Abstract
The theory of chaos originated in the works by H. Poincaré is related to the problem of integrability of Hamiltonian systems under small periodic perturbations, i.e., to the fundamental problem of dynamics (see, Sect. 6.2.1). Chaos of magnetic field lines in magnetic fusion devices is an excellent example of dynamical chaos in Hamiltonian systems with one-and-half-degrees of freedom. In this chapter we discuss the some mathematical aspects of the onset of chaos in Hamiltonian systems.
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Notes
- 1.
Arnold (1963a) gave the proof of this theorem under assumption that the Hamiltonian \(H(I,\vartheta )\) is an analytical function of its variables \(I,\vartheta \), while Moser had proved it for smooth Hamiltonian functions of class \(C^r\), \(r>2N\), i.e., for the functions with a finite number \(r\) of continuous derivatives. Initially, for the case \(N=1\) Moser proved that \(r \ge 333\), but later Herman showed that it is sufficient \(r\ge 3\).
- 2.
The perturbation function \(\psi _1(\psi _t,\vartheta , \varphi )\) should belong to the class \(C^k\), \(k\ge 2\). This condition on the smoothness of perturbations, perhaps, is satisfied for any magnetic perturbations.
- 3.
There are also topological obstacles to the integrability of Hamiltonian systems related with its complex phase space structure (Kozlov 1983; see also Kozlov 1996; Arnold et al. 2006).
- 4.
As asserted in Arnold et al. (2006) “the existence of infinitely many periodic solutions can be derived from Kolmogorov’s theorem on the conservation conditionally periodic motions and the Poincaré–Birkhoff geometric theorem” (see also Moser 1973).
- 5.
This formula has been first derived by Melnikov (1963b) (see also Guckenheimer and Holmes 1983; Arnold et al. 2006; Treschev 2006).
- 6.
There are examples of Hamiltonian systems in which stable periodic orbits and the KAM stability islands appear even inside the lobes, the regions enclosed by the stable and unstable manifolds (Abdullaev 2000; Simó and Treschev 2008; Simó and Vieiro 2011). Particularly, these islands are responsible for the regular motions (even flights) of orbits in areas located deep inside the stochastic layer (Abdullaev 2000).
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Abdullaev, S. (2014). Onset of Dynamical Chaos: Mathematical Aspects. In: Magnetic Stochasticity in Magnetically Confined Fusion Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-01890-4_7
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DOI: https://doi.org/10.1007/978-3-319-01890-4_7
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