Nonlinear Dynamics

, Volume 67, Issue 4, pp 2317–2342 | Cite as

Finding attractors of continuous-time systems by parameter switching

  • Marius-F. Danca
  • Miguel Romera
  • Gerardo Pastor
  • Fausto Montoya


The review presents a parameter switching algorithm and his applications which allows numerical approximation of any attractor of a class of continuous-time dynamical systems depending linearly on a real parameter. The considered classes of systems are modeled by a general initial value problem which embeds dynamical systems continuous and discontinuous with respect to the state variable, of integer, and fractional order. The numerous results, presented in several papers, are systematized here on four representative known examples representing the four classes. The analytical proof of the algorithm convergence for the systems belonging to the continuous class is presented briefly, while for the other categories of systems, the convergence is numerically verified via computational tools. The utilized numerical tools necessary to apply the algorithm are contained in Appendices A, B, C, D and E.


Parameter switching Global attractors Local attractors Fractional systems Discontinuous systems Filippov regularization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Danca, M.-F., Tang, W.K.S., Chen, G.: A switching scheme for synthesizing attractors of dissipative chaotic systems. Appl. Math. Comput. 201, 1–2 (2008) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Danca, M.-F.: Random parameter-switching synthesis of a class of hyperbolic attractors. Chaos 18, 033111 (2008) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Danca, M.-F.: Synthesizing the Lü attractor by parameter-switching. Int. J. Bifurc. Chaos 21(1), 323–331 (2011) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Danca, M.-F.: Finding stable attractors of a class of dissipative dynamical systems by numerical parameter switching. Dyn. Syst. 25(2), 189–201 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Danca, M.-F., Wang, Q.: Synthesizing attractors of Hindmarsh-Rose neuronal systems. Nonlinear Dyn. 62(1), 437–446 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Danca, M.-F., Morel, C.: Attractors synthesis of a class of networks. Dyn. Contin. Discrete Impuls. Syst. B (2010, accepted) Google Scholar
  7. 7.
    Danca, M.-F.: Attractors synthesis for a Lotka-Volterra-like system. Appl. Math. Comput. 216(7), 2107–2117 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Almeida, J., Peralta-Salas, D., Romera, M.: Can two chaotic systems give rise to order? Physica D 200, 124–132 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Romera, M., Small, M., Danca, M.-F.: Deterministic and random synthesis of discrete chaos. Appl. Math. Comput. 192(1), 283–297 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Danca, M.-F., Romera, M., Pastor, G.: Alternated Julia sets and connectivity properties. Int. J. Bifurc. Chaos 19(6), 2123–2129 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Mao, Y., Tang, W.K.S., Danca, M.-F.: An averaging model for the chaotic system with periodic time-varying parameter. Appl. Math. Comput. 217(1), 355–362 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Lempio, F.: Difference methods for differential inclusions. Lect. Notes Econ. Math. Syst. 378, 236–273 (1992) Google Scholar
  13. 13.
    Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Diethelm, K.: The Analysis of Fractional Differential Equations an Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010) zbMATHGoogle Scholar
  15. 15.
    Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000) CrossRefGoogle Scholar
  16. 16.
    Lü, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354, 305–311 (2006) CrossRefGoogle Scholar
  17. 17.
    Brown, R.: Generalizations of the Chua equations. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 40(11), 878–883 (1993) zbMATHCrossRefGoogle Scholar
  18. 18.
    Kapitanski, L., Rodnianski, I.: Shape and Morse theory of attractors. Commun. Pure Appl. Math. 53(2), 218–242 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Stuart, A., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge Monographs on Applied and Computational Mathematics, vol. 2. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  20. 20.
    Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99(2), 177–195 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68. Springer, New York (1988) zbMATHCrossRefGoogle Scholar
  22. 22.
    Hirsch, W.M., Pugh, C.: Stable manifolds and hyperbolic sets. In: American Mathematical Society. Proc. Symp. Pure Math., vol. 14, pp. 133–164 (1970) Google Scholar
