Numerical approaches for investigating the chaotic behavior of ... (H. Skokos, 29/6/2023)
Channel: Academy of Athens
Date: 2023-11-09

Numerical approaches for investigating the chaotic behavior of multidimensional Hamiltonian systems

H. 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, focusing our analysis on the chaotic evolution of initially localized energy excitations in the disordered Klein-Gordon (DKG) oscillator chain in one spatial dimension. Nonlinear interactions in the system 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 [1-4]. Quantifying the strength of chaos through the computation of the maximum Lyapunov exponent (MLE, see for example [5] and references therein), we observe that the index exhibits power law decays, with different exponents for the weak and strong chaos regimes, whose values are distinct from -1 seen in the case of regular motion [6, 7]. 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 [6, 7]. Furthermore, the implementation of the SALI/GALI2 chaos indicator [4, 8-10] permits the efficient discrimination between localized and spreading chaos [11]. 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 [12], we reveal several characteristics of the dynamics related to the location of highly chaotic oscillators and the propagation of chaos [13].

References
1. Flach, Krimer, Skokos, 2009, PRL, 102, 024101
2. Skokos, Krimer, Komineas, Flach, 2009, PRE, 79, 056211
3. Laptyeva, Bodyfelt, Krimer, Skokos, Flach, 2010, EPL, 91, 30001
4. Bountis Skokos, 2012, “Complex Hamiltonian Dynamics”, Synergetics series of Springer Verlag.
5. Skokos, 2010, Lect. Notes Phys., 790, 63
6. Skokos, Gkolias, Flach, 2013, PRL, 111, 064101
7. Senyange, Many Manda, Skokos, 2018, PRE, 98, 052229
8. Skokos, 2001, J. Phys. A, 34, 10029
9. Skokos, Bountis, Antonopoulos, 2007, Physica D, 231, 30
10.Skokos, Manos, 2016, Lect. Notes Phys., 915, 129
11.Senyange, Skokos, 2022, Physica D, 432, 133154
12.Laskar, 1990, Icarus, 88, 266
13.Skokos, Gerlach, Flach, 2022, IJBC, 32, 2250074