.R.H., Moghtadaei, M., Kingni, S.T.: The relationship between chaotic maps and some chaotic systems...
  • 21. Jafari, S., Sprott, J., Nazarimehr, F.: Recent new examples of hidden attractors. Eur. Phys. J. Special Topics 224, 1469–1476 (2015)
  • 22. Khennaoui, A.A., Ouannas, A., Boulaaras, S., Pham, V.-T., Azar, A.T.: A fractional map with hidden attractors: chaos and control. Eur....
  • 23. Kloeden, P., Li, Z.: Li-Yorke chaos in higher dimensions: a review. J. Difference Equ. Appl. 12, 247–269 (2006)
  • 24. Krstic, M., Smyshlyaev, A.: Boundary control of PDEs: A course on backstepping design. SIAM, Philadelphia (2008)
  • 25. Smyshlyaev, A., Krstic, M., Guo, B.Z.: Boundary controllers and observers for the linearized Schrödinger equation. SIAM J. Control. Optim....
  • 26. Kuznetsov, N., Leonov, G., Vagaitsev, V.: Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc....
  • 27. Leonov, G., Kuznetsov, N.: Hidden attractors in dynamical systems. From hidden oscillations in HilbertKolmogorov, Aizerman, and Kalman...
  • 28. Leonov, G., Kuznetsov, N., Vagaitsev, V.: Localization of hidden Chua’s attractors. Phys. Lett. A 375, 2230–2233 (2011)
  • 29. Leonov, G., Vagaitsev, V., Kuznetsov, N.: Algorithm for localizing Chua attractors based on the harmonic linearization method. Doklady...
  • 30. Li, L.: Chaotic oscillations of 1d wave equation with or without energy-injections. Electronic Res. Arch. 30, 2600–2617 (2022)
  • 31. Li, L., Huang, Y.: Growth rates of total variations of snapshots of 1d linear wave equations with nonlinear right-end boundary conditions....
  • 32. Liao, K., Shih, C.: Snapback repellers and homoclinic orbits for multi-dimensional maps. J. Math. Anal. Appl. 386, 387–400 (2012)
  • 33. Logan, R.W., McClung, C.A.: Rhythms of life: circadian disruption and brain disorders across the lifespan. Nat. Rev. Neurosci. 20, 49–65...
  • 34. Marco, M., Forti, M., Pancioni, L., Tesi, A.: Snap-back repellers and chaos in a class of discrete-time memristor circuits. Nonlinear...
  • 35. Marotto, F.R.: Snap-back repellers imply chaos in Rn. J. Math. Anal. Appl. 63, 199–223 (1978)
  • 36. Molaie, M., Jafari, S., Sprott, J., Golpayegani, M.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurcation Chaos 23, 1350188...
  • 37. Panahi, S., Sprott, J.C., Jafari, S.: Two simplest quadratic chaotic maps without equilibrium. Int. J. Bifurcation Chaos 28, 1850144 (2018)
  • 38. Shi, Y., Chen, G.: Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons Fractals 22, 555–571 (2004)
  • 39. Shi, Y., Chen, G.: Discrete chaos in Banach spaces. Sci. China, Ser. A Math. 34, 595–609 (2004)
  • 40. Shi, Y., Yu, P.: On chaos of the logistic maps. Dyn. Contin. Discrete Impuls. Syst. Ser. B 14, 175–195 (2007)
  • 41. Shi, Y., Yu, P.: Chaos induced by regular snap-back repellers. J. Math. Anal. Appl. 337, 1480–1494 (2008)
  • 42. Smyshlyaev, A., Krstic, M.: Boundary control of an anti-stable wave equation with anti-damping on the uncontrollered boundary. Sys. Control...
  • 43. Solis, F.J., Jódar, L., Chen, B.: Chaos in the one-dimensional wave equation. Appl. Math. Lett. 18, 85–90 (2005)
  • 44. Sprott, J.: Some simple chaotic flows. Phys. Rev. E 50, R647–R650 (1994)
  • 45. Vagaitsev, V., Kuznetsov, N., Leonov, G.: Localization of hidden attractors of the generalized Chua system based on the method of harmonic...
  • 46. Wang, F., Wang, J., Feng, Z.: Chaotic dynamical behavior of coupled one-dimensional wave equations. Int. J. Bifurcation Chaos 31, 2150115...
  • 47. Wang, X., Chen, G.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71, 429–436 (2013)
  • 48. Wang, X., Kuznetsov, N.V., Chen, G. (eds.): Chaotic Systems with Multistability and Hidden Attractors. Springer Nature Switzerland AG,...
  • 49. Xiang, Q., Yang, Q.: Nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear...
  • 50. Yue, X., Lv, G., Zhang, Y.: Rare and hidden attractors in a periodically forced Duffing system with absolute nonlinearity. Chaos, Solitons...
  • 51. Zelik, S.V.: Attractors. then and now. Russ. Math. Surv. 78, 635–777 (2023)
  • 52. Zhang, L., Shi, Y., Zhang, X.: Chaotic dynamical behaviors of a one-dimensional wave equation. J. Math. Anal. Appl. 369, 623–634 (2010)
  • 53. Zhang, X., Chen, G.: Polynomial maps with hidden complex dynamics. Discrete Continuous Dyn. Syst. Ser. B 24, 2941–2954 (2019)
  • 54. Zhang, X., Li, C., Lei, T., Liu, Z., Tao, C.: A symmetric controllable hyperchaotic hidden attractor. Symmetry 12, 550 (2020)
  • 55. Zhang, X., Shi, Y., Chen, G.: Constructing chaotic polynomial maps. Int. J. Bifurcation Chaos 19, 531–543 (2009)
  • 56. Zhu, X., Zhong, C.: Uniform attractors for reaction-diffusion equations with a larger class of external forces. J. Math. Phys. 63, 011508...