Chaotic oscillations of 1D wave equation with or without energy-injections

Liangliang Li; Liangliang Li

doi:10.3934/era.2022133

Electronic Research Archive

2022, Volume 30, Issue 7: 2600-2617. doi: 10.3934/era.2022133

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Chaotic oscillations of 1D wave equation with or without energy-injections

Liangliang Li ^,

Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China

Received: 31 December 2020 Revised: 28 February 2022 Accepted: 10 March 2022 Published: 16 May 2022

It is interesting and challenging to study chaotic phenomena in partial differential equations. In this paper, we mainly study the problems for oscillations governed by 1D wave equation with general nonlinear feedback control law and energy-conserving or energy-injecting effects at the boundaries. We show that i) energy-injecting effect at the boundary is the necessary condition for the onset of chaos when the nonlinear feedback law is an odd function; ii) chaos never occurs if the nonlinear feedback law is an even function; iii) when one of the two ends is fixed, only the effect of self-regulation at the other end can still cause the onset of chaos; whereas if one of the two ends is free, there will never be chaos for any feedback control law at the other end. In addition, we give a sufficient condition about the general feedback law at one of two ends to ensure the occurrence of chaos. Numerical simulations are provided to demonstrate the effectiveness of the theoretical outcomes.
- chaotic vibrations,
- wave equation,
- wave propagation,
- generalized boundary condition
Citation: Liangliang Li. Chaotic oscillations of 1D wave equation with or without energy-injections[J]. Electronic Research Archive, 2022, 30(7): 2600-2617. doi: 10.3934/era.2022133

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Abstract

It is interesting and challenging to study chaotic phenomena in partial differential equations. In this paper, we mainly study the problems for oscillations governed by 1D wave equation with general nonlinear feedback control law and energy-conserving or energy-injecting effects at the boundaries. We show that i) energy-injecting effect at the boundary is the necessary condition for the onset of chaos when the nonlinear feedback law is an odd function; ii) chaos never occurs if the nonlinear feedback law is an even function; iii) when one of the two ends is fixed, only the effect of self-regulation at the other end can still cause the onset of chaos; whereas if one of the two ends is free, there will never be chaos for any feedback control law at the other end. In addition, we give a sufficient condition about the general feedback law at one of two ends to ensure the occurrence of chaos. Numerical simulations are provided to demonstrate the effectiveness of the theoretical outcomes.

References

[1]	Y.Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection, Int. J. Bifurcation Chaos, 13 (2003), 1183–1195. https://doi.org/10.1142/S0218127403007138 doi: 10.1142/S0218127403007138
[2]	T. Y. Li, J. A.Yorke, Period Three Implies Chaos, Am. Math. Mon., 82 (1975), 985–992. https://doi.org/10.1007/978-0-387-21830-4
[3]	M. Štefánková, Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval, Discrete Contin. Dyn. Syst., 36 (2016), 3435–3443. https://doi.org/10.3934/dcds.2016.36.3435 doi: 10.3934/dcds.2016.36.3435
[4]	J. S. Cánocas, T. Puu, M. R. Marín, Detecting chaos in a duopoly model via symbolic dynamics, Discrete Contin. Dyn. Syst.-B, 13 (2010), 269–278. https://doi.org/10.3934/dcdsb.2010.13.269 doi: 10.3934/dcdsb.2010.13.269
[5]	G. Chen, S. B. Hsu, J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part I: controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265–4311. https://doi.org/10.1090/S0002-9947-98-02022-4 doi: 10.1090/S0002-9947-98-02022-4
[6]	G. Chen, , S.B. Hsu, J. Zhou, , 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 Chaos, 8 (1998), 423–445. https://doi.org/10.1142/S0218127498000280 doi: 10.1142/S0218127498000280
[7]	Y. Huang, J. Luo, Z. L. Zhou, Rapid fluctuations of snapshots of one-dimensional linear wave equations with a van der Pol nonlinear boundary conditions, Int. J. Bifurcation Chaos, 15 (2005), 567–580. https://doi.org/10.1142/S0218127405012223 doi: 10.1142/S0218127405012223
[8]	L. L. Li, Y. Huang, Growth of total varitions of snapshots of 1D linear wave equations with nonlinear right-end boundary conditions, J. Math. Anal. Appl., 361 (2010), 69–85. https://doi.org/10.1016/j.jmaa.2009.09.011 doi: 10.1016/j.jmaa.2009.09.011
[9]	J. Liu, Y. Huang, H. Sun, M. Xiao, Numerical methods for weak solution of wave equation with van der Pol type nonlinear boundary conditions, Numer. Meth. Part. D. E., 32 (2016), 373–398. https://doi.org/10.1002/num.21997 doi: 10.1002/num.21997
[10]	P. C. Etter, Underwater Acoustic Modeling and Simulation, Spon Press, London, New Yoek, 2003. https://doi.org/10.4324/9780203417652
[11]	L. S. Block, W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, Springer-Verlag, NY, Heidelberg Berlin, 1992. https://doi.org/10.1007/BFb0084762
[12]	G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Int. J. Bifurcation Chaos, 14 (2004), 2161–2186. https://doi.org/10.1142/S0218127404010540 doi: 10.1142/S0218127404010540

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