Nonlinear Dynamics

, Volume 56, Issue 1–2, pp 145–157 | Cite as

Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation

  • G. Diaz
  • C. F. M. Coimbra
Original Paper


We investigate the dynamics and control of a nonlinear oscillator that is described mathematically by a Variable Order Differential Equation (VODE). The dynamic problem in question arises from the dynamical analysis of a variable viscoelasticity oscillator. The dynamics of the model and the behavior of the variable order differintegrals are shown in variable phase space for different parameters. Two different controllers are developed for the VODEs under study in order to track an arbitrary reference function. A generalization of the van der Pol equation using the VODE formulation is analyzed under the light of the methods introduced in this work.


Fractional derivatives and integrals Variable order differential equations Control of nonlinear oscillators van der Pol equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) MATHGoogle Scholar
  2. 2.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  3. 3.
    Podlubni, I.: Fractional Differential Equations. Academic Press, San Diego (1999) Google Scholar
  4. 4.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000) MATHGoogle Scholar
  5. 5.
    Hu, Y.: Integral Transformations and Anticipative Calculus for Fractional Brownian Motions, Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2005) Google Scholar
  6. 6.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Amsterdam (2006) Google Scholar
  7. 7.
    Coimbra, C.F.M., L’Esperance, D., Lambert, A., Trolinger, J.D., Rangel, R.H.: An experimental study on the history effects in high-frequency Stokes flows. J. Fluid Mech. 504, 353–363 (2004) MATHCrossRefGoogle Scholar
  8. 8.
    L’Esperance, D., Coimbra, C.F.M., Trolinger, J.D., Rangel, R.H.: Experimental verification of fractional history effects on the viscous dynamics of small spherical particles. Exp. Fluids 38, 112–116 (2005) CrossRefGoogle Scholar
  9. 9.
    Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91(8), 134–147 (1971) CrossRefGoogle Scholar
  10. 10.
    Coimbra, C.F.M., Rangel, R.H.: General solution of the particle equation of motion in unsteady Stokes flows. J. Fluid Mech. 370, 53–72 (1998) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Coimbra, C.F.M., Kobayashi, M.H.: On the viscous motion of a small particle in a rotating cylinder. J. Fluid Mech. 469, 257–286 (2002) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Trans. Autom. Control 37(9), 1465–1470 (1992) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Podlubni, I.: Fractional-order systems and PI λ D μ-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999) CrossRefGoogle Scholar
  14. 14.
    Petras, I.: Control of fractional-order Chua’s system. J. Electr. Eng. 53(7–8), 219–222 (2002) Google Scholar
  15. 15.
    Hwang, C., Leu, J.-F., Tsay, S.-Y.: A note on time-domain simulation of feedback fractional-order systems. IEEE Trans. Autom. Control 47(4), 625–631 (2002) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hartley, T.T., Lorenzo, C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29, 201–233 (2002) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ahmad, W.M., El-Khazali, R., Al-Assaf, Y.: Stabilization of generalized fractional order chaotic systems using state feedback control. Chaos Solitons Fractals 22, 141–150 (2004) MATHCrossRefGoogle Scholar
  18. 18.
    Cao, J.-Y., Liang, J., Cao, B.-G.: Optimization of fractional order PID controllers based on genetic algorithms. In: Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, Guangzhou, pp. 18–21 (2005) Google Scholar
  19. 19.
    Ladaci, S., Charef, A.: On fractional adaptive control. Nonlinear Dyn. 43, 365–378 (2006) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Barbosa, R.S., Machado, J.A.T., Ferreira, I.M., Tar, J.K.: Dynamics of the fractional-order Van der Pol oscillator. In: Proceedings of the IEEE International Conference on Computational Cybernetics (ICCC04), Vienna, Austria, CD-ROM Google Scholar
  21. 21.
    Barbosa, R.S., Machado, J.A.T., Vinagre, B.M., Calderon, A.J.: Analysis of the Van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13(9–10), 1291–1301 (2007) MATHCrossRefGoogle Scholar
  22. 22.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Momani, S.: A numerical scheme for the solution of multi-order fractional differential equations. Appl. Math. Comput. 182, 761–770 (2006) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Samko, S.K., Ross, B.: Integration and differentiation to a variable fractional order. Integral Transforms Special Funct. 1(4), 277–300 (1993) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Ross, B., Samko, S.K.: Fractional integration operator of variable order in the Holder space H λ(x). Int. J. Math. Math. Sci. 18(4), 777–788 (1995) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Ingman, D., Suzdalnitsky, J., Zeifman, M.: Constitutive dynamic-order model for nonlinear contact phenomena. J. Appl. Mech. 67, 383–390 (2000) MATHCrossRefGoogle Scholar
  27. 27.
    Ingman, D., Suzdalnitsky, J.: Control of damping oscillations by fractional differential operator with time-dependent order. Comput. Methods Appl. Mech. Eng. 193, 5585–5595 (2004) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ingman, D., Suzdalnitsky, J.: Application of differential operator with servo-order function in model of viscoelastic deformation process. J. Eng. Mech. 131, 763–767 (2005) CrossRefGoogle Scholar
  30. 30.
    Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12(11–12), 692–703 (2003) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Soon, C.M., Coimbra, C.F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Ann. Phys. 14(6), 378–389 (2005) MATHCrossRefGoogle Scholar
  32. 32.
    Ramirez, L.E.S., Coimbra, C.F.M.: A variable order constitutive relation for viscoelasticity. Ann. Phys. 16(7–8), 543–552 (2007) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Atkinson, K.E.: The numerical solution of Fredholm integral equations of the second kind. SIAM J. 4(3), 337–348 (1967) MATHGoogle Scholar
  34. 34.
    Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.: Algorithms for the fractional calculus: A selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194(6–8), 743–773 (2005) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Balachandran, K., Park, J.Y., Anandhi, E.R.: Local controllability of quasilinear integrodifferential evolution systems in Banach spaces. J. Math. Anal. Appl. 258, 309–319 (2001) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Calvet, J.-P., Arkun, Y.: Stabilization of feedback linearized nonlinear processes under bounded perturbations. In: Proceedings of the American Control Conference, pp. 747–752 (1989) Google Scholar
  38. 38.
    Calvet, J.-P., Arkun, Y.: Design of P and PI stabilizing controllers for quasi-linear systems. Comput. Chem. Eng. 14(4–5), 415–426 (1990) CrossRefGoogle Scholar
  39. 39.
    Sun, Z., Tsao, T.-C.: Control of linear systems with nonlinear disturbance dynamics. In: Proceedings of the American Control Conference, pp. 3049–3054 (2001) Google Scholar
  40. 40.
    Williams II, R.L., Lawrence, D.A.: Linear State-Space Control Systems. Wiley, New Jersey (2007) CrossRefGoogle Scholar
  41. 41.
    van der Pol, B.: On relaxation-oscillations. Philos. Mag. 7(2), 978–992 (1926) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of EngineeringUniversity of California MercedMercedUSA

Personalised recommendations