Nonlinear Dynamics

, Volume 54, Issue 4, pp 345–354 | Cite as

Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems

Original Paper

Abstract

This paper presents analysis of Lyapunov type stability for multiplicative dynamical systems. It has been formally defined and numerical simulations were performed to explore nonlinear dynamics. Chaotic behavior manifested for exemplary multiplicative dynamical systems has been confirmed by calculated Lyapunov exponent values.

Keywords

Multiplicative calculus Lyapunov stability Lyapunov exponent 

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References

  1. 1.
    Rybaczuk, M., Kedzia, A., Zielinski, W.: The concept of physical and fractal dimension, II:  the differential calculus in dimensional spaces. Chaos Solitons Fractals 12, 2537–2552 (2001) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Rybaczuk, M., Stoppel, P.: The fractal growth of fatigue defects in materials. Int. J. Fract. 103, 71–94 (2000) CrossRefGoogle Scholar
  3. 3.
    Kasprzak, W., Lysik, B., Rybaczuk, M.: Measurements, Dimensions, Invariants Models and Fractals. Ukrainian Society on Fracture Mechanics Publishing House/SPOLOM, Lviv/Wroclaw (2004) Google Scholar
  4. 4.
    Volterra, V., Hostinsky, B.: Operations Infinitesimales Lineares. Herman, Paris (1938) Google Scholar
  5. 5.
    Lyapunov, A.M.: Stability of Motion. Academic Press, New York (1966) MATHGoogle Scholar
  6. 6.
    Aniszewska, D., Rybaczuk, M.: Analysis of the multiplicative Lorenz system. Chaos Solitons Fractals 25, 79–90 (2005) MATHCrossRefGoogle Scholar
  7. 7.
    Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer, New York (1982) MATHGoogle Scholar
  8. 8.
    Aniszewska, D.: Multiplicative Runge–Kutta method. Nonlinear Dyn. 50, 265–272 (2007) CrossRefGoogle Scholar
  9. 9.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute of Materials Science and Applied MechanicsWroclaw University of TechnologyWroclawPoland

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