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

@article{Chen1998SnapbackRA,
  title={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},
  author={Goong Chen and Sze-Bi Hsu and Jianxin Zhou},
  journal={Journal of Mathematical Physics},
  year={1998},
  volume={39},
  pages={6459-6489}
}
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 wish to study chaotic wave propagation in the system. A cause of chaos by snapback repellers has been identified. These snapback repellers are repelling fixed points… 
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References

SHOWING 1-10 OF 15 REFERENCES
Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis
The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos
CHAOTIC VIBRATIONS OF THE ONE-DIMENSIONAL WAVE EQUATION DUE TO A SELF-EXCITATION BOUNDARY CONDITION. III. NATURAL HYSTERESIS MEMORY EFFECTS
The nonlinear reflection curve due to a van der Pol type boundary condition at the right end becomes a multivalued relation when one of the parameters (α) exceeds the characteristic impedance value
Chaotic Vibrations of the One-Dimensional Wave Equation Due to a Self-Excitation Boundary Condition.
Consider the initial-boundary value problem of the linear wave equation wtt-wxx=0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system,
LINEAR SUPERPOSITION OF CHAOTIC AND ORDERLY VIBRATIONS ON TWO SERIALLY CONNECTED STRINGS WITH A VAN DER POL JOINT
Two identical vibrating strings are serially coupled end-to-end with nonlinear joints that behave like a Van der Pol oscillator. This coupled PDE system has an infinite dimensional center manifold of
Exponential stability analysis of Xa long chain of coupled vibrating strings with dissipative linkage
Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers’
Introduction to Applied Nonlinear Dynamical Systems and Chaos
Equilibrium Solutions, Stability, and Linearized Stability * Liapunov Functions * Invariant Manifolds: Linear and Nonlinear Systems * Periodic Orbits * Vector Fields Possessing an Integral * Index
Introduction to Chaotic Dynamical Systems
TLDR
This thesis develops some of the current definitions of chaos and discusses several quantitative measures of chaos, and serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.
Chaotic Dynamics in Two-Dimensional Noninvertible Maps
Part 1: Generalities and definition one-dimensional endomorphisms the simplest type of two-dimensional endomorphisms. Part 2: Properties of the phase plane simplest type of two-dimensional
Vibration and damping in distributed systems
The WKB Method for Eigenvalue Problems I: Space Dimension One. The WKB Method for Eigenvalue Problems II: Multidimensional Spaces. Miscellaneous Methods. Visualization. Remarks on Experimental
...
...