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Stability of limit cycles in autonomous nonlinear systems

  • Nonlinear Dynamics and Control of Composites for Smart Engi design
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Abstract

Periodical solutions or limit cycles (LC) comprise a significant family among the response types of nonlinear autonomous systems. Their identification and stability assessment is of a great importance during the analysis of an unknown system. A new analytical/iterative method of LC identification and portrait investigation was presented recently. The current study proposes a novel technique for their stability assessment. This strategy facilitates the distinction of stable and unstable LCs, thereby allowing the definition of attractive and repulsive response fields. A narrow toroidal domain is constructed around the LC, which is arithmetized by an orthogonal system that is positioned by tangential and normal vectors to the LC. The stability of the LC is investigated using the transformed differential system of the normal components of the response, which are functions of the coordinate along the LC trajectory. Exponential LC stability criteria are also proposed, which are based on the first degree of the perturbation procedure. Theoretical considerations are illustrated using single and two degree of freedom systems including demonstrations with specific systems. The strengths, future steps, and shortcomings of this method are evaluated.

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References

  1. Náprstek J, Fischer C (2009) Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper. Comput Struct 87:1204–1215

    Article  Google Scholar 

  2. Abarbanel HDI, Brown R, Kadtke JB (1990) Prediction in chaotic nonlinear systems: methods for time series with broadband Fourier spectra. Phys Rev A 41:1782–1807

    Article  ADS  MathSciNet  Google Scholar 

  3. Lewis A (2006) A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation. Int J Nonlin Mech 41:644–656

    Article  MATH  Google Scholar 

  4. Yan-Qian Ye (1986) Theory of limit cycles. American Mathematical Society, Boston.

    MATH  Google Scholar 

  5. Song Y, Sato H, Iwata Y, Komatsuzaki T (2003) The response of a dynamic vibration absorber system with a parametrically excited pendulum. J Sound Vib 259:747–759

    Article  ADS  Google Scholar 

  6. Ji JC (2006) Nonresonant Hopf bifurcations of a controlled van der Pol-Duffing oscillator. J Sound Vib 297:183–199

    Article  ADS  MATH  Google Scholar 

  7. Bajaj AK, Chang SI, Johnson JM (1994) Amplitude modulated dynamics of a resonantly excited autoparametric two degree of freedom system. Nonlinear Dyn 5:433–457

    Article  Google Scholar 

  8. Doyle MM, Sri Namachchivaya N, Van Roessel HJ (1997) Asymptotic stability of structural systems based on Lyapunov exponents and moment Lyapunov exponents. Int J Nonlin Mech 32:681–692

    Article  MATH  Google Scholar 

  9. Shahrzad P, Mahzoon M (2002) Limit cycle flutter of airfoils in steady and unsteady flows. J Sound Vib 256:213–225

    Article  ADS  Google Scholar 

  10. Thomas JP, Dowell EH, Hall KC (2002) Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA J Propuls Power 40:638–646

    Article  ADS  Google Scholar 

  11. Přibylová L (2009) Bifurcation routes to chaos in an extended van der Pol’s equation applied to economic models. Electron J Differ Equation 2009:1-21

    Google Scholar 

  12. Mees AI (1973) Limit cycle stability. IMA J Appl Math 11:281–295

    Article  MATH  MathSciNet  Google Scholar 

  13. Dong H, Zeng J, Xie JH, Jia L (2013) Bifurcation \ instability forms of high speed railway vehicles. Sci China Ser E 56:1685–1696

    Article  Google Scholar 

  14. Paak M, Païdoussis MP, Misra AK (2013) Nonlinear dynamics and stability of cantilevered circular cylindrical shells conveying fluid. J Sound Vib 332:3474-3498

    Article  ADS  Google Scholar 

  15. Markus L (1980) Lectures in differentiable dynamics. American Mathematical Society, Cleveland.

    Google Scholar 

  16. Henry D (1981) Geometric theory of semilinear parabolic equations. Springer, Berlin.

    MATH  Google Scholar 

  17. Xu G-Q, Yung SP (2003) Lyapunov stability of abstract nonlinear dynamic system in Banach space. IMA J Math Control I 20:105–127

    Article  MATH  MathSciNet  Google Scholar 

  18. Nabergoj R, Tondl A, Virag Z (1994) Auto-parametric resonance in an externally excited system. Chaos Soliton Fract 4:263–273

    Article  ADS  MATH  Google Scholar 

  19. Tondl A (1997) To the analysis of auto-parametric systems. ZAMM Z Angew Math Me 77:407–418

    Article  MATH  Google Scholar 

  20. Tondl A, Ruijgrok T, Verhulst F, Nabergoj R (2000) Auto-parametric resonance in mechanical systems. Cambridge University Press, Cambridge

    Google Scholar 

  21. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems. Springer, New York

    MATH  Google Scholar 

  22. Verhulst F (2005) Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics. Springer-Verlag, New York

    Book  MATH  Google Scholar 

  23. Bakri T, Nabergoj R, Tondl A, Verhulst F (2004) Parametric excitation in non-linear dynamics. Int J Non-Linear Mech 39:311–329

