Dynamical Systems

Currently a system in DynamicalSystems.jl can be either continuous

or discrete

where $p$ contains the parameters of the system. In addition to the above equations of motion, information about the Jacobian of the system is also part of a "dynamical system".

Keep in mind that almost all functions of DynamicalSystems.jl assume that $\vec{f}$ is differentiable!

Creating a Dynamical System

#DynamicalSystemsBase.DynamicalSystem — Type.

DynamicalSystem

The central structure of DynamicalSystems.jl. All functions of the suite that can use known equations of motion expect an instance of this type.

Constructing a DynamicalSystem

DiscreteDynamicalSystem(eom, state, p [, jacobian [, J0]]; t0::Int = 0)
ContinuousDynamicalSystem(eom, state, p [, jacobian [, J0]]; t0 = 0.0)

with eom the equations of motion function (see below). p is a parameter container, which we highly suggest to use a mutable object like Array, LMArray or a dictionary. Pass nothing in the place of p if your system does not have parameters.

t0, J0 allow you to choose the initial time and provide an initialized Jacobian matrix. See CDS_KWARGS for the default options used to evolve continuous systems (through OrdinaryDiffEq).

Equations of motion

The are two "versions" for DynamicalSystem, depending on whether the equations of motion (eom) are in-place (iip) or out-of-place (oop). Here is how to define them:

• oop : The eommust be in the form eom(x, p, t) -> SVector which means that given a state x::SVector and some parameter container p it returns an SVector (from the StaticArrays module) containing the next state.
• iip : The eommust be in the form eom!(xnew, x, p, t) which means that given a state x::Vector and some parameter container p, it writes in-place the new state in xnew.

t stands for time (integer for discrete systems). iip is suggested for big systems, whereas oop is suggested for small systems. The break-even point at around 100 dimensions, and for using functions that use the tangent space (like e.g. lyapunovs or gali), the break-even point is at around 10 dimensions.

The constructor deduces automatically whether eom is iip or oop. It is not possible however to deduce whether the system is continuous or discrete just from the equations of motion, hence the 2 constructors.

Jacobian

The optional argument jacobian for the constructors is a function and (if given) must also be of the same form as the eom, jacobian(x, p, n) -> SMatrix for the out-of-place version and jacobian!(xnew, x, p, n) for the in-place version.

If jacobian is not given, it is constructed automatically using the module ForwardDiff. It is heavily advised to provide a Jacobian function, as it gives multiple orders of magnitude speedup.

Interface to DifferentialEquations.jl

Continuous systems are solved using DifferentialEquations.jl. The following two interfaces are provided:

ContinuousDynamicalSystem(prob::ODEProblem [, jacobian [, J0]])
ODEProblem(continuous_dynamical_system, tspan, args...)

where in the second case args stands for the standard extra arguments of ODEProblem: callback, mass_matrix.

If you want to use callbacks with tangent_integrator or parallel_integrator, then invoke them with extra arguments as shown in the Advanced Documentation.

Relevant Functions

trajectory, set_parameter!.

Definition Table

Here is a handy table that summarizes in what form should be the functions required for the equations of motion and the Jacobian, for each system type:

General Functions

The following functions are defined for convenience for any dynamical system:

#DynamicalSystemsBase.dimension — Function.

dimension(thing) -> D

Return the dimension of the thing, in the sense of state-space dimensionality.

#DynamicalSystemsBase.jacobian — Function.

jacobian(ds::DynamicalSystem, u = ds.u0, t = ds.t0)

Return the jacobian of the system at u, at t.

#DynamicalSystemsBase.set_parameter! — Function.

set_parameter!(ds::DynamicalSystem, index, value)
set_parameter!(ds::DynamicalSystem, values)

Change one or many parameters of the system by setting p[index] = value in the first case and p .= values in the second.

The same function also works for any integrator.

Examples

Continuous, out-of-place

Let's see an example for a small system, which is a case where out-of-place equations of motion are preferred.

using DynamicalSystems # also exports relevant StaticArrays names
# Lorenz system
# Equations of motion:
@inline @inbounds function loop(u, p, t)
    σ = p[1]; ρ = p[2]; β = p[3]
    du1 = σ*(u[2]-u[1])
    du2 = u[1]*(ρ-u[3]) - u[2]
    du3 = u[1]*u[2] - β*u[3]
    return SVector{3}(du1, du2, du3)
end
# Jacobian:
@inline @inbounds function loop_jac(u, p, t)
    σ, ρ, β = p
    J = @SMatrix [-σ  σ  0;
    ρ - u[3]  (-1)  (-u[1]);
    u[2]   u[1]  -β]
    return J
end

ds = ContinuousDynamicalSystem(loop, rand(3), [10.0, 28.0, 8/3], loop_jac)

3-dimensional continuous dynamical system
 state:     [0.068248, 0.828095, 0.0743729]
 e.o.m.:    loop
 in-place?  false
 jacobian:  loop_jac

Discrete, in-place

The following example is only 2-dimensional, and thus once again it is "correct" to use out-of-place version with SVector. For the sake of example though, we use the in-place version.

