Detecting order and chaos in Hamiltonian systems by the SALI method
arXiv:nlin/0404058 · doi:10.1088/0305-4470/37/24/006
Abstract
We use the Smaller Alignment Index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behavior of the SALI for the two cases: the index fluctuates around non--zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behavior for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents , i.e. . Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of 2 and 3 degrees of freedom.
18 pages, 10 figures, accepted for publication in J. Phys. A
References in corpus (2)
Cited by in corpus (68)
- On the Implementation of the 0-1 Test for Chaos
- The Lyapunov Characteristic Exponents and their computation
- Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method
- Extremes and Recurrence in Dynamical Systems
- Lyapunov indices with two nearby trajectories in a curved spacetime
- Numerical integration of variational equations
- Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
- Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate
- A comparison of different indicators of chaos based on the deviation vectors. Application to symplectic mappings
- Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models
- Self-consistent models of cuspy triaxial galaxies with dark matter haloes
- Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems
- Stability of Simple Periodic Orbits and Chaos in a Fermi -- Pasta -- Ulam Lattice
- Nonlinear dynamics of weakly dissipative optomechanical systems
- Application of the SALI chaos detection method to accelerator mappings
- Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map
- Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice
- Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method
- Quantum chaos in the nuclear collective model: I. Classical-quantum correspondence
- Chaos detection tools: application to a self-consistent triaxial model
- Orbit classification in the meridional plane of a disk galaxy model with a spherical nucleus
- Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems
- A Hamiltonian system of three degrees of freedom with eight channels of escape: The Great Escape
- Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator
- Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model
- Comparative study of variational chaos indicators and ODEs' numerical integrators
- Weak Chaos and the "Melting Transition" in a Confined Microplasma System
- Generalized Extreme Value distribution parameters as dynamical indicators of Stability
- Orbital classification in an N-body bar
- Boxy Orbital Structures in Rotating Bar Models
- Orbital and escape dynamics in barred galaxies - II. The 3D system: Exploring the role of the normally hyperbolic invariant manifolds
- Quantifying chaos using Lagrangian descriptors
- Efficient control of accelerator maps
- Chaotic dynamics of string around charged black brane with hyperscaling violation
- Effects of chaos on the detectability of stellar streams
- Characterizing weak chaos in nonintegrable Hamiltonian systems: the fundamental role of stickiness and initial conditions
- Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
- Nonlinear dynamics and onset of chaos in a physical model of a damper pressure relief valve
- Orbit classification in an equal-mass non-spinning binary black hole pseudo-Newtonian system
- Models of cuspy triaxial stellar systems. III: The effect of velocity anisotropy on chaoticity
- The Fast Norm Vector Indicator (FNVI) method: A new dynamical parameter for detecting order and chaos in Hamiltonian systems
- Orbital and escape dynamics in barred galaxies - III. The 3D system: Correlations between the basins of escape and the NHIMs
- Determining the nature of orbits in disk galaxies with non spherical nuclei
- Kinetic versus Magnetic Chaos in Toroidal Plasmas: A systematic quantitative comparison
- Labyrinth chaos: Revisiting the elegant, chaotic and hyperchaotic walks
- Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
- Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies
- From Manifolds to Lagrangian Coherent Structures in galactic bar models
- Diffusion transitions in a 2D periodic lattice
- Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space
- Elucidating the escape dynamics of the four hill potential
- Mixing properties in the advection of passive tracers via recurrences and extreme value theory
- Symplectic integration of deviation vectors and chaos determination. Application to the Hénon-Heiles model and to the restricted three-body problem
- Comparing the escape dynamics in tidally limited star cluster models
- Effect of Multipole Moments in the Weak Field Limit of a Black Hole Plus Halo Potential
- New phenomenons in the spatial isosceles three-body problem
- Energy-recurrence Breakdown and Chaos in Disordered Fermi-Pasta-Ulam-Tsingou Lattices
- Finding how many isolating integrals of motion an orbit obeys
- Orbit classification in a disk galaxy model with a pseudo-Newtonian central black hole
- Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems
- On the Integrability and Chaos of an N=2 Maxwell-Chern-Simons-Higgs Mechanical Model
- High Order Three Part Split Symplectic Integration Schemes
- A dynamical survey of the trans-Neptunian region II.: On the nature of chaotic diffusion
- Chaotic dynamics of off-equatorial orbits around pseudo-Newtonian compact objects with dipolar halos
- Stability analysis of planetary systems via second-order Rényi entropy
- Exploring the origin, the nature and the dynamical behaviour of distant stars in galaxy models
- Nonlinear dynamics and energy transfer for two rotating dipoles in an external field: A three-dimensional analysis
- Arnold web and dynamical tunneling in a four-site Bose-Hubbard model