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Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

Liangliang Li

Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Guangzhou, P. R. China

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Yu Huang

Department of Mathematics, Sun Yat-Sen University, Guangzhou, P. R. China

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Goong Chen

Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar

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, and

Tingwen Huang

Science Program, Texas A&M University at Qatar, Education City, Doha, Qatar

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https://doi.org/10.1142/S0218127415300323Cited by:7 (Source: Crossref)

Abstract

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

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