ChaosTools.jl

ChaosTools.jl

A Julia module that offers various tools for analysing nonlinear dynamics and chaotic behaviour. It can be used as a standalone package, or as part of DynamicalSystems.jl.

To install it, run import Pkg; Pkg.add("ChaosTools").

All further information is provided in the documentation, which you can either find online or build locally by running the docs/make.jl file.

ChaosTools.jl is the jack-of-all-trades package of the DynamicalSystems.jl library: methods that are not extensive enough to be a standalone package are added here. You should see the full DynamicalSystems.jl library for other packages that may contain functionality you are looking for but did not find in ChaosTools.jl.

DynamicalSystem

DynamicalSystem is an abstract supertype encompassing all concrete implementations of what counts as a "dynamical system" in the DynamicalSystems.jl library.

All concrete implementations of DynamicalSystem can be iteratively evolved in time via the step! function. Hence, most library functions that evolve the system will mutate its current state and/or parameters. See the documentation online for implications this has on for parallelization.

DynamicalSystem is further separated into two abstract types: ContinuousTimeDynamicalSystem, DiscreteTimeDynamicalSystem. The simplest and most common concrete implementations of a DynamicalSystem are DeterministicIteratedMap or CoupledODEs.

Description

A ds::DynamicalSystemrepresentes a flow Φ in a state space. It mainly encapsulates three things:

1. A state, typically referred to as u, with initial value u0. The space that u occupies is the state space of ds and the length of u is the dimension of ds (and of the state space).
2. A dynamic rule, typically referred to as f, that dictates how the state evolves/changes with time when calling the step! function. f is a standard Julia function, see below.
3. A parameter container p that parameterizes f. p can be anything, but in general it is recommended to be a type-stable mutable container.

In sort, any set of quantities that change in time can be considered a dynamical system, however the concrete subtypes of DynamicalSystem are much more specific in their scope. Concrete subtypes typically also contain more information than the above 3 items.

In this scope dynamical systems have a known dynamic rule f defined as a standard Julia function. Observed or measured data from a dynamical system are represented using StateSpaceSet and are finite. Such data are obtained from the trajectory function or from an experimental measurement of a dynamical system with an unknown dynamic rule.

Construction instructions on f and u

Most of the concrete implementations of DynamicalSystem, with the exception of ArbitrarySteppable, have two ways of implementing the dynamic rule f, and as a consequence the type of the state u. The distinction is done on whether f is defined as an in-place (iip) function or out-of-place (oop) function.

• oop : fmust be in the form f(u, p, t) -> out which means that given a state u::SVector{<:Real} and some parameter container p it returns the output of f as an SVector{<:Real} (static vector).
• iip : fmust be in the form f!(out, u, p, t) which means that given a state u::AbstractArray{<:Real} and some parameter container p, it writes in-place the output of f in out::AbstractArray{<:Real}. The function must return nothing as a final statement.

t stands for current time in both cases. iip is suggested for systems with high dimension and oop for small. The break-even point is between 10 to 100 dimensions but should be benchmarked on a case-by-case basis as it depends on the complexity of f.

API

The API that the interface of DynamicalSystem employs is the functions listed below. Once a concrete instance of a subtype of DynamicalSystem is obtained, it can quieried or altered with the following functions.

The main use of a concrete dynamical system instance is to provide it to downstream functions such as lyapunovspectrum from ChaosTools.jl or basins_of_attraction from Attractors.jl. A typical user will likely not utilize directly the following API, unless when developing new algorithm implementations that use dynamical systems.

API - information

• ds(t) with ds an instance of DynamicalSystem: return the state of ds at time t. For continuous time systems this interpolates and extrapolates, while for discrete time systems it only works if t is the current time.
• current_state
• initial_state
• current_parameters
• initial_parameters
• isdeterministic
• isdiscretetime
• dynamic_rule
• current_time
• initial_time
• isinplace
• succesful_step

API - alter status

• reinit!
• set_state!
• set_parameter!
• set_parameters!

