Chaos

My work in chaos started at Cornell with my doctoral study on an oscillator with dry friction.  At the root of this oscillator's dynamics was a simple iterated map much like the famous logistic (quadratic) map, a paradigm of chaos analyzed by the chaos pioneers who discovered period doubling and other universal behavior.

Chaos can be studied from an established model, either by analytical treatment or simulation of the model.  Steve Shaw and I teach a course (ME 961) with this perspective, involving local bifurcation theory, global bifurcations due to homoclinic tangles, symbol dynamics and the horseshoe map, and perhaps routes to chaos (period doubling, intermittency, torus wrinkling, torus doubling, and quasiperiodicity), or "codimension-2" bifurcations.  A related course is ME 863 (nonlinear oscillations), which focuses less on chaos and more on nonlinear resonances and self-excitation.  Chaos can also be studied by analyzing data from a chaotic process.  When I get a chance, I do a course (ME 960) on the treatment of chaotic data.  The idea is to take measurements of a chaotic process and recreate the geometry of what's actually going on.  Ultimately, the process can be characterized by its topology, fractal dimension, Lyapunov exponents, and entropy.  Methods of "chaos theory" can be used to distinguish deterministic from random processes, uncover a model of the process, and lead to nonlinear prediction.

Chaos applications include, for example, the mixing on non-turbulent fluids, coding and communication, system identification, and control  Chaos is advantageous for system identification because chaotic data has more information than periodic data.  Such dynamical information can be exploited to obtain concrete information, such as parameter values, about the system at hand.  Finally, the reality is that nonlinear phenomena are present in a vast number of interesting systems.  Understanding chaos is essential for understanding the behavior of many of these nonlinear systems.  Chaos theory allows us to know the way things work.

The papers below deal with descriptions of chaotic behavior and nonlinear phenomena, developments of chaos tools, and modeling.

Support:  NASA Langley Research Center, 4/01-11/02 (Parametric System Identification), National Science Foundation CMS-9624347 (CAREER, primarily on Friction Dynamics), 8/96-7/00. (Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.)

