ODESystem

struct ODESystem <: ModelingToolkit.AbstractODESystem

A system of ordinary differential equations.

Fields

• tag

: tag: a tag for the system. If two system have the same tag, then they are structurally identical.

• eqs

: The ODEs defining the system.

• iv

: Independent variable.

• states

: Dependent (state) variables. Must not contain the independent variable.

N.B.: If torn_matching !== nothing, this includes all variables. Actual
ODE states are determined by the SelectedState() entries in `torn_matching`.

• ps

: Parameter variables. Must not contain the independent variable.

• tspan

: Time span.

• var_to_name

: Array variables.

• ctrls

: Control parameters (some subset of ps).

• observed

: Observed states.

• tgrad

: Time-derivative matrix. Note: this field will not be defined until calculate_tgrad is called on the system.

• jac

: Jacobian matrix. Note: this field will not be defined until calculate_jacobian is called on the system.

• ctrl_jac

: Control Jacobian matrix. Note: this field will not be defined until calculate_control_jacobian is called on the system.

• Wfact

: Wfact matrix. Note: this field will not be defined until generate_factorized_W is called on the system.

• Wfact_t

: Wfact_t matrix. Note: this field will not be defined until generate_factorized_W is called on the system.

• name

: Name: the name of the system

• systems

: systems: The internal systems. These are required to have unique names.

• defaults

: defaults: The default values to use when initial conditions and/or parameters are not supplied in ODEProblem.

• torn_matching

: torn_matching: Tearing result specifying how to solve the system.

• connector_type

: connector_type: type of the system

• preface

: preface: inject assignment statements before the evaluation of the RHS function.

• continuous_events

: continuous_events: A Vector{SymbolicContinuousCallback} that model events. The integrator will use root finding to guarantee that it steps at each zero crossing.

• discrete_events

: discrete_events: A Vector{SymbolicDiscreteCallback} that models events. Symbolic analog to SciMLBase.DiscreteCallback that exectues an affect when a given condition is true at the end of an integration step.

• metadata

: metadata: metadata for the system, to be used by downstream packages.

• tearing_state

: tearing_state: cache for intermediate tearing state

• substitutions

: substitutions: substitutions generated by tearing.

• complete

: complete: if a model sys is complete, then sys.x no longer performs namespacing.

• discrete_subsystems

: discrete_subsystems: a list of discrete subsystems

Example

using ModelingToolkit

@parameters σ ρ β
@variables t x(t) y(t) z(t)
D = Differential(t)

eqs = [D(x) ~ σ*(y-x),
       D(y) ~ x*(ρ-z)-y,
       D(z) ~ x*y - β*z]

@named de = ODESystem(eqs,t,[x,y,z],[σ,ρ,β],tspan=(0, 1000.0))

ode_order_lowering(sys::ODESystem) -> Any

Takes a Nth order ODESystem and returns a new ODESystem written in first order form by defining new variables which represent the N-1 derivatives.

dae_index_lowering(sys::ODESystem; kwargs...) -> ODESystem

Perform the Pantelides algorithm to transform a higher index DAE to an index 1 DAE. kwargs are forwarded to pantelides!. End users are encouraged to call structural_simplify instead, which calls this function internally.

liouville_transform(sys::ModelingToolkit.AbstractODESystem)

