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OGY method for a class of discontinuous dynamical systems

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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.

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Notes

  1. Meanwhile, several modified variants for higher-order systems were developed.

  2. For autonomous systems, like those modeled by (1), T is considered to be the smallest time τ, for which a trajectory starting at a point x on the Poincaré surface, pierces again the surface in a small neighborhood of x [33].

  3. The Poincaré map is defined on the entire section Σ only if the underlying system is nonautonomous and presents a periodically forced vector field (see [40]).

  4. The reason of the appearance of the Floquet multiplier 1 in the eigenvalues spectrum relies on geometric reasons. The reminder multipliers are identical to the eigenvalues of the linearization of the Poincaré map. [46, Theorem 1.6, p. 30].

  5. Higher dimensional cases, such as R 3, are less common for discontinuous systems. However, they can be treated similarly.

References

  1. Wiercigroch, M., de Kraker, B.: Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  2. Popp, K., Stelter, P.: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. Lond. A 332, 89–105 (1990)

    Article  MATH  Google Scholar 

  3. Chillingworth, D.R.J.: Discontinuity geometry for an impact oscillator. Dyn. Syst. 17, 389–420 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cho, Y.-H., Pisano, A.P., Howe, R.T.: Viscous damping model for laterally oscillating microstructures. J. Microelectromech. Syst. 3, 81–87 (1994)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  8. Sprott, J.C., Linz, S.J.: Algebraically simple chaotic flows. Int. J. Chaos Theory Appl. 5, 1–20 (2000)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  11. Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. Springer, New York (2004)

    MATH  Google Scholar 

  12. Clarke, F.H.: Optimization and Non-smooth Analysis. Wiley-Interscience, New York (1983)

    Google Scholar 

  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)

    Article  Google Scholar 

  14. Dasgupta, P., Maskin, E.: The existence of equilibrium in discontinuous economic games, I: theory. Rev. Econ. Stud. 53, 1–26 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Edwards, C., Spurgeon, S.K.: On the development of discontinuous observers. Int. J. Control 59, 1211–1229 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. De Santos, P.G., Jimenez, M.A.: Generation of discontinuous gaits for quadruped walking vehicles. J. Robot. Syst. 12, 599–611 (1995)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  18. Buhite, J.L., Owen, D.R.: An ordinary differential equation from the theory of plasticity. Arch. Ration. Mech. Anal. 71, 357–383 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Utkin, V.I.: Sliding Regimes in Optimization and Control Problems. Nauka, Moscow (1981). (in Russian)

    Google Scholar 

  22. Ott, E., Grebogi, C., Yorke, A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Güémez, J., Matías, M.A.: Control of chaos in unidimensional maps. Phys. Lett. A 181, 29–32 (1993)

    Article  MathSciNet  Google Scholar 

  24. Matías, M.A., Güémez, J.: Stabilization of chaos by proportional pulses in system variables. Phys. Rev. Lett. 72, 1455–1458

  25. Danca, M.-F.: Controlling chaos in discontinuous dynamical systems. Chaos Solitons Fractals 22, 605–612 (2004)

    Article  MATH  Google Scholar 

  26. Aubin, J.-P., Cellina, A.: Set-Valued Maps and Viability Theory. Springer, Berlin (1984)

    MATH  Google Scholar 

  27. Aubin J.-P., Frankowska, H.: Set-Valued Analysis. Systems and Control: Foundations and Applications, vol. 2. Birkhäuser, Boston (1984)

    Google Scholar 

  28. Danca, M.-F.: Synchronization of switch dynamical systems. Int. J. Bifurc. Chaos 12, 1813–1826 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Filippov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic, Dordrecht (1988)

    MATH  Google Scholar 

  30. Danca, M.-F.: On a class of discontinuous dynamical systems. Math. Notes 2, 103–116 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Dontchev A., Lempio, F.: Difference methods for differential inclusions: a survey. SIAM Rev. 34, 263–294 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kastner-Maresch, A.: Implicit Runge–Kutta methods for differential inclusions. Numer. Funct. Anal. Optim. 10–11, 937–958 (1991)

    MathSciNet  Google Scholar 

  33. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  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

  35. Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)

    Article  MATH  Google Scholar 

  36. Diecia, L., Lopez, L.: Fundamental matrix solutions of piecewise smooth differential systems. Math. Comput. Simul. 81, 932–953 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Nusse, H.E., Yorke, J.A.: A procedure for finding numerical trajectories on chaotic saddles. Physica D 36, 137–156 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    MATH  Google Scholar 

  42. Hénon, M.: On the numerical computation of Poincaré maps. Physica D 5, 412–414 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tucker, W.: Computing accurate Poincaré maps. Physica D 171, 127–137 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Palaniyandi, P.: On computing Poincaré map by Hénon method. Chaos Solitons Fractals 39, 1877–1882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Palis J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  46. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995)

    MATH  Google Scholar 

  47. Lai, Y.-C., Grebogi, C., Yorke, J.A., Kan, I.: How often are chaotic saddles nonhyperbolic? Nonlinearity 6, 779–797 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hsu, G.-H., Ott, E., Grebogi, C.: Strange saddles and the dimensions of their invariant manifolds. Phys. Lett. A 127, 199–204 (1988)

    Article  MathSciNet  Google Scholar 

  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. Hobson, D.: An efficient method for computing invariant manifolds. J. Comput. Phys. 104, 14–22 (1991)

    Article  MathSciNet  Google Scholar 

  51. You, Z., Kostelich, E.J., Yorke, J.A.: Calculating stable and unstable manifolds. Int. J. Bifurc. Chaos 1, 605–623 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  52. Danca, M.-F., Codreanu, S.: On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals 13, 681–691 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  53. Andrievskii, B.R., Fradkov, A.L.: Control of chaos: methods and applications. I methods. Autom. Remote Control 64, 673–713 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, Avram Iancu University, 400380, Cluj-Napoca, Romania

    Marius-F. Danca

  2. Romanian Institute of Science and Technology, 400487, Cluj-Napoca, Romania

    Marius-F. Danca

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Danca, MF. OGY method for a class of discontinuous dynamical systems. Nonlinear Dyn 70, 1523–1534 (2012). https://doi.org/10.1007/s11071-012-0552-6

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