DeterministicIteratedMap <: DynamicalSystem
DeterministicIteratedMap(f, u0, p = nothing; t0 = 0)

A deterministic discrete time dynamical system defined by an iterated map as follows:

\[\vec{u}_{n+1} = \vec{f}(\vec{u}_n, p, n)\]

An alias for DeterministicIteratedMap is DiscreteDynamicalSystem.

Optionally configure the parameter container p and initial time t0.

For construction instructions regarding f, u0 see DynamicalSystem.

CoupledODEs <: ContinuousTimeDynamicalSystem
CoupledODEs(f, u0 [, p]; diffeq, t0 = 0.0)

A deterministic continuous time dynamical system defined by a set of coupled ordinary differential equations as follows:

\[\frac{d\vec{u}}{dt} = \vec{f}(\vec{u}, p, t)\]

An alias for CoupledODE is ContinuousDynamicalSystem.

Optionally provide the parameter container p and initial time as keyword t0.

For construction instructions regarding f, u0 see DynamicalSystem.

DifferentialEquations.jl keyword arguments and interfacing

The ODEs are evolved via the solvers of DifferentialEquations.jl. When initializing a CoupledODEs, you can specify the solver that will integrate f in time, along with any other integration options, using the diffeq keyword. For example you could use diffeq = (abstol = 1e-9, reltol = 1e-9). If you want to specify a solver, do so by using the keyword alg, e.g.: diffeq = (alg = Tsit5(), reltol = 1e-6). This requires you to have been first using OrdinaryDiffEq to access the solvers. The default diffeq is:

(alg = Tsit5(; stagelimiter! = triviallimiter!, steplimiter! = triviallimiter!, thread = static(false),), abstol = 1.0e-6, reltol = 1.0e-6)

diffeq keywords can also include callback for event handling , however the majority of downstream functions in DynamicalSystems.jl assume that f is differentiable.

The convenience constructor CoupledODEs(prob::ODEProblem, diffeq) and CoupledODEs(ds::CoupledODEs, diffeq) are also available.

Dev note: CoupledODEs is a light wrapper of ODEIntegrator from DifferentialEquations.jl. The integrator is available as the field integ, and the ODEProblem is integ.sol.prob. The convenience syntax ODEProblem(ds::CoupledODEs, tspan = (t0, Inf)) is available.

CoreDynamicalSystem

Union type meaning either DeterministicIteratedMap or CoupledODEs, which are the core systems whose dynamic rule f is known analytically.

This type is used for deciding whether a creation of a TangentDynamicalSystem is possible or not.

StroboscopicMap <: DiscreteTimeDynamicalSystem
StroboscopicMap(ds::CoupledODEs, period::Real) → smap
StroboscopicMap(period::Real, f, u0, p = nothing; kwargs...)

A discrete time dynamical system that produces iterations of a time-dependent (non-autonomous) CoupledODEs system exactly over a given period. The second signature first creates a CoupledODEs and then calls the first.

StroboscopicMap follows the DynamicalSystem interface. In addition, the function set_period!(smap, period) is provided, that sets the period of the system to a new value (as if it was a parameter). As this system is in discrete time, current_time and initial_time are integers. The initial time is always 0, because current_time counts elapsed periods. Call these functions on the parent of StroboscopicMap to obtain the corresponding continuous time. In contrast, reinit! expects t0 in continuous time.

The convenience constructor

StroboscopicMap(T::Real, f, u0, p = nothing; diffeq, t0 = 0) → smap

is also provided.

See also PoincareMap.

PoincareMap <: DiscreteTimeDynamicalSystem
PoincareMap(ds::CoupledODEs, plane; kwargs...) → pmap

A discrete time dynamical system that produces iterations over the Poincaré map^{[DatserisParlitz2022]} of the given continuous time ds. This map is defined as the sequence of points on the Poincaré surface of section, which is defined by the plane argument.

See also StroboscopicMap, poincaresos.

