Controlling homoclinic orbits

  • A. M. Bloch
  • J. E. Marsden
Article

Abstract

In this paper we analyze various control-theoretic aspects of a nonlinear control system possessing homoclinic or heteroclinic orbits. In particular, we show that for a certain class of nonlinear control system possessing homoclinic orbits, a control can be found such that the system exhibits arbitrarily long periods in a neighborhood of the homoclinic. We then apply these ideas to bursting phenomena in the near wall region of a turbulent boundary layer. Our analysis is based on a recently developed finite-dimensional model of this region due to Aubry, Holmes, Lumley, and Stone.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Armbruster, J. Guckenheimer, and P. Holmes (1988). Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. Phys. D, 29, 257–282.Google Scholar
  2. V. Arnold (1978). Mathematical Methods in Classical Mechanics. Springer-Verlag, New York.Google Scholar
  3. N. Aubry, P. Holmes, J.L. Lumley, and E. Stone (1988). The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech., 192, 115–173.Google Scholar
  4. N. Aubry, J.L. Lumley, and P. Holmes (1989). The effect of drag reduction on the wall region. Theoret. Comput. Fluid Dynamics, submitted.Google Scholar
  5. S. Barnett (1975). Introduction to Mathematical Control Theory. Clarendon Press, Oxford.Google Scholar
  6. R.W. Brockett (1970). Finite-Dimensional Linear Systems. Wiley, New York.Google Scholar
  7. R.W. Brockett (1972). Systems theory on group manifolds and coset spaces. SIAM J. Control Optim., 10, 265–284.Google Scholar
  8. R. Hermann and A.J. Krener (1977). Nonlinear controllability and observability. IEEE Trans. Automat. Control, 22, 725–740.Google Scholar
  9. C. Jones and M. Proctor (1987). Strong spatial resonance and travelling waves in Bernard convection. Phys. Lett. A, 121, 224–227.Google Scholar
  10. V. Jurdjevic and J. Quinn (1978). Controllability and stability. J. Differential Equations, 28, 381–389.Google Scholar
  11. I. Kubo and J.L. Lumley (1980). A study to assess the potential for using long chain polymers dissolved in water to study turbulence. An. Rep. NASA-AMES Grant No. NSG-2382, Cornell University.Google Scholar
  12. H. Kwakernaak and R. Sivan (1972). Linear Optimal Control Systems. Wiley-Interscience, New York.Google Scholar
  13. J.L. Lumley (1967). The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A.M. Yaglom and V.I. Tatarskii), pp. 166–178. Nauka, Moscow.Google Scholar
  14. J.L. Lumley (1970). Stochastic Tools in Turbulence. Academic Press, New York.Google Scholar
  15. J.L. Lumley (1981). Coherent structures in turbulence. In Transition and Turbulence (ed. R.E. Meyer), pp. 215–242. Academic Press, New York.Google Scholar
  16. J.L. Lumley and I. Kubo (1984). Turbulent drag reduction by polymer additives; a survey. In The Influence of Polymer Additives on Velocity and Temperature Fields. IUTAM Symposium, Essen, 1984 (ed. B. Gampert), pp. 3–21, Springer-Verlag, Berlin.Google Scholar
  17. J.L. Lumley, J.M. Guckenheimer, J.E. Marsden, P.J. Holmes, and S. Leibovich (1988). Structure and control of the wall region in a turbulent boundary layer. Proposal No. 88-NA-321, August, 1988.Google Scholar
  18. B. Nicolaenko (1986). Lecture delivered at workshop on Computational Aspects of Dynamical Systems, Mathematical Sciences Institute, Cornell University, September 8–10, 1986.Google Scholar
  19. L.P. Silnikov (1967). The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus. Soviet Math. Dokl., 8, 54–58.Google Scholar
  20. J.J. Stoker (1950). Nonlinear Vibrations. Wiley, New York.Google Scholar
  21. E. Stone and P. Holmes (1988). Random perturbations of heteroclinic attractors. Preprint.Google Scholar
  22. H.J. Sussman and V. Jurdjevic (1972). Controllability of nonlinear systems. J. Differential Equations, 12, 95–116.Google Scholar
  23. A.J. van der Schaft (1986). Stabilization of Hamiltonian systems. Nonlinear Anal. Theory Methods Applic., 10, 1021–1035.Google Scholar
  24. S. Wiggins (1988). Global Bifurcations and Chaos. Springer-Verlag, New York.Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • A. M. Bloch
    • 1
    • 2
  • J. E. Marsden
    • 3
    • 4
  1. 1.MSI, Cornell UniversityIthacaUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  4. 4.Cornell UniversityIthacaUSA

Personalised recommendations