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Analytical and numerical investigation of two families of Lorenz-like dynamical systems

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  • Panchev, S.
  • Spassova, T.
  • Vitanov, N.K.

Abstract

We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke–Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to ϵ=±1, m=0,1. (I)ω=ϵσr,ϵ=±1,m=1,0The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal (t→∞) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters σ, r, b, ω and m.

Suggested Citation

  • Panchev, S. & Spassova, T. & Vitanov, N.K., 2007. "Analytical and numerical investigation of two families of Lorenz-like dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1658-1671.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:5:p:1658-1671
    DOI: 10.1016/j.chaos.2006.03.037
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    References listed on IDEAS

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    1. Martínez-Guerra, R. & Cruz-Victoria, J.C. & Gonzalez-Galan, R. & Aguilar-Lopez, R., 2006. "A new reduced-order observer design for the synchronization of Lorenz systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 511-517.
    2. Delfino, Doriana & Simmons, Peter J., 2005. "Dynamics of tuberculosis and economic growth," Environment and Development Economics, Cambridge University Press, vol. 10(6), pages 719-743, December.
    3. Dimitrova, Zlatinka I. & Vitanov, Nikolay K., 2001. "Adaptation and its impact on the dynamics of a system of three competing populations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(1), pages 91-115.
    4. Wang, Ruiqi & Deng, Jin & Jing, Zhujun, 2006. "Chaos control in duffing system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 249-257.
    5. Chen, Zhi-Min & Price, W.G., 2006. "On the relation between Rayleigh–Bénard convection and Lorenz system," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 571-578.
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    Cited by:

    1. Shu, Yonglu & Xu, Hongxing & Zhao, Yunhong, 2009. "Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2852-2857.
    2. Vitanov, Nikolay K. & Sakai, Kenshi & Dimitrova, Zlatinka I., 2008. "SSA, PCA, TDPSC, ACFA: Useful combination of methods for analysis of short and nonstationary time series," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 187-202.
    3. Vitanov, Nikolay K. & Hoffmann, Norbert P. & Wernitz, Boris, 2014. "Nonlinear time series analysis of vibration data from a friction brake: SSA, PCA, and MFDFA," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 90-99.

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