Stability of Dynamical Systems: Linear Approach

  • Vadim S. Anishchenko
  • Tatyana E. Vadivasova
  • Galina I. Strelkova
Part of the Springer Series in Synergetics book series (SSSYN)


Our understanding of the stability of a particular operating mode of a dynamical system is formed intuitively as we build up our experience and understanding of everyday life and nature. The first steps of a small child give him or her very real representations of the stability of walking, although these representations may not yet enter consciousness. Looking at the famous painting entitled Young Acrobat on a Ball by P. Picasso, we have a distinct feeling that the girl’s equilibrium is not quite stable. As adults, we can already discuss the stability of a ship on a stormy sea, the stability of economic trends in relation to the actions of managers and politicians, the stability of our nervous system with regard to stressful perturbation, etc. In each case, we talk about different properties that are specific to the considered systems. However, if we think about it carefully, we can find something in common, inherent in any system. The common feature is that, when we talk about stability, we understand the way the dynamical system reacts to a small perturbation of its state. If arbitrarily small changes in the system state begin to grow in time, the system is unstable. Otherwise, small perturbations decay with time and the system is stable.


Periodic Solution Lyapunov Exponent Linearization Matrix Unstable Manifold Initial Perturbation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Andronov, A.A., Vitt, E.A., Khaikin, S.E.: Theory of Oscillations. Pergamon, Oxford (1966)Google Scholar
  2. 2.
    Anishchenko, V.S.: Dynamical Chaos – Models and Experiments. World Scientific, Singapore (1995)Google Scholar
  3. 3.
    Anishchenko, V.S., Astakhov, V.V., Neiman, A.B., Vadivasova, T.E., Schimansky-Geier, L.: Nonlinear Dynamics of Chaotic and Stochastic Systems. Springer, Berlin (2002)Google Scholar
  4. 4.
    Barreira, L., Pesin, Y.: Nonuniform Hyperbolicity: Dynamics of Systems With Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2007)Google Scholar
  5. 5.
    Drazin, P.G.: Nonlinear Systems. Cambridge University Press, Cambridge (1992)Google Scholar
  6. 6.
    Glendinning, P.: Stability, Instability, and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press, Cambridge (1994)Google Scholar
  7. 7.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)Google Scholar
  8. 8.
    Hilborn, R.C.: Chaos and Nonlinear Dynamics. An Introduction for Scientists and Engineers. Oxford University Press, Oxford (2002/2004)Google Scholar
  9. 9.
    Marek, M., Schreiber, I.: Chaotic Behaviour of Deterministic Dissipative Systems. Cambridge University Press, Cambridge (1991/1995)Google Scholar
  10. 10.
    Marsden, L.E., McCraken, V.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)Google Scholar
  11. 11.
    Moon, F.C.M.: Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers. Wiley, New York (1992)Google Scholar
  12. 12.
    Moon, F.C.M.: Chaotic Vibration: An Introduction for Applied Scientists and Engineers. Wiley, New York (2004)Google Scholar
  13. 13.
    Nicolis, G.: Introduction to Nonlinear Science. Cambridge University Press, Cambridge (1995)Google Scholar
  14. 14.
    Ogorzalek, M.J.: Chaos and Complexity in Nonlinear Electronic Circuits. World Scientific, Singapore (1997)Google Scholar
  15. 15.
    Schuster, H.G.: Deterministic Chaos. Physik-Verlag, Weinheim (1988)Google Scholar
  16. 16.
    Seydel, R.: Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos. Springer, New York (1994/2009)Google Scholar
  17. 17.
    Thompson, J.M.T., Stewart, H.B.: Nonlinear Dynamics and Chaos. Wiley, New York (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Vadim S. Anishchenko
    • 1
  • Tatyana E. Vadivasova
    • 1
  • Galina I. Strelkova
    • 1
  1. 1.Department of Physics Instiute of Nonlinear DynamicsSaratov State UniversitySaratovRussia

Personalised recommendations