Skip to main content

Chaos in Oscillators

  • Chapter
  • First Online:
  • 2064 Accesses

Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

Abstract

The problem of chaotic motion is widely investigated for the oscillators with negative linear and cubic nonlinear term

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Balthazar, J. M., Mook, D. T., Weber, H. I., Brasil, R. M. L. R. F., Fenili, A., Beltano, D., et al. (2003). An overview on non-ideal vibrations. Meccanica, 38, 613–621.

    Article  MATH  Google Scholar 

  • Bogolubov, N. N., & Mitropolski, Ju A. (1974). Asimptoticheskie metodi v teorii nelinejnih kolebanij. Moscow: Nauka.

    Google Scholar 

  • Chacon, R. (1999). General result on chaos suppression for biharmonically driven dissipative systems. Physics Letters A, 257, 293–300.

    Article  ADS  Google Scholar 

  • Chen, G. R., & Yu, X. H. (1999). On time-delayed feedback control of chaotic systems. IEEE Transactions on Circuit Systems, 46, 767–772.

    Article  MATH  Google Scholar 

  • Cveticanin, L. (1993). Extension of Melnikov criterion for the differential equation with complex function. Nonlinear Dynamics, 4, 139–152.

    Google Scholar 

  • Cveticanin, L., & Zukovic, M. (2009). Melnikov’s criteria and chaos in systems with fraction order deflection. Journal of Sound and Vibration, 326, 768–779.

    Article  ADS  Google Scholar 

  • Dantas, M. H., & Balthazar, J. M. (2003). On the appearance of a Hopf bifurcation in a non-ideal mechanical system. Mechanics Research Communications, 30, 493–503.

    Article  MathSciNet  MATH  Google Scholar 

  • Dantas, M. H., & Balthazar, J. M. (2006). A comment on a non-ideal centrifugal vibrator machine behavior with soft and hard springs. International Journal of Bifurcation and Chaos, 16, 1083–1088.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dimentberg, M. F., McGovern, L., Norton, R. L., Chapdelaine, J., & Harrison, R. (1997). Dynamics of an unbalanced shaft interacting with a limited power supply. Nonlinear Dynamics, 13, 171–187.

    Article  MATH  Google Scholar 

  • Fang, T., & Dowell, E. H. (1987). Numerical simulations of periodic and chaotic responses in a stable Duffing system. International Journal of Non-Linear Mechanics, 22, 401–425.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gradstein, I. S., & Rjizhik, I. M. (1971). Tablici integralov, summ, rjadov i proizvedenij. Moscow: Nauka.

    Google Scholar 

  • Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer Verlag.

    Google Scholar 

  • Holmes, P. (1979). A nonlinear oscillator with a strange attractor. Philosophical Transactions of the Royal Society of London, Series A, Mathematical and Physical Sciences, 292, 419–448.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Holmes, P., & Marsden, J. (1981). A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam. Archives for Rational Mechanics and Analysis, 76, 135–166.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kononenko, V. O. (1969). Vibrating system with a limited power supply. London: Illife.

    Google Scholar 

  • Mahmoud, G. M., Mohamed, A. A., & Aly, S. A. (2001). Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Physica A, 292, 193–206.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Melnikov, V. K. (1963). On the stability of the center for time periodic perturbations. Transactions of the Moscow Mathematical Society, 12, 1–57.

    Google Scholar 

  • Nayfeh, A. H., & Mook, D. T. (1976). Nonlinear oscillations. New York: Wiley.

    Google Scholar 

  • Pezeshki, C., & Dowell, E. H. (1988). On chaos and fractal behaviour in a general Duffing’s system. Physica D, 32, 194–209.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pyragas, K. (1992). Continuous control of chaos by self controlling feedback. Physics Letters A, 170, 421–428.

    Article  ADS  Google Scholar 

  • Pyragas, K. (1995). Control of chaos via extended delay feedback. Physics Letters A, 206, 323–330.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Pyragas, K. (1996). Continuous control of chaos by self-controlling feedback. In Controlling chaos (pp. 118–123). San Diego: Academic Press.

