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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 span. (English) Zbl 0959.37027

Summary: A wave equation on a one-dimensional interval \(I\) has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is fixed. At exactly the midpoint of the interval \(I\), energy is injected into the system through a pair of transmission conditions in the feedback form of anti-damping. We study chaotic wave propagation in the system. A cause of chaos of snapback repellers has been identified. These snapback repellers are repelling fixed points possessing homoclinic orbits of the noninvertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middle-of-the-span points. Existing literature [F. R. Marotto, J. Math. Anal. Appl. 63, 199-223 (1978; Zbl 0381.58004)] on snapback repellers contains an error. We clarify the error and give a refined theorem that snapback repellers imply chaos. Numerical simulations of chaotic vibration are also illustrated.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
35L70 Second-order nonlinear hyperbolic equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations

Citations:

Zbl 0381.58004
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References:

[1] DOI: 10.1142/S0218127496000898 · Zbl 0886.73026
[2] DOI: 10.1142/S0218127498000280 · Zbl 0938.35088
[3] DOI: 10.1142/S0218127498000292 · Zbl 0938.35089
[4] DOI: 10.1137/0149102 · Zbl 0685.93054
[5] DOI: 10.1016/0022-247X(78)90115-4 · Zbl 0381.58004
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