Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Dynamical Systems

  • Chapter
  • First Online:
Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics

Part of the book series: SpringerBriefs in Molecular Science ((BRIEFSMAGNET))

  • 930 Accesses

Abstract

The basic theory of dynamical systems is introduced in this chapter. Invariant phase space structures—equilibria, periodic orbits, tori, normally hyperbolic invariant manifolds and stable/unstable manifolds—are defined mainly with graphs produced by numerically solving the equations of motion of 1, 2 and 3 degrees of freedom model Hamiltonian systems. Stability analysis and elementary bifurcations of equilibria and periodic orbits are discussed. The center-saddle, pitchfork, period doubling and complex instability elementary bifurcations encountered in continuation diagrams of equilibria and periodic orbits by varying a parameter in the potential function or the energy of the system are investigated. Methods of analysing non-periodic orbits, regular and chaotic, such as Poincaré surfaces of section, maximal Lyapunov exponent and autocorrelation functions are introduced and explained.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For general dynamical systems this elementary bifurcation is called saddle-node.

  2. 2.

    To avoid using many symbols we use \((q, p)\) to ascribe both the internal and the normal coordinates and conjugate momenta.

  3. 3.

    Examples of stable/unstable manifolds (separatrices) are shown in Figs. 3.1d and 3.4, where the arrows explain the directions of the linearized flow in the neighbourhood of the unstable equilibrium.

References

  1. Bensimon D, Kadanoff LP (1984) Extended chaos and disappearance of KAM trajectories. Physica D 13:82–89

    Article  Google Scholar 

  2. Birkhoff GD, Lewis DC (1933) On the periodic motions near a given periodic motion of a dynamical system. Ann Mat Pura Appl 12:117–133

    Article  Google Scholar 

  3. Contopoulos G, Farantos SC, Papadaki H, Polymilis C (1994) Complex unstable periodic orbits and their manifestation in classical and quantum dynamics. Phys Rev E 50(5):4399–4403

    Article  CAS  Google Scholar 

  4. Duarte P (1999) Abundance of elliptic isles at conservative bifurcations. Dyn Stab Sys 14(4):339–356

    Article  Google Scholar 

  5. Founargiotakis M, Farantos SC, Contopoulos G, Polymilis C (1989) Periodic orbits, bifurcations and quantum mechanical eigenfunctions and spectra. J Chem Phys 91(1):1389–1402

    Article  Google Scholar 

  6. Gomez Llorente JM, Taylor HS (1989) Spectra in the chaotic region: a classical analysis for the sodium trimer. J Chem Phys 91:953–962

    Article  CAS  Google Scholar 

  7. Gomez Llorente JM, Pollak E (1992) Classical dynamics methods for high energy vibrational spectroscopy. Ann Rev Phys Chem 43:91–126

    Article  CAS  Google Scholar 

  8. Gonchenko SV, Silnikov LP (2000) On two-dimensional area-preserving diffeomorphisms with infinitely many elliptic islands. J Stat Phys 101(1/2):321–356

    Article  Google Scholar 

  9. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, Berlin

    Book  Google Scholar 

  10. Hanssmann H (2007) Local and semi-local bifurcations in Hamiltonian dynamical systems: results and examples. Springer, Berlin

    Google Scholar 

  11. Hartman P (1964) Ordinary differential equations. Wiley, New York

    Google Scholar 

  12. Hénon M (1982) On the numerical computation of Poincaré maps. Physica D 5:412–414

    Article  Google Scholar 

  13. Krasnosel’skii MA, Zabreiko PP (1984) Geometrical methods of nonlinear analysis. A series of comprehensive mathematics, Springer, Berlin

    Google Scholar 

  14. Mackay RS, Meiss JD, Percival IC (1984) Transport in Hamiltonian systems. Physica D 13:55–81

    Article  Google Scholar 

  15. Moser J (1976) Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun Pure Appl Math 29:727–747

    Article  Google Scholar 

  16. Newhouse SE (1979) The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ Math IHES 50:101–151

    Article  Google Scholar 

  17. Rabinowitz PH (1984) Periodic solutions of Hamiltonian systems: a survey. SIAM J Math Anal 13:343

    Article  Google Scholar 

  18. Skokos C (2010) The Lyapunov characteristic exponents and their computation. Lec Notes Phys 790:63–135

    Article  Google Scholar 

  19. Waalkens H, Burbanks A, Wiggins S (2004) Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions. J Chem Phys 121(13):6207–6225

    Article  CAS  Google Scholar 

  20. Weinstein A (1973) Normal modes for nonlinear Hamiltonian systems. Inv Math 20:47–57

    Article  Google Scholar 

  21. Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Springer, New York

    Google Scholar 

  22. Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients. Wiley, New York

    Google Scholar 

  23. Yoshizawa T (1975) Stability theory and the existence of periodic solutions., Applied mathematical sciences, Springer, Berlin

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry, University of Crete, Iraklion, Greece

    Stavros C. Farantos

  2. Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, Iraklion, Greece

    Stavros C. Farantos

Authors
  1. Stavros C. Farantos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stavros C. Farantos .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Author(s)

About this chapter

Cite this chapter

Farantos, S.C. (2014). Dynamical Systems. In: Nonlinear Hamiltonian Mechanics Applied to Molecular Dynamics. SpringerBriefs in Molecular Science(). Springer, Cham. https://doi.org/10.1007/978-3-319-09988-0_3

Download citation

Publish with us

Policies and ethics

Search

Navigation