A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: Application to a two-well oscillator

An experimental study of local and global bifurcations in a driven two-well magneto-mechanical oscillator is presented. A detailed picture of the local bifurcation structure of the system is obtained using an automated bifurcation data acquisition system. Basins of attractions for the system are obtained using a new experimental technique: an ensemble of initial conditions is generated by switching between stochastic and deterministic excitation. Using this stochastic interrogation method, we observe the evolution of basins of attraction in the nonlinear oscillator as the forcing amplitude is increased, and find evidence for homoclinic bifurcation before the onset of chaos. Since the entire transient is collected for each initial condition, the same data can be used to obtain pictures of the flow of points in phase space. Using Liouville's Theorem, we obtain damping estimates by calculating the contraction of volumes under the action of the Poincaré map, and show that they are in good agreement with the results of more conventional damping estimation methods. Finally, the stochastic interrogation data is used to estimate transition probability matrices for finite partitions of the Poincaré section. Using these matrices, the evolution of probability densities can be studied.

[1]  C. Hsu,et al.  Cell-To-Cell Mapping A Method of Global Analysis for Nonlinear Systems , 1987 .

[2]  R. Mañé,et al.  On the dimension of the compact invariant sets of certain non-linear maps , 1981 .

[3]  Jose Antonio Coarasa Perez,et al.  Direct observation of crises of the chaotic attractor in a nonlinear oscillator , 1983 .

[4]  E. Dowell,et al.  An examination of initial condition maps for the sinusoidally excited buckled beam modeled by the Duffing's equation , 1987 .

[5]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[6]  Leonard Meirovitch,et al.  Elements Of Vibration Analysis , 1986 .

[7]  Leon O. Chua,et al.  Bifurcation diagrams and fractal brain boundaries of phase-locked loop circuits , 1990 .

[8]  Fractal basin boundary of a two-dimensional cubic map , 1985 .

[9]  J. Yorke,et al.  Fractal Basin Boundaries, Long-Lived Chaotic Transients, And Unstable-Unstable Pair Bifurcation , 1983 .

[10]  Celso Grebogi,et al.  Basin boundary metamorphoses: changes in accessible boundary orbits , 1987 .

[11]  J. M. T. Thompson,et al.  Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design , 1991 .

[12]  Philip Holmes,et al.  Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum , 1987 .

[13]  Grebogi,et al.  Metamorphoses of basin boundaries in nonlinear dynamical systems. , 1986, Physical review letters.

[14]  Jose Antonio Coarasa Perez,et al.  Evidence for universal chaotic behavior of a driven nonlinear oscillator , 1982 .

[15]  Global sensitivity analysis of a nonlinear system using animated basins of attraction , 1992 .

[16]  Li,et al.  Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential. , 1985, Physical review letters.

[17]  M. Mackey,et al.  Probabilistic properties of deterministic systems , 1985, Acta Applicandae Mathematicae.

[18]  P. Grassberger,et al.  Characterization of Strange Attractors , 1983 .

[19]  Van Buskirk R,et al.  Observation of chaotic dynamics of coupled nonlinear oscillators. , 1985, Physical review. A, General physics.

[20]  J. Yorke,et al.  Final state sensitivity: An obstruction to predictability , 1983 .

[21]  F. Moon.  Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors , 1980 .

[22]  Philip Holmes,et al.  A magnetoelastic strange attractor , 1979 .

[23]  Robert Shaw,et al.  The Dripping Faucet As A Model Chaotic System , 1984 .

[24]  J. Yorke,et al.  Fractal basin boundaries , 1985 .

[25]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[26]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .