Modeling with Stochasticity

All models with ODESystem are deterministic. SDESystem adds another element to the model: randomness. This is a stochastic differential equation which has a deterministic (drift) component and a stochastic (diffusion) component. Let's take the Lorenz equation from the first tutorial and extend it to have multiplicative noise by creating @brownian variables in the equations.

using ModelingToolkit, StochasticDiffEq
using ModelingToolkit: t_nounits as t, D_nounits as D

# Define some variables
@parameters σ ρ β
@variables x(t) y(t) z(t)
@brownian a
eqs = [D(x) ~ σ * (y - x) + 0.1a * x,
    D(y) ~ x * (ρ - z) - y + 0.1a * y,
    D(z) ~ x * y - β * z + 0.1a * z]

@mtkbuild de = System(eqs, t)

u0map = [
    x => 1.0,
    y => 0.0,
    z => 0.0
]

parammap = [
    σ => 10.0,
    β => 26.0,
    ρ => 2.33
]

prob = SDEProblem(de, u0map, (0.0, 100.0), parammap)
sol = solve(prob, LambaEulerHeun())

retcode: Success
Interpolation: 1st order linear
t: 2438-element Vector{Float64}:
   0.0
   5.106983219771961e-7
   1.0852339342015416e-6
   1.7315864979539305e-6
   2.458733132175368e-6
   3.2767730956744856e-6
   4.197068054610992e-6
   5.232399883414562e-6
   6.397148190818579e-6
   7.707490036648098e-6
   ⋮
  99.76011142893856
  99.79338170109914
  99.83081075727979
  99.87291844548304
  99.90063877565261
  99.93182414709338
  99.96690768996424
  99.9885992355981
 100.0
u: 2438-element Vector{Vector{Float64}}:
 [1.0, 0.0, 0.0]
 [0.9999475332012283, 1.1898673915221327e-6, 3.0383094182944587e-13]
 [1.0000090510122377, 2.528630297436693e-6, 1.3720333866760973e-12]
 [1.000031435839958, 4.0347548056177184e-6, 3.493226220668491e-12]
 [1.00004324588966, 5.729159330790694e-6, 7.043252643182728e-12]
 [1.0000320956606195, 7.635238938531201e-6, 1.2509538458335753e-11]
 [1.0001249796200822, 9.780575470213636e-6, 2.0525213281515313e-11]
 [1.00023159759304, 1.2194602345196305e-5, 3.190527348136014e-11]
 [1.00051596934932, 1.4913475154958532e-5, 4.7708347677359926e-11]
 [1.0006149508671258, 1.7970118444815603e-5, 6.926927522225845e-11]
 ⋮
 [5.6811288592891245, 5.68209902134136, 1.259681180516766]
 [5.726107076899268, 5.738348273371732, 1.2654214233383025]
 [5.784729611252468, 5.8030833702839075, 1.2837507921471758]
 [6.0167017558166265, 6.030821296202833, 1.3673873756234085]
 [5.980109977357129, 5.984322282632437, 1.3679191228258072]
 [6.130414465613056, 6.123560318758714, 1.4201979747562499]
 [5.916278999262875, 5.899593215460238, 1.358514778772065]
 [5.872807522420231, 5.857679642723908, 1.3399201069510303]
 [5.902303464345322, 5.888186624012328, 1.3437290066799519]