Advertisement

Nonlinear Dynamics

, Volume 70, Issue 2, pp 1523–1534 | Cite as

OGY method for a class of discontinuous dynamical systems

  • Marius-F. Danca
Original Paper
  • 180 Downloads

Abstract

In this paper, we prove that the OGY method to control unstable periodic orbits (UPOs) of continuous-time systems can be applied to a class of systems discontinuous with respect the state variable, by using a generalized derivative. Because the discontinuous problem may have not classical solutions, the initial value problem is transformed into a set-valued problem via Filippov regularization. The existence of the ingredients necessary to apply OGY method (UPO, Poincaré map and stable and unstable directions) is proved and the numerically implementation is explained. Another possible way analyzed in this paper is the continuous approximation of the underlying initial value problem, via Cellina’s theorem for differential inclusions. Thus, the problem is approximated by a continuous initial value problem, and the OGY method can be applied as usual.

Keywords

Generalized derivative OGY method Poincaré map Filippov regularization Cellina’s theorem 

References

  1. 1.
    Wiercigroch, M., de Kraker, B.: Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities. World Scientific, Singapore (2000) MATHGoogle Scholar
  2. 2.
    Popp, K., Stelter, P.: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. Lond. A 332, 89–105 (1990) MATHCrossRefGoogle Scholar
  3. 3.
    Chillingworth, D.R.J.: Discontinuity geometry for an impact oscillator. Dyn. Syst. 17, 389–420 (2002) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cho, Y.-H., Pisano, A.P., Howe, R.T.: Viscous damping model for laterally oscillating microstructures. J. Microelectromech. Syst. 3, 81–87 (1994) CrossRefGoogle Scholar
  5. 5.
    Runesson, K., Ottosen, N.S., Dunja, P.: Discontinuous bifurcations of elastic-plastic solutions at plane stress and plane strain. Int. J. Plast. 7, 99–121 (1991) MATHCrossRefGoogle Scholar
  6. 6.
    Govardham, R., Williamson, C.H.K.: Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85–130 (2000) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Salerno, M., Samuelsen, M.R., Filatrella, G., Pagano, S., Parmentier, R.D.: Microwave phase locking of Josephson-junction fluxon oscillators. Phys. Rev. B 41, 6641–6654 (1990) CrossRefGoogle Scholar
  8. 8.
    Sprott, J.C., Linz, S.J.: Algebraically simple chaotic flows. Int. J. Chaos Theory Appl. 5, 1–20 (2000) Google Scholar
  9. 9.
    Heemels, W.P.M.H., Çamlýbel, M.K., Schumacher, J.M.: On the dynamic analysis of piecewise-linear networks. IEEE Trans. Circuits Syst. I, Regul. Pap. 49, 315–327 (2002) CrossRefGoogle Scholar
  10. 10.
    Sun, J., Mitchell, D.M., Greuel, M.F., Krein, P.T., Bass, R.M.: Averaged modeling of PWM converters operating in discontinuous conduction mode. IEEE Trans. Power Electron. 16, 482–492 (2001) CrossRefGoogle Scholar
  11. 11.
    Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, New York (2004) MATHGoogle Scholar
  12. 12.
    Clarke, F.H.: Optimization and Non-smooth Analysis. Wiley-Interscience, New York (1983) Google Scholar
  13. 13.
    Barton, P.I., Allgor, R.J., Feehery, W.F., Galán, S.: Dynamic optimization in a discontinuous. Ind. Eng. Chem. Res. 37, 966–981 (1998) CrossRefGoogle Scholar
  14. 14.
    Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, I: theory. Rev. Econ. Stud. 53, 1–26 (1986) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Edwards, C., Spurgeon, S.K.: On the development of discontinuous observers. Int. J. Control 59, 1211–1229 (1994) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    De Santos, P.G., Jimenez, M.A.: Generation of discontinuous gaits for quadruped walking vehicles. J. Robot. Syst. 12, 599–611 (1995) MATHCrossRefGoogle Scholar
  17. 17.
    Mitchell, M.A., McRury, I.D., Everett, T.H., Li, H., Mangrum, J.M., Haines, D.E.: Morphological and physiological characteristics of discontinuous linear atrial ablations during atrial pacing and atrial fibrillation. J. Cardiovasc. Electrophysiol. 10, 378–386 (1999) CrossRefGoogle Scholar
  18. 18.
    Buhite, J.L., Owen, D.R.: An ordinary differential equation from the theory of plasticity. Arch. Ration. Mech. Anal. 71, 357–383 (1979) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Deimling, K.: Multivalued differential equations, friction problems. In: Fink, A.M., Miller, R.K., Kliemann, W. (eds.) Proc. Conf. Differential & Delay Equations, Ames, Iowa, 1991, pp. 99–106. World Scientific, Singapore (1992). Google Scholar
  20. 20.
    Schilling, K.: An algorithm to solve boundary value problem for differential equations and applications in optimal control. Numer. Funct. Anal. Optim. 10, 733–764 (1989) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Utkin, V.I.: Sliding Regimes in Optimization and Control Problems. Nauka, Moscow (1981). (in Russian) Google Scholar
  22. 22.
