Kybernetika 48 no. 6, 1089-1099, 2012

Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems

Didier Henrion

Abstract:

Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of one-dimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software.

Keywords:

invariant measures, semidefinite programming, dynamical systems

Classification:

37-04, 37L40, 90C22

paper.pdf

References:

  1. K. T. Alligood, T. D. Sauer and J. A. Yorke: Chaos - An Introduction to Dynamical Systems. Springer, 1997.   CrossRef
  2. D. Barkley, I. G. Kevrekidis and A. M. Stuart: The moment map: nonlinear dynamics of density evolution via a few moments. SIAM J. Appl. Dynam. Systems 5 (2006), 3, 403-434.   CrossRef
  3. S. Boyd and L. Vandenberghe: Convex Optimization. Cambridge Univ. Press, 2005.   CrossRef
  4. D. Campbell, J. Crutchfield, D. Farmer and E. Jen: Experimental mathematics: the role of computation in nonlinear science. Comm. ACM 28 (1985), 4, 374-384.   CrossRef
  5. M. Dellnitz and O. Junge: On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (1999), 2, 491-515.   CrossRef
  6. P. Diaconis and D. Freedman: Iterated random functions. SIAM Rev. 41 (1999), 1, 45-76.   CrossRef
  7. P. Góra and A. Boyarsky: Why computers like Lebesgue measure. Comput. Math. Appl. 16 (1988), 4, 321-329.   CrossRef
  8. P. Góra, A. Boyarsky, M. D. S. Islam and W. Bahsoun: Absolutely continuous invariant measures that cannot be observed experimentally. SIAM J. Appl. Dynam. Systems 5 (2006), 1, 84-90.   CrossRef
  9. O. Hernández-Lerma and J. B. Lasserre: Markov Chains and Invariant Probabilities. Birkhauser, 2003.   CrossRef
  10. A. Lasota and M. C. Mackey: Probabilistic Properties of Deterministic Systems. Cambridge Univ. Press, 1985.   CrossRef
  11. J. B. Lasserre: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (2001), 3, 796-817.   CrossRef
  12. J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat: Nonlinear optimal control via occupation measures and LMI relaxations. SIAM J. Control Optim. 47 (2008), 4, 1643-1666.   CrossRef
  13. H. Peyrl and P. A. Parrilo: A theorem of the alternative for SOS Lyapunov functions. In: Proc. IEEE Conference on Decision and Control, 2007.   CrossRef
  14. A. Rantzer: A dual to Lyapunov's stability theorem. Systems Control Lett. 42 (2001), 3, 161-168.   CrossRef
  15. U. Vaidya and P. G. Mehta: Lyapunov measure for almost everywhere stability. IEEE Trans. Automat. Control 53 (2008), 1, 307-323.   CrossRef