Dynamical Systems

  • Stavros C. Farantos
Part of the SpringerBriefs in Molecular Science book series (BRIEFSMOLECULAR)


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.


Periodic Orbit Unstable Manifold Fundamental Matrix Monodromy Matrix Maximal Lyapunov Exponent 
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  1. 1.
    Bensimon D, Kadanoff LP (1984) Extended chaos and disappearance of KAM trajectories. Physica D 13:82–89CrossRefGoogle Scholar
  2. 2.
    Birkhoff GD, Lewis DC (1933) On the periodic motions near a given periodic motion of a dynamical system. Ann Mat Pura Appl 12:117–133CrossRefGoogle Scholar
  3. 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):4399–4403CrossRefGoogle Scholar
  4. 4.
    Duarte P (1999) Abundance of elliptic isles at conservative bifurcations. Dyn Stab Sys 14(4):339–356CrossRefGoogle Scholar
  5. 5.
    Founargiotakis M, Farantos SC, Contopoulos G, Polymilis C (1989) Periodic orbits, bifurcations and quantum mechanical eigenfunctions and spectra. J Chem Phys 91(1):1389–1402CrossRefGoogle Scholar
  6. 6.
    Gomez Llorente JM, Taylor HS (1989) Spectra in the chaotic region: a classical analysis for the sodium trimer. J Chem Phys 91:953–962CrossRefGoogle Scholar
  7. 7.
    Gomez Llorente JM, Pollak E (1992) Classical dynamics methods for high energy vibrational spectroscopy. Ann Rev Phys Chem 43:91–126CrossRefGoogle Scholar
  8. 8.
    Gonchenko SV, Silnikov LP (2000) On two-dimensional area-preserving diffeomorphisms with infinitely many elliptic islands. J Stat Phys 101(1/2):321–356CrossRefGoogle Scholar
  9. 9.
    Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, BerlinCrossRefGoogle Scholar
  10. 10.
    Hanssmann H (2007) Local and semi-local bifurcations in Hamiltonian dynamical systems: results and examples. Springer, BerlinGoogle Scholar
  11. 11.
    Hartman P (1964) Ordinary differential equations. Wiley, New YorkGoogle Scholar
  12. 12.
    Hénon M (1982) On the numerical computation of Poincaré maps. Physica D 5:412–414CrossRefGoogle Scholar
  13. 13.
    Krasnosel’skii MA, Zabreiko PP (1984) Geometrical methods of nonlinear analysis. A series of comprehensive mathematics, Springer, BerlinGoogle Scholar
  14. 14.
    Mackay RS, Meiss JD, Percival IC (1984) Transport in Hamiltonian systems. Physica D 13:55–81CrossRefGoogle Scholar
  15. 15.
    Moser J (1976) Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun Pure Appl Math 29:727–747CrossRefGoogle Scholar
  16. 16.
    Newhouse SE (1979) The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ Math IHES 50:101–151CrossRefGoogle Scholar
  17. 17.
    Rabinowitz PH (1984) Periodic solutions of Hamiltonian systems: a survey. SIAM J Math Anal 13:343CrossRefGoogle Scholar
  18. 18.
    Skokos C (2010) The Lyapunov characteristic exponents and their computation. Lec Notes Phys 790:63–135CrossRefGoogle Scholar
  19. 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):6207–6225CrossRefGoogle Scholar
  20. 20.
    Weinstein A (1973) Normal modes for nonlinear Hamiltonian systems. Inv Math 20:47–57CrossRefGoogle Scholar
  21. 21.
    Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Springer, New YorkGoogle Scholar
  22. 22.
    Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients. Wiley, New YorkGoogle Scholar
  23. 23.
    Yoshizawa T (1975) Stability theory and the existence of periodic solutions., Applied mathematical sciences, Springer, BerlinGoogle Scholar

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© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of CreteIraklionGreece
  2. 2.Institute of Electronic Structure and LaserFoundation for Research and Technology-HellasIraklionGreece

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