Publications involving chaos

• J. L. Quinby and B. F. Feeny, 2009, "Low-frequency behavior in a frictionally excited beam," Journal of Sound and Vibration 325, 884-893. (preprint, with Movie 1, Movie 2, and Movie 3.)
• G. Lin, B. F. Feeny, and T. Das, 2009, "Fractional Derivative Reconstruction of Forced Oscillators," Nonlinear Dynamics 55 (3) 239-250. (preprint)
• Y. Liang and B. F. Feeny, 2008,  “Parametric identification of a chaotic base-excited double pendulum experiment,”  Nonlinear Dynamics 52, 181-197.  Conference version.
• B. F. Feeny and F. C. Moon, 2007, "Empirical friction modeling in forced oscillators using chaos," Nonlinear Dynamics 47 (1-3) 129-141.  Preprint.
• Y. Liang and B. F. Feeny, 2006, "Parametric identification of a base-excited single pendulum," Nonlinear Dynamics 46 (1-2) 17-29. Preprint.
• J. L. Quinby and B. F. Feeny, 2004, "Low frequency phenomena in a frictionally excited beam", proceedings of the ASME IDETC, Anaheim, September, on CD-ROM. (Preprint)
• B. F. Feeny and G. Lin, 2004, "Fractional derivatives applied to phase-space reconstructions," Nonlinear Dynamics 38 (1-4) 85-99. (Preprint)
• B. F. Feeny, G. Lin, and T. Tas, 2003, "Reconstructing the Phase Space with Fractional Derivatives,” proceedings of the ASME IDETC’03, September 5-9, Chicago, on CD-ROM.
• G. Kerschen, B.F. Feeny and J.C. Golinval, 2003, "On the exploitation of chaos to build reduced-order models," Computer Methods in Applied Mechanics and Engineering 192, 1785-1795.  (preprint)
• B. F. Feeny, C.-M. Yuan, and J. Cusumano, 2001, "Parametric Identification of a Magneto-Elastic Oscillator," Journal of Sound and Vibration 247(5) 785-806. (preprint)
• B. F. Feeny, 2000, "Fast multifractal analysis by recursive box covering," International Journal of Bifurcation and Chaos 10 (9) 2277-2287. (preprint, figures)
• B. F. Feeny and F. C. Moon, 2000, "Quenching stick-slip chaos with dither," Journal of Sound and Vibration 237 (1) 173-180.  (preprint, figures).
• R. Kappagantu and B. F. Feeny, 2000, "Part 1: Dynamical characterization of a frictionally excited beam," Nonlinear Dynamics 22 (4) 317-333. (Preprint).
• R. Kappagantu and B. F. Feeny, 2000, "Part 2: Proper orthogonal modeling of a frictionally excited beam," Nonlinear Dynamics 23 (1) 1-11. (Preprint).
• B. F. Feeny and J.-W. Liang, 2000, "Stick-slip and the phase-space reconstruction," in Applied Nonlinear Dynamcis and Chaos of Mechanical Systems with Discontinuities, M. Wiercigroch and B. de Kraker (eds.), pp. 261-291, World Scientific, Singapore. (preprint)
• R. Kappagantu and B. F. Feeny, 1999, "An optimal modal reduction of a system with frictional excitation," Journal of Sound and Vibration 224 (5) 863-877.
• C.-M. Yuan and B. F. Feeny, 1998, "Parametric identification of chaotic systems," Journal of Vibration and Control 4 (4) 405-426. (preprint)
• B. F. Feeny and J. W. Liang, 1997, "Phase-space reconstructions and stick-slip," Nonlinear Dynamics 13 (1), 39-57. (preprint)
• S. W. Shaw and B. F. Feeny, 1997, "Nonlinear dynamics with impact and friction: new analysis methods," Dynamics and Chaos in Manufacturing Processes, F. C. Moon, ed., Wiley, 241-264.
• B. F. Feeny, 1996, "The nonlinear dynamics of oscillators with stick-slip friction," in Dynamics with Friction, A. Guran, F. Pfeiffer and K. Popp (eds.), pp. 36-92, World Scientific, River Edge. (preprint)
• B. F. Feeny, F. C. Moon, P. Y. Chen, and S. Mukherjee, 1995, "Chaotic mixing in rigid, perfectly plastic material," International Journal of Bifurcation and Chaos 5 (1) 133-144. (preprint)
• B. F. Feeny and F. C. Moon, 1994, "Chaos in a forced dry-friction oscillator: experiment and numerical modeling," Journal of Sound and Vibration 170, 303-323. (preprint)
• B. F. Feeny and F. C. Moon, 1993, "Bifurcation sequences of a Coulomb friction oscillator," Nonlinear Dynamics 4, 25-37. (preprint)
• B. F. Feeny, 1992, "A nonsmooth Coulomb friction oscillator," Physica D 59, 25-38. (preprint)
• B. F. Feeny and F. C. Moon, 1989, "Autocorrelation on symbol dynamics for a chaotic dry-friction oscillator," Physics Letters A 141 (8,9) 397-400. (preprint)
• J. L. Quinby, 2003, Nonlinear Dynamics of a Frictionally Excited Beam, MS Thesis, Michigan State University, East Lansing.
• G. Lin, 2001, Alternative Methods of Phase Space Reconstruction, MS Thesis, Michigan State University, East Lansing.
• Z. Al-Zamel, 1999, Unstable Periodic Orbit Extraction Error and its Effect on Nonlinear System Parametric Identification, PhD thesis, Michigan State University, East Lansing.
• C. M. Yuan, 1995, A Method of Parametric Identification of Chaotic Systems, PhD thesis, Michigan State University, East Lansing.
• R. Kappagantu, 1997, An Optimal Modal Reduction for Frictionally Excited Systems, PhD thesis, Michigan State University.