  23. 23.
    Hirsch, W.M., Smale, S., Devaney, L.R.: Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. Elsevier/Academic Press, London (2004) zbMATHGoogle Scholar
  24. 24.
    Foias, C., Jolly, M.S.: On the numerical algebraic approximation of global attractors. Nonlinearity 8, 295–319 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Lempio, F.: Euler’s method revisited. Proc. Steklov Inst. Math. Mosc. 211, 473–494 (1995) MathSciNetGoogle Scholar
  26. 26.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) zbMATHGoogle Scholar
  27. 27.
    Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2000) zbMATHGoogle Scholar
  28. 28.
    Ahmed, E., Elgazzar, A.S.: On fractional order differential equations model for nonlocal epidemics. Physica A 379, 607–614 (2007) MathSciNetCrossRefGoogle Scholar
  29. 29.
    El-Sayed, A.M.A., El-Mesiry, A.E.M., El-Saka, H.A.A.: On the fractional-order logistic equation. Appl. Math. Lett. 20(7), 817–823 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Podlubny, I., Petrás, I., Vinagre, B.M., O’Leary, P., Dorcák, L.: Analogue realization of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002) zbMATHCrossRefGoogle Scholar
  31. 31.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Lü, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurc. Chaos 12(12), 2917–2926 (2002) zbMATHCrossRefGoogle Scholar
  33. 33.
    Danca, M.-F.: Chaotic behavior of a class of discontinuous dynamical systems of fractional-order. Nonlinear Dyn. 60(4), 525–534 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Danca, M.-F., Diethlem, K.: Fractional-order attractors synthesis via parameter switchings. Commun. Nonlinear Sci. Numer. Simul. 15(12), 3745–3753 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Popp, K., Stelter, P.: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. Lond. A 332(1624), 89–105 (1990) zbMATHCrossRefGoogle Scholar
  36. 36.
    Deimling, K.: Multivalued differential equations and dry friction problems. In: Fink, A.M., Miller, R.K., Kliemann, W. (eds.) Proc. Conf. Differential & Delay Equations, Ames, Iowa, pp. 99–106. World Scientific, Singapore (1992) Google Scholar
  37. 37.
    Wiercigroch, M., de Kraker, B.: Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities. World Scientific, Singapore (2000) zbMATHCrossRefGoogle Scholar
  38. 38.
    Danca, M.-F.: On a class of non-smooth dynamical systems: a sufficient condition for smooth vs nonsmooth solutions. Regul. Chaotic Dyn. 12(1), 1–11 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Danca, M.-F.: Numerical approximation of a class of discontinuous systems of fractional order. Nonlinear Dyn. (2010). doi: 10.1007/s11071-010-9915-z Google Scholar
  40. 40.
    Aubin, J.-P., Cellina, A.: Differential Inclusions Set-Valued Maps and Viability Theory. Springer, Berlin (1984) zbMATHGoogle Scholar
  41. 41.
    Danca, M.-F., Codreanu, S.: On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals 13(4), 681–691 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Li, S., Chen, G., Mou, X.: On the dynamical degradation of digital piecewise linear chaotic maps. Int. J. Bifurc. Chaos 15(10), 3119–3151 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (1990) zbMATHGoogle Scholar
  44. 44.
    Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer, New York (1985) zbMATHGoogle Scholar
  45. 45.
    Dacunha, J.J.: Transition matrix and generalized matrix exponential via the Peano-Baker Series. J. Differ. Equ. Appl. 11(15), 1245–1264 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992) Google Scholar
  47. 47.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic, Dordrecht (1988) Google Scholar
  48. 48.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990) zbMATHGoogle Scholar
  49. 49.
    Danca, M.-F.: On a class of discontinuous dynamical system. Miskolc Math. Notes 2(2), 103–116 (2001) MathSciNetzbMATHGoogle Scholar
  50. 50.
    Kastner-Maresch, A., Lempio, F.: Difference methods with selection strategies for differential inclusions. Numer. Funct. Anal. Optim. 14(5–6), 555–572 (1993) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Marius-F. Danca
    • 1
    • 2
  • Miguel Romera
    • 3
  • Gerardo Pastor
    • 3
  • Fausto Montoya
    • 3
  1. 1.Department of Mathematics and Computer ScienceAvram Iancu UniversityCluj-NapocaRomania
  2. 2.Romanian Institute of Science and TechnologyCluj-NapocaRomania
  3. 3.Instituto de Física AplicadaConsejo Superior de Investigaciones CientíficasMadridSpain

Personalised recommendations