    Article  MATH  MathSciNet  Google Scholar 

  24. Tondl A, Ecker H (2003) On the problem of self-excited vibration quenching by means of parametric excitation. Arch Appl Mech 72:923–932

    MATH  Google Scholar 

  25. Dohnal F, Verhulst F (2008) Averaging in vibration suppression by parametric stiffness excitation. Nonlinear Dyn 54:231–248

    Article  MATH  MathSciNet  Google Scholar 

  26. Kovacic I (2013) Harmonically excited generalized van der Pol oscillators: Entrainment phenomenon. Meccanica 48:2415–2425

    Article  MathSciNet  Google Scholar 

  27. Beyn WJ, Champneys A, Doedel E, Govaerts W, Kuznetsov YA, Sandstede B (2002) Numerical continuation and computation of normal forms. In: Fiedler B. (ed) Handbook of dynamical systems. Elsevier, Amsterdam, 149–219.

  28. Doedel EJ, Paffenroth RC, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X (2000) Auto 2000: continuation and bifurcation software for ordinary differential equations. http://www.dam.brown.edu/people/sandsted/publications/auto2000.pdf.

  29. Kuznetsov YA, Levitin VV (1997) Content, a multiplatform environment for analysing dynamical systems. Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, Amsterdam. http://www.math.uu.nl/people/kuznet/CONTENT.

  30. Govaerts W, Kuznetsov YA (2013) The program Matcont. http://sourceforge.net/projects/matcont/.

  31. Weiqin Y, Fangqi C (2013) Global bifurcations and homoclinic trees in motion of a thin rectangular plate on a nonlinear elastic foundation. Meccanica 48:1251–1261.

    Article  MathSciNet  Google Scholar 

  32. Golub GH, van Loan CF (1996) Matrix computations, 3rd edn. John Hopkins U.P, Baltimore.

    MATH  Google Scholar 

  33. Benettin G, Galgani L, Giorgilli A, Strelcyn M (1980) Lyapunovcharacteristic exponents for smooth dynamical systems andHamiltonian systems: a method for computing all of them. Part charact 2, numerical application. Meccanica 15:21-30.

    Article  Google Scholar 

  34. Ahmadian M, Inman DJ (1985) On the stability of general dynamic systems using a Lyapunov’s direct method approach. Comput Struct, 20:287-292

    Article  MATH  Google Scholar 

  35. Wu Q, Thornton-Trump AB, Sepehri N (1998) Lyapunov stability control of inverted pendulums with general base point motion. Int J Nonlin Mech 33:801–818

    Article  MATH  MathSciNet  Google Scholar 

  36. Chan HSY, Chung KW, Xu Z (1996) A perturbation-incremental method for strongly non-linear oscillators. Int J Nonlin Mech 31:59–72

    Article  MATH  MathSciNet  Google Scholar 

  37. Náprstek J, Pospíšil S (2009) Stable and unstable limit cycles and nonlinear quasiperiodic response of aeroelastic structure. In: Borri C et al (eds)Proceedings 5th European and African Conference on Wind Engineering, Univ.di Firenze, Firenze, CD ROM paper 90, p 8.

  38. Parkinson GV, Smith JD (1963) The square prism as an aeroelastic non-linear oscillator. Quart J Mech Appl Math 17:225–239.

    Article  Google Scholar 

  39. Lumbantobing A, Haaker TI (2004) On the parametric excitation of some nonlinear aeroelastic oscillators. J Fluids Struct 19:221–237

    Article  Google Scholar 

  40. Náprstek J, Pospíšil S, Fischer C (2008) Post-critical limit cycles and nonstationary response types of aeroelastic system. In: Zolotarev I, Horáček J (eds) Proceedings 9th International Conference on Flow-Induced Vibrations, Prague, 41–47

  41. Chung KW, Chan CL, Xu Z, Mahmoud GM (2002) Perturbation-incremental method for strongly non-linear autonomous oscillators with many degrees of freedom. Nonlinear Dyn 28:243–259

    Article  MATH  MathSciNet  Google Scholar 

  42. Mees AI (1972) The describing function matrix. IMA J Appl Math 10:49–67

    Article  MATH  Google Scholar 

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Acknowledgements

The kind support of the Czech Science Foundation Project No. 103/09/0094, GC13-34405J, Grant Agency of the ASCR No. A200710902, and of the RVO 68378297 institutional support are gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics, AS CR, v.v.i., Prosecká 76, 190 00, Prague 9, Czech Republic

    Jiří Náprstek & Cyril Fischer

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  1. Jiří Náprstek
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  2. Cyril Fischer
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Correspondence to Jiří Náprstek.

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Náprstek, J., Fischer, C. Stability of limit cycles in autonomous nonlinear systems. Meccanica 49, 1929–1943 (2014). https://doi.org/10.1007/s11012-014-9899-8

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