Skip to main content
SpringerLink
Log in

Complex dynamics in the stretch-twist-fold flow

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The system associated with fluid particle motions of the stretch-twist-fold (STF) flow has displayed rich and attractive dynamic properties. Detailed research on the system has been done in this work. By using a high-dimensional generalization of the Melnikov method, the explicit parametric conditions for the existence of periodic solutions in the system can be determined. Then, by using the new-KAM-like theorems for perturbations of a three-dimensional generalized Hamiltonian system, the criteria for the existence of invariant tori in the STF flow have been obtained. In addition, one new first integral is found. On the basis of it, nonexistence of chaos in the system at α=0 is rigorously proved. Nonexistence of homoclinic orbits is also proved in the system if some conditions hold. And an interesting phenomenon is found, where the unit circle in the (y,z)-plane is filled with heteroclinic orbits of the system at α=0. The system with α=0 is also successfully reduced to a generalized Hamiltonian system, and further transformed to slowly varying oscillators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Vainshtein, S.I., Zel’dovich, Y.A.B.: Origin of magnetic fields in astrophysics (turbulent “dynamo” mechanisms). Sov. Phys., Usp. 15, 159–172 (1972)

    Article  Google Scholar 

  2. Moffatt, H.K., Proctor, M.R.E.: Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493–507 (1985)

    Article  MATH  Google Scholar 

  3. Childress, S., Gilbert, A.D.: Stretch, Twist, Fold: The Fast Dynamo. Springer, Berlin/Heidelberg (1995)

    MATH  Google Scholar 

  4. Bajer, K.: Flow kinematics and magnetic equilibria, Ph.D. thesis, University of Cambridge, Cambridge (1989)

  5. Bajer, K.: Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos Solitons Fractals 4, 895–911 (1994)

    Article  MATH  Google Scholar 

  6. Bajer, K., Moffatt, H.K.: On a class of steady confined stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bajer, K., Moffatt, H.K., Nex, F.H.: Steady confined stokes flows with chaotic streamlines. In: Moffatt, H.K., Tsinober, A. (eds.) Topological Fluid Mechanics: Proceedings of the IUTAM Symposium, pp. 459–466. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  8. Vainshtein, S.I., Sagdeev, R.Z., Rosner, R., et al.: Fractal properties of the stretch-twist-fold magnetic dynamo. Phys. Rev. E 53, 4729–4744 (1996)

    Article  Google Scholar 

  9. Vainshtein, D.L., Vasiliev, A.A., Neishtadt, A.I.: Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow. Chaos 6, 67–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Vainshtein, S.I., Sagdeev, R.Z., Rosner, R.: Stretch-twist-fold and ABC nonlinear dynamos: restricted chaos. Phys. Rev. E 56, 1605–1622 (1997)

    Article  MathSciNet  Google Scholar 

  11. Wiggins, S., Holmes, P.: Periodic orbits in slowly varying oscillators. SIAM J. Math. Anal. 18, 592–611 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, Y., Yi, Y.F.: Persistence of invariant tori in generalized Hamiltonian systems. Ergod. Theory Dyn. Syst. 22, 1233–1261 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    MATH  Google Scholar 

  14. Klausmeier, C.A.: Floquet theory: a useful tool for understanding nonequilibrium dynamics. Theor. Ecol. 1, 153–161 (2008)

    Article  Google Scholar 

  15. Escalona, J.L., Chamorro, R.: Stability analysis of vehicles on circular motions using multibody dynamics. Nonlinear Dyn. 53, 237–250 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gao, P.Y.: Hamiltonian structure and first integrals for the Lotka–Volterra systems. Phys. Lett. A 273, 85–96 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. S̆ilnikov, L.P.: A case of the existence of a countable number of periodic motions. Sov. Math. Dockl. 6, 163–166 (1965)

    Google Scholar 

  18. S̆ilnikov, L.P.: A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddlefocus type. Math. USSR Sb. 10, 91–102 (1970)

    Article  Google Scholar 

  19. Silva, C.P.: S̆ilnikov theorem—a tutorial. IEEE Trans. Circuits Syst. I Fundam Theory 40, 675–682 (1993)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510641, P.R. China

    Jianghong Bao & Qigui Yang

Authors
  1. Jianghong Bao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qigui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianghong Bao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bao, J., Yang, Q. Complex dynamics in the stretch-twist-fold flow. Nonlinear Dyn 61, 773–781 (2010). https://doi.org/10.1007/s11071-010-9686-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-010-9686-6

Keywords

Search

Navigation