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On dynamics of infinite-equilibrium systems

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Abstract

In this paper, dynamical systems with infinite-equilibriums is discussed through the local analysis. A computational method for equilibriums in nonlinear dynamical systems is developed which extends the Newton–Raphson method for solving nonlinear algebraic equations. The generalized normal forms of nonlinear dynamical systems at equilibriums are presented for a better understanding of singularity in nonlinear dynamical systems. At bifurcation points of equilibriums, the corresponding dynamical systems may become infinite-equilibrium systems. The dynamics of infinite-equilibrium dynamical systems is discussed through a few examples for demonstration of complexity and singularity in infinite-equilibrium systems.

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References

  1. Poincare H (1885) L’Equilibre Diune masses fluide animmee d’un movement de rotation. Acta Math 7:259–380

    Article  MathSciNet  Google Scholar 

  2. Poincare H (1890) Sur La Probleme des Trois Corps et les Equations de la Dynamique. Acta Math 13:1–271

    MathSciNet  MATH  Google Scholar 

  3. Pioncare H (1892) Les Methods Nouvelles de la Mechanique Celeste. Gau-thier-Villars, Paris

    Google Scholar 

  4. Birkhoff GD (1913) Proof of Poincare’s geometric theorem. Trans Am Math Soc 14:14–22

    MathSciNet  MATH  Google Scholar 

  5. Birkhoff GD (1927) Dynamical systems. American Mathematical Society, New York

    MATH  Google Scholar 

  6. Melnikov VK (1962) On the behavior of trajectories of system near to autonomous Hamiltonian systems. Sov Math Doklady 3:109–112

    Google Scholar 

  7. Melnikov VK (1963) On the stability of the center for time periodic perturbations. Trans Moscow Math Soc 12:1–57

    MathSciNet  Google Scholar 

  8. Shilnikov LP (1965) A case of the existence of a denumerable set of periodic motions. Sov Math Doklady 6:163–166

    Google Scholar 

  9. Shilnikov LP (1967) The existence of denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus. Sov Math Doklady 8:54–58

    Google Scholar 

  10. Bruno AD (1989) Local methods in nonlinear differential equations. Springer, Berlin

    Book  Google Scholar 

  11. Murdock J (2003) Normal forms and unfoldings for local dynamical systems. Springer, New York

    Book  Google Scholar 

  12. Luo ACJ (2019) On stability and bifurcation of equilibriums in nonlinear systems. J Vib Test Syst Dyn 3:147–232

    Google Scholar 

  13. Luo ACJ (2019) The stability and bifurcation of equilibriums in \( (2m){\text{th}} \)-degree polynomial systems. J Vib Test Syst Dyn (in press)

  14. Luo ACJ (2019) The stability and bifurcation of equilibriums in \( (2m + 1){\text{th}} \)-degree polynomial systems. J Vib Test Syst Dyn (in press)

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1805, USA

    Albert C. J. Luo

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  1. Albert C. J. Luo
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Correspondence to Albert C. J. Luo.

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Luo, A.C.J. On dynamics of infinite-equilibrium systems. Int. J. Dynam. Control 8, 21–43 (2020). https://doi.org/10.1007/s40435-019-00539-4

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  • DOI: https://doi.org/10.1007/s40435-019-00539-4

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