Criteria for chaos of a three-well potential oscillator with homoclinic and heteroclinic orbits

Criteria for chaotic vibrations of a non-linear elastic beam having three stable and two unstable equilibrium positions are compared. The responses of the system exhibit chaotic behavior in certain regions in the driving frequency and amplitude plane. With the use of the Melnikov technique, necessary conditions for chaos are derived based on homoclinic and heteroclinic bifurcations of the system. Numerical simulations of the stable and unstable manifolds, Poincaré maps, Lyapunov exponents and fractal basin boundaries are performed. The results indicate that both homoclinic and heteroclinic bifurcations are necessary conditions for chaos; thus, at least two necessary conditions must be satisfied for any chaotic responses. However, the final steady state of motion for many initial conditions becomes unpredictable (in the sense of fractal basin boundaries) if at least one homoclinic or heteroclinic bifurcation occurs. Finally, an experimental criterion for chaotic motion, and an heuristic criterion based on classical methods in non-linear dynamics are also obtained and compared with the bifurcation criteria, as a function of forcing amplitude and frequency.