    Google Scholar 

  • Pyragas, K. (2001). Control of chaos via an unstable delayed feedback controller. Physics Review Letters, 86, 2265–2268.

    Article  ADS  Google Scholar 

  • Pyragas, K. (2006). Delayed feedback control of chaos. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 364, 2309–2334.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Sandri, M. (1996). Numerical calculation of Lyapunov exponents. The Mathematical Journal, 6, 78–84.

    Google Scholar 

  • Souza, S. L. T., Caldas, I. L., Viana, R. L., Balthazar, J. M., Brasil, R. M. L. R. F. (2005a). Impact dampers for controlling chaos in systems with limited power supply. Journal of Sound and Vibration, 279, 955–965.

    Google Scholar 

  • Souza, S. L. T., Caldas, I. L., Viana, R. L., Balthazar, J. M., Brasil, R. M. L. R. F. (2005b). Basins of attraction changes by amplitude constraining of oscillators with limited power supply. Chaos, Solitons and Fractals, 26, 1211–1220.

    Google Scholar 

  • Tseng, W. Y., & Dugundji, J. (1971). Nonlinear vibrations of a buckled beam under harmonic excitation. Journal of Applied Mechanics, 38, 467–476.

    Article  ADS  MATH  Google Scholar 

  • Tsuchida, M., Guilherme, K. L., Balthazar, J. M., Silva, G. N., & Cheshankov, B. I. (2003). On regular and irregular vibrations of a non-ideal system with two degrees of freedom: 1:1 resonance. Journal of Sound and Vibration, 260, 949–960.

    Article  ADS  Google Scholar 

  • Tsuchida, M., Guilherme, K. L., & Balthazar, J. M. (2005). On chaotic vibrations of a non-ideal system with two degree of freedom: 1:2 resonance and Sommerfeld effect. Journal of Sound and Vibration, 282, 1201–1207.

    Article  ADS  Google Scholar 

  • Ueda, Y. (1985). Random phenomena resulting from nonlinearity in the system described by Duffing’s equation. International Journal of Non-Linear Mechanics, 20, 481–491.

    Article  ADS  Google Scholar 

  • Ushio, T. (1996). Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Transactions on Circuit Systems, 43, 815–816.

    Article  Google Scholar 

  • Van Dooren, R., & Janssen, H. (1996). A continuation algorithm for discovering new chaotic motions in forced Duffing systems. Journal of Computational and Applied Mathematics, 66, 527–541.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, R., Deng, J., & Jing, Z. (2006). Chaos, control in Duffing system. Chaos, Solitons and Fractals, 23, 249–257.

    Article  ADS  MathSciNet  Google Scholar 

  • Warminski, J., Balthazar, J. M., & Brasil, R. M. L. R. F. (2001). Vibrations of a non-ideal parametrically and self-excited model. Journal of Sound and Vibration, 245, 363–374.

    Article  ADS  MATH  Google Scholar 

  • Wiggins, S. (1988). Global bifurcations and chaos: Analytical methods. New York: Springer Verlag.

    Google Scholar 

  • Wolf, A., Swift, J., Swinney, H., & Vastano, J. (1985). Determining Lyapunov exponents from a time series. Physica D, 16, 285–317.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Yang, J., & Jing, Z. (2008). Control of chaos in a three-well duffing system. Chaos, Solitons and Fractals,. doi:10.1016/j.chaos.2008.05018.

    MathSciNet  Google Scholar 

  • Zhu, J., & Tian, Y.-P. (2005). Necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control. Physics Letters A, 343, 95–107.

    Article  ADS  MATH  Google Scholar 

  • Zukovic, M., & Cveticanin, L. (2007). Chaotic responses in a stable Duffing system of non-ideal type. Journal of Vibration and Control, 13, 751–767.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Livija Cveticanin .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Cveticanin, L. (2014). Chaos in Oscillators. In: Strongly Nonlinear Oscillators. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-05272-4_8

Download citation

Publish with us

Policies and ethics