    Ott, E., Grebogi, C., Yorke, A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Güémez, J., Matías, M.A.: Control of chaos in unidimensional maps. Phys. Lett. A 181, 29–32 (1993) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Matías, M.A., Güémez, J.: Stabilization of chaos by proportional pulses in system variables. Phys. Rev. Lett. 72, 1455–1458 Google Scholar
  25. 25.
    Danca, M.-F.: Controlling chaos in discontinuous dynamical systems. Chaos Solitons Fractals 22, 605–612 (2004) MATHCrossRefGoogle Scholar
  26. 26.
    Aubin, J.-P., Cellina, A.: Set-Valued Maps and Viability Theory. Springer, Berlin (1984) MATHGoogle Scholar
  27. 27.
    Aubin J.-P., Frankowska, H.: Set-Valued Analysis. Systems and Control: Foundations and Applications, vol. 2. Birkhäuser, Boston (1984) Google Scholar
  28. 28.
    Danca, M.-F.: Synchronization of switch dynamical systems. Int. J. Bifurc. Chaos 12, 1813–1826 (2002) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Filippov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic, Dordrecht (1988) MATHGoogle Scholar
  30. 30.
    Danca, M.-F.: On a class of discontinuous dynamical systems. Math. Notes 2, 103–116 (2001) MathSciNetMATHGoogle Scholar
  31. 31.
    Dontchev A., Lempio, F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34, 263–294 (1992) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Kastner-Maresch, A.: Implicit Runge–Kutta methods for differential inclusions. Numer. Funct. Anal. Optim. 10–11, 937–958 (1991) MathSciNetGoogle Scholar
  33. 33.
    Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989) MATHCrossRefGoogle Scholar
  34. 34.
    Colombo, A., di Bernardo, M., Hogan, S.J., Jeffrey, M.R.: Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems. Physica D. Available online 8 October 2011 Google Scholar
  35. 35.
    Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000) MATHCrossRefGoogle Scholar
  36. 36.
    Diecia, L., Lopez, L.: Fundamental matrix solutions of piecewise smooth differential systems. Math. Comput. Simul. 81, 932–953 (2011) CrossRefGoogle Scholar
  37. 37.
    di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B., Tost, G.O., Piiroinen, P.T.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50, 629–701 (2008) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    di Bernardo, M., Kowalczyk P., Nordmark, A.: Bifurcations of dynamical systems with sliding: derivation of normal form mappings. Physica D 170, 175–205 (2002) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Nusse, H.E., Yorke, J.A.: A procedure for finding numerical trajectories on chaotic saddles. Physica D 36, 137–156 (1989) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    England, J.P., Krauskopf, B., Osinga, H.M.: Computing one-dimensional global manifolds of Poincaré maps by continuation. SIAM J. Appl. Dyn. Syst. 4, 1008–1041 (2005) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) MATHGoogle Scholar
  42. 42.
    Hénon, M.: On the numerical computation of Poincaré maps. Physica D 5, 412–414 (1982) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Tucker, W.: Computing accurate Poincaré maps. Physica D 171, 127–137 (2002) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Palaniyandi, P.: On computing Poincaré map by Hénon method. Chaos Solitons Fractals 39, 1877–1882 (2009) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Palis J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, Berlin (1982) MATHCrossRefGoogle Scholar
  46. 46.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995) MATHGoogle Scholar
  47. 47.
    Lai, Y.-C., Grebogi, C., Yorke, J.A., Kan, I.: How often are chaotic saddles nonhyperbolic? Nonlinearity 6, 779–797 (1993) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Hsu, G.-H., Ott, E., Grebogi, C.: Strange saddles and the dimensions of their invariant manifolds. Phys. Lett. A 127, 199–204 (1988) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Macau, E.E.N., Grebogi, C.: Control of chaos and its relevancy to spacecraft steering. Phys. Lett. A 127, 199–204 (2006) Google Scholar
  50. 50.
    Hobson, D.: An efficient method for computing invariant manifolds. J. Comput. Phys. 104, 14–22 (1991) MathSciNetCrossRefGoogle Scholar
  51. 51.
    You, Z., Kostelich, E.J., Yorke, J.A.: Calculating stable and unstable manifolds. Int. J. Bifurc. Chaos 1, 605–623 (1991) MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Danca, M.-F., Codreanu, S.: On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals 13, 681–691 (2002) MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Andrievskii, B.R., Fradkov, A.L.: Control of chaos: methods and applications. I methods. Autom. Remote Control 64, 673–713 (2003) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Dept. of Mathematics and Computer ScienceAvram Iancu UniversityCluj-NapocaRomania
  2. 2.Romanian Institute of Science and TechnologyCluj-NapocaRomania

Personalised recommendations