GAIO.jl

Currently under development -- breaking changes may occur at any time ...

About

GAIO.jl is a Julia package for set oriented computations. Sets are represented by collections of boxes (i.e. cubes). GAIO.jl provides algorithms for

• dynamical systems
  • invariant sets (maximal invariant set, chain recurrent set, (relative) attractor, (un-)stable manifold)
  • almost invariant and coherent sets
  • finite time Lyapunov exponents
  • entropy and box dimension
• root finding problems
• multi-objective optimization problems
• computing implicitly defined manifolds

Documentation

Installation

The package requires Julia 1.10 or later. In Julia's package manager, type

pkg> add https://github.com/gaioguys/GAIO.jl.git

Getting started

The following script computes the attractor of the Hénon map within the box [-3,3)²:

using GAIO
using Plots: plot

center, radius = (0,0), (3,3)
Q = Box(center, radius)                       # domain for the computation
P = BoxGrid(Q, (2, 2))                   # 2 x 2 partition of Q
S = cover(P, :)                               # Set of all boxes in P

f((x,y)) = (1 - 1.4*x^2 + y, 0.3*x)           # the Hénon map
F = BoxMap(f, P)                              # ... turned into a map on boxes

R = relative_attractor(F, S, steps = 18)      # subdivison algorithm computing
                                              # the attractor relative to Q
plot(R)                                       # plot R

For more examples, see the examples\ folder.

License

See LICENSE for GAIO.jl's licensing information.

References

• Dellnitz, M.; Froyland, G.; Junge, O.: The algorithms behind GAIO - Set oriented numerical methods for dynamical systems, in: B. Fiedler (ed.): Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, 2001.
• Dellnitz, M.; Junge, O.: On the approximation of complicated dynamical behavior, SIAM Journal on Numerical Analysis, 36 (2), 1999.
• Dellnitz, M.; Hohmann, A.: A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik 75, pp. 293-317, 1997.