Skip to main content

Onset of Dynamical Chaos: Physical Aspects

  • 946 Accesses

Part of the Springer Series on Atomic, Optical, and Plasma Physics book series (SSAOPP,volume 78)


The physical aspects of dynamical chaos in Hamiltonian systems was developed by Chirikov (1959). It is based on the phenomena nonlinear resonance and the interaction of resonances (see Zaslavsky and Chirikov 1971, 1979). These physical principles of dynamical chaos were instrumental in the first studies of the problems of stability and destruction of magnetic surfaces in tokamaks and stellarators caused by the magnetic perturbations (Kerst 1962; Rosenbluth et al. 1966; Filonenko et al. 1967; Freis et al. 1973; Hamzeh 1974; Finn 1975; Matsuda and Yoshikawa 1975; Boozer and Rechester 1978). Chaos of magnetic field lines in a generic example of chaotic dynamics in Hamiltonian systems with one-and-half degrees of freedom. The nonlinear resonance corresponds to the formation of magnetic islands due to destruction of resonant magnetic surfaces in the presence of non-axisymmetric magnetic perturbations. The interaction of magnetic islands with the growing of a perturbations leads global chaotic behavior of field lines in a certain region of plasma.


  • Hamiltonian System
  • Field Line
  • Safety Factor
  • Magnetic Surface
  • Perturbation Amplitude

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-01890-4_8
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-01890-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   179.99
Price excludes VAT (USA)
Fig. 8.1
Fig. 8.2
Fig. 8.3
Fig. 8.4
Fig. 8.5
Fig. 8.6
Fig. 8.7
Fig. 8.8
Fig. 8.9
Fig. 8.10
Fig. 8.11
Fig. 8.12
Fig. 8.13
Fig. 8.14


  1. 1.

    One should note that the bifurcation of a hyperbolic fixed point into an elliptic fixed point occurs also in the problem of the stability of the upper position of a pendulum with a vibrating suspension point. However, in this phenomenon discovered by Stephenson (1908), Bogolyubov (1950), Kapitsa (1951) (see a book by AKN (2006) for more details and additional references), such a bifurcation appears when the system is subjected to the external time–periodic perturbation, while in our case it occurs in a fully integrable autonomous system.

  2. 2.

    Here the notations \(I_{z},{\vartheta _{z}},\) and \(\omega _{z}\) stand for \(J,{\vartheta },\) and \(\omega ,\) respectively, used in (5.17).

  3. 3.

    We should note that in some cases the Chirikov criterion cannot be applied to study onset of global chaos. Particularly, since it uses the pendulum approximation (see Sect. 8.1.1) it is not valid near the region with small or zero shears \(dq/d\psi \) (see, the article “Chirikov criterion” by Shepelyansky 2009, and also Sect. 8.4).

  4. 4.

    There are rigorous approaches to determine the onset of chaos: the Greene’s residue criterion conjectured by Greene (1979) and the renormalization procedure developed by Escande (1982) and MacKay (1983) (see also Lichtenberg and Lieberman (1992), MacKay (1992), Elskens and Escande (2003)).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Sadrilla Abdullaev .

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Abdullaev, S. (2014). Onset of Dynamical Chaos: Physical Aspects. In: Magnetic Stochasticity in Magnetically Confined Fusion Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 78. Springer, Cham.

Download citation