Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method

The Classical Action as a Tool to Visualise the Phase Space of Hamiltonian Systems

“…In this two-dof integrable system, M x S is the Lagrangian descriptor associated with the elliptic motion and M y S is the Lagrangian descriptor associated with the hyperbolic motion. Meanwhile, S x + , S x − , S y + , and S y − are the actions that form those Lagrangian descriptors, like in the Equation (10). For the sake of simplicity, we consider the extremes of the integration interval t 0 = 0 and τ − = 0 for the next analytical calculations.…”

“…Some remarkable examples of the statistical approach based on time series analysis are Renormalization Group [1], Mutual Information [2,3], and Multifractal Metrics [4,5]. In the geometrical approach, new tools have been developed to study the phase space structure of multidimensional systems like Fast Lyapunov Exponents [6,7], Mean Exponential Growth Factor of Nearby Orbits [8], Smaller Alignment Indices, generalised Alignment Indices [9,10], Determinant of Scattering Functions [11,12], Delay Time [13], Shannon Entropy [14], Birkhof Averages [15], and based on Geometric properties of Hamiltonian systems [16,17]. Those phase space structure indicators are scalar fields constructed with the trajectories of the system.…”

Introduction to special issue: Chaos Indicators, Phase Space and Chemical Reaction Dynamics

“…The indicators are an alternative to studying some properties of high dimensional systems where it is not easy the visualisation of their phase space. The work [17] is a study of properties of trajectories of the nonlinear disordered Klein -Gordon lattice chain in one spatial dimension using GALI. This indicator can characterise the behaviour of regular and chaotic trajectories in an efficient way.…”

Frequency map analysis of spatiotemporal chaos in the nonlinear disordered Klein-Gordon lattice

“…The nonlinear disordered Klein-Gordon (DKG) lattice of coupled anharmonic oscillators has been extensively used in studies of the effect of nonlinearity on the energy propagation in disordered media, mainly in one (1D) [Flach et al, 2009;Skokos et al, 2009;Laptyeva et al, 2010;Flach, 2010;Bodyfelt et al, 2011a,b;Skokos et al, 2013;Antonopoulos et al, 2014Antonopoulos et al, , 2017Senyange & Skokos, 2021] but also in two (2D) [Laptyeva et al, 2012[Laptyeva et al, , 2014Many Manda et al, 2020] spatial dimensions. In these studies numerical evidences of the destruction of 'Anderson localization' [Anderson, 1958;Kramer & MacKinnon, 1993] (i.e.…”