Numerical investigation of spatiotemporal chaos in multidimensional Hamiltonian systems

Haris Skokos

Nonlinear Dynamics and Chaos Group,

Department of Mathematics and Applied Mathematics,

University of Cape Town, Rondebosch, 7701, Cape Town, South Africa

Abstract: We discuss various numerical approaches for studying the chaotic dynamics of multidimensional Hamiltonian systems. In particular, we focus our analysis on determining the characteristics of chaotic evolution of initially localized energy excitations in a Hamiltonian lattice model describing the disordered Klein-Gordon (DKG) oscillator chain in one spatial dimension. The linear modes of the system are exponentially localized by disorder and consequently, energy localization (usually called Anderson localization [1]) is observed in the absence of nonlinearity. On the other hand, nonlinear interactions result to the destruction of the initial energy localization, leading to the eventual subdiffusive spreading of wave packets in two different dynamical regimes (the so-called ‘weak’ and ‘strong chaos’ spreading regimes), which are characterized by particular power-law increases of the wave packet’s second moment and participation number [2-6].

Quantifying the strength of chaos through the computation of the maximum Lyapunov exponent (MLE, see for example [7] and references therein), we observe that the MLE exhibit a power-law decays, with different exponents for the weak and strong chaos, whose values are distinct from -1 seen in the case of regular motion [8-10]. The spatiotemporal evolution of the coordinates’ distribution of the deviation vector used to compute the MLE (the so-called deviation vector distribution – DVD) reveals that chaos is spreading through the random oscillation of localized chaotic hot spots in the excited part of the wave packet [8-10]. Furthermore, the implementation of the SALI/GALI2 chaos indicator [11-13] permits the efficient discrimination between localized and spreading chaos, with the former dominating the dynamics for lower energy values, for which the system is approaching its linear limit [14]. In addition, by computing the time variation of the fundamental frequencies of the motion of each oscillator in the lattice, i.e. the so-called frequency map analysis (FMA) technique [15-17], we reveal several characteristics of the dynamics for both the weak and strong chaos regimes [18]. More specifically, we find that in both regimes chaos is more intense at the central regions of the wave packet, where also the energy content is higher, while the oscillators at the wave packet’s edges exhibit regular motion up until the time they gain enough energy to become part of the highly excited portion of the wave packet. Furthermore, we find that in the strong chaos regime the chaotic component of the wave packet is not only more extended than in the weak chaos one, but in addition the fraction of strongly chaotic oscillators is much higher.

References

  1. Anderson, 1958, Phys. Rev., 109, 1492
  2. Flach, Krimer, Skokos, 2009, Phys. Rev. Let., 102, 024101
  3. Skokos, Krimer, Komineas, Flach, 2009, Phys. Rev. E, 79, 056211
  4. Skokos, Flach S., 2010, Phys. Rev. E, 82, 016208
  5. Laptyeva, Bodyfelt, Krimer, Skokos, Flach, 2010, Europhys. Lett., 91, 30001
  6. Bodyfelt, Laptyeva, Skokos, Krimer, Flach, 2011, Phys. Rev. E, 84, 016205
  7. Skokos, 2010, Lect. Notes Phys., 790, 63
  8. Skokos, Gkolias, Flach, 2013, Phys. Rev. Let., 111, 064101
  9. Senyange, Many Manda, Skokos, 2018, Phys. Rev. E, 98, 052229
  10. Many Manda, Senyange, Skokos, 2020, Phys. Rev. E, 101, 032206
  11. Skokos, 2001, J. Phys. A, 34, 10029
  12. Skokos, Bountis, Antonopoulos, 2007, Physica D, 231, 30
  13. Skokos, Manos, 2016, Lect. Notes Phys., 915, 129
  14. Senyange, Skokos, 2022, Physica D, 432, 133154
  15. Laskar, 1990, Icarus, 88, 266
  16. Laskar, Froeschlé, Celletti, 1992, Physica D, 56, 253
  17. Laskar, 1993, Physica D, 67, 257
  18. Skokos, Gerlach, Flach, 2022, Int. J. Bifurc. Chaos (in press), ArXiv:2112.04190