Keyword arguments

• direction = -1: Only crossings with sign(direction) are considered to belong to the surface of section. Negative direction means going from less than $b$ to greater than $b$.
• u0 = nothing: Specify an initial state.
• rootkw = (xrtol = 1e-6, atol = 1e-8): A NamedTuple of keyword arguments passed to find_zero from Roots.jl.
• Tmax = 1e3: The argument Tmax exists so that the integrator can terminate instead of being evolved for infinite time, to avoid cases where iteration would continue forever for ill-defined hyperplanes or for convergence to fixed points, where the trajectory would never cross again the hyperplane. If during one step! the system has been evolved for more than Tmax, then step!(pmap) will terminate and error.

Description

The Poincaré surface of section is defined as sequential transversal crossings a trajectory has with any arbitrary manifold, but here the manifold must be a hyperplane. PoincareMap iterates over the crossings of the section.

If the state of ds is $\mathbf{u} = (u_1, \ldots, u_D)$ then the equation defining a hyperplane is

\[a_1u_1 + \dots + a_Du_D = \mathbf{a}\cdot\mathbf{u}=b\]

where $\mathbf{a}, b$ are the parameters of the hyperplane.

In code, plane can be either:

• A Tuple{Int, <: Real}, like (j, r): the plane is defined as when the jth variable of the system equals the value r.
• A vector of length D+1. The first D elements of the vector correspond to $\mathbf{a}$ while the last element is $b$.

PoincareMap uses ds, higher order interpolation from DifferentialEquations.jl, and root finding from Roots.jl, to create a high accuracy estimate of the section.

PoincareMap follows the DynamicalSystem interface with the following adjustments:

1. dimension(pmap) == dimension(ds), even though the Poincaré map is effectively 1 dimension less.
2. Like StroboscopicMap time is discrete and counts the iterations on the surface of section. initial_time is always 0 and current_time is current iteration number.
3. A new function current_crossing_time returns the real time corresponding to the latest crossing of the hyperplane, which is what the current_state(ds) corresponds to as well.
4. For the special case of plane being a Tuple{Int, <:Real}, a special reinit! method is allowed with input state of length D-1 instead of D, i.e., a reduced state already on the hyperplane that is then converted into the D dimensional state.

Example

using DynamicalSystemsBase
ds = Systems.rikitake(zeros(3); μ = 0.47, α = 1.0)
pmap = poincaremap(ds, (3, 0.0))
step!(pmap)
next_state_on_psos = current_state(pmap)

TangentDynamicalSystem <: DynamicalSystem
TangentDynamicalSystem(ds::CoreDynamicalSystem; kwargs...)

A dynamical system that bundles the evolution of ds (which must be an CoreDynamicalSystem) and k deviation vectors that are evolved according to the dynamics in the tangent space (also called linearized dynamics or the tangent dynamics).

The state of dsmust be an AbstractVector for TangentDynamicalSystem.

TangentDynamicalSystem follows the DynamicalSystem interface with the following adjustments:

• reinit! takes an additional keyword Q0 (with same default as below)
• The additional functions current_deviations and set_deviations! are provided for the deviation vectors.

Keyword arguments

• k or Q0: Q0 represents the initial deviation vectors (each column = 1 vector). If k::Int is given, a matrix Q0 is created with the first k columns of the identity matrix. Otherwise Q0 can be given directly as a matrix. It must hold that size(Q, 1) == dimension(ds). You can use orthonormal for random orthonormal vectors. By default k = dimension(ds) is used.
• u0 = current_state(ds): Starting state.
• J and J0: See section "Jacobian" below.

Description

Let $u$ be the state of ds, and $y$ a deviation (or perturbation) vector. These two are evolved in parallel according to

\[\begin{array}{rcl} \frac{d\vec{x}}{dt} &=& f(\vec{x}) \\ \frac{dY}{dt} &=& J_f(\vec{x}) \cdot Y \end{array} \quad \mathrm{or}\quad \begin{array}{rcl} \vec{x}_{n+1} &=& f(\vec{x}_n) \\ Y_{n+1} &=& J_f(\vec{x}_n) \cdot Y_n. \end{array}\]

for continuous or discrete time respectively. Here $f$ is the dynamic_rule(ds) and $J_f$ is the Jacobian of $f$.