# Henon map.
# equations of motion:
function hiip(dx, x, p, n)
    dx[1] = 1.0 - p[1]*x[1]^2 + x[2]
    dx[2] = p[2]*x[1]
    return
end
# Jacobian:
function hiip_jac(J, x, p, n)
    J[1,1] = -2*p[1]*x[1]
    J[1,2] = 1.0
    J[2,1] = p[2]
    J[2,2] = 0.0
    return
end
ds = DiscreteDynamicalSystem(hiip, zeros(2), [1.4, 0.3], hiip_jac)

2-dimensional discrete dynamical system
 state:     [0.0, 0.0]
 e.o.m.:    hiip
 in-place?  true
 jacobian:  hiip_jac

Or, if you don't want to write a Jacobian and want to use the auto-differentiation capabilities of DynamicalSystems.jl, which use the module ForwardDiff:

ds = DiscreteDynamicalSystem(hiip, zeros(2), [1.4, 0.3])

2-dimensional discrete dynamical system
 state:     [0.0, 0.0]
 e.o.m.:    hiip
 in-place?  true
 jacobian:  ForwardDiff

Complex Example

In this example we will go through the implementation of the coupled standard maps from our Predefined Systems. It is the most complex implementation and takes full advantage of the flexibility of the constructors. The example will use a Functor as equations of motion, as well as a sparse matrix for the Jacobian.

Coupled standard maps is a big mapping that can have arbitrary number of equations of motion, since you can couple Nstandard maps which are 2D maps, like:

To model this, we will make a dedicated struct, which is parameterized on the number of coupled maps:

struct CoupledStandardMaps{N}
    idxs::SVector{N, Int}
    idxsm1::SVector{N, Int}
    idxsp1::SVector{N, Int}
end

(what these fields are will become apparent later)

We initialize the struct with the amount of standard maps we want to couple, and we also define appropriate parameters:

M = 5  # couple number
u0 = 0.001rand(2M) #initial state
ks = 0.9ones(M) # nonlinearity parameters
Γ = 1.0 # coupling strength
p = (ks, Γ) # parameter container

# Create struct:
SV = SVector{M, Int}
idxs = SV(1:M...) # indexes of thetas
idxsm1 = SV(circshift(idxs, +1)...)  #indexes of thetas - 1
idxsp1 = SV(circshift(idxs, -1)...)  #indexes of thetas + 1
# So that:
# x[i] ≡ θᵢ
# x[[idxsp1[i]]] ≡ θᵢ+₁
# x[[idxsm1[i]]] ≡ θᵢ-₁
csm = CoupledStandardMaps{M}(idxs, idxsm1, idxsp1);

We will now use this struct to define a functor, a Type that also acts as a function.

function (f::CoupledStandardMaps{N})(xnew::AbstractVector, x, p, n) where {N}
    ks, Γ = p
    @inbounds for i in f.idxs

        xnew[i+N] = mod2pi(
            x[i+N] + ks[i]*sin(x[i]) -
            Γ*(sin(x[f.idxsp1[i]] - x[i]) + sin(x[f.idxsm1[i]] - x[i]))
        )

        xnew[i] = mod2pi(x[i] + xnew[i+N])
    end
    return nothing
end

We will use the samestruct to create a function for the Jacobian:

function (f::CoupledStandardMaps{M})(
    J::AbstractMatrix, x, p, n) where {M}

    ks, Γ = p
    # x[i] ≡ θᵢ
    # x[[idxsp1[i]]] ≡ θᵢ+₁
    # x[[idxsm1[i]]] ≡ θᵢ-₁
    @inbounds for i in f.idxs
        cosθ = cos(x[i])
        cosθp= cos(x[f.idxsp1[i]] - x[i])
        cosθm= cos(x[f.idxsm1[i]] - x[i])
        J[i+M, i] = ks[i]*cosθ + Γ*(cosθp + cosθm)
        J[i+M, f.idxsm1[i]] = - Γ*cosθm
        J[i+M, f.idxsp1[i]] = - Γ*cosθp
        J[i, i] = 1 + J[i+M, i]
        J[i, f.idxsm1[i]] = J[i+M, f.idxsm1[i]]
        J[i, f.idxsp1[i]] = J[i+M, f.idxsp1[i]]
    end
    return nothing
end

The only reason that this is possible, is because the eom always takes a AbstractVector as first argument, while the Jacobian always takes an AbstractMatrix. Therefore we can take advantage of multiple dispatch!

Notice in addition, that the Jacobian function accesses only half the elements of the matrix. This is intentional, and takes advantage of the fact that the other half is constant. We can leverage this further, by making the Jacobian a sparse matrix. Because the DynamicalSystem constructors allow us to give in a pre-initialized Jacobian matrix, we take advantage of that and create:

J = zeros(eltype(u0), 2M, 2M)
# Set ∂/∂p entries (they are eye(M,M))
# And they dont change they are constants
for i in idxs
    J[i, i+M] = 1
    J[i+M, i+M] = 1
end
sparseJ = sparse(J)

csm(sparseJ, u0, p, 0) # apply Jacobian to initial state

And finally, we are ready to create our dynamical system:

ds = DiscreteDynamicalSystem(csm, u0, p, csm, sparseJ)

10-dimensional discrete dynamical system
 state:       [0.000803001, 0.00092095, 0.000313022, …, 3.07769e-5, 0.000670152]
 e.o.m.:      CoupledStandardMaps
 in-place?    true
 jacobian:    CoupledStandardMaps
 parameters:  Tuple