Jacobian

The keyword J provides the Jacobian function. It must be a Julia function in the same form as f, the dynamic_rule. Specifically, J(u, p, n) -> M::SMatrix for the out-of-place version or J(M, u, p, n) for the in-place version acting in-place on M. in both cases M is a matrix whose columns are the deviation vectors.

By default J = nothing. In this case J is constructed automatically using the module ForwardDiff, hence its limitations also apply here. Even though ForwardDiff is very fast, depending on your exact system you might gain significant speed-up by providing a hand-coded Jacobian and so it is recommended. Additionally, automatic and in-place Jacobians cannot be time dependent.

The keyword J0 allows you to pass an initialized Jacobian matrix J0. This is useful for large in-place systems where only a few components of the Jacobian change during the time evolution. J0 can be a sparse or any other matrix type. If not given, a matrix of zeros is used. J0 is ignored for out of place systems.

ParallelDynamicalSystem <: DynamicalSystem
ParallelDynamicalSystem(ds::DynamicalSystem, states::Vector{<:AbstractArray})

A struct that evolves several states of a given dynamical system in parallel at exactly the same times. Useful when wanting to evolve several different trajectories of the same system while ensuring that they share parameters and time vector.

This struct follows the DynamicalSystem interface with the following adjustments:

• The function current_state is called as current_state(pds, i::Int = 1) which returns the ith state. Same for initial_state.
• Similarly, set_state! obtains a third argument i::Int = 1 to set the i-th state.
• current_states and initial_states can be used to get all parallel states.
• reinit! takes in a vector of states (like states) for u.

ProjectedDynamicalSystem <: DynamicalSystem
ProjectedDynamicalSystem(ds::DynamicalSystem, projection, complete_state)

A dynamical system that represents a projection of an existing ds on a (projected) space.

The projection defines the projected space. If projection isa AbstractVector{Int}, then the projected space is simply the variable indices that projection contains. Otherwise, projection can be an arbitrary function that given the state of the original system ds, returns the state in the projected space. In this case the projected space can be equal, or even higher-dimensional, than the original.

complete_state produces the state for the original system from the projected state. complete_state can always be a function that given the projected state returns a state in the original space. However, if projection isa AbstractVector{Int}, then complete_state can also be a vector that contains the values of the remaining variables of the system, i.e., those not contained in the projected space. In this case the projected space needs to be lower-dimensional than the original.

Notice that ProjectedDynamicalSystem does not require an invertible projection, complete_state is only used during reinit!. ProjectedDynamicalSystem is in fact a rather trivial wrapper of ds which steps it as normal in the original state space and only projects as a last step, e.g., during current_state.

Examples

Case 1: project 5-dimensional system to its last two dimensions.

ds = Systems.lorenz96(5)
projection = [4, 5]
complete_state = [0.0, 0.0, 0.0] # completed state just in the plane of last two dimensions
pds = ProjectedDynamicalSystem(ds, projection, complete_state)
reinit!(pds, [0.2, 0.4])
step!(pds)
get_state(pds)

Case 2: custom projection to general functions of state.

ds = Systems.lorenz96(5)
projection(u) = [sum(u), sqrt(u[1]^2 + u[2]^2)]
complete_state(y) = repeat([y[1]/5], 5)
pds = # same as in above example...

reinit!(ds::DynamicalSystem, u = initial_state(ds); kwargs...) → ds

Reset the status of ds, so that it is as if it has be just initialized with initial state u. Practically every function of the ecosystem that evolves ds first calls this function on it. Besides the new initial state u, you can also configure the keywords t0 = initial_time(ds) and p = current_parameters(ds).

Note the default settings: the state and time are the initial, but the parameters are the current.

The special method reinit!(ds, ::Nothing; kwargs...) is also available, which does nothing and leaves the system as is. This is so that downstream functions that call reinit! can still be used without resetting the system but rather continuing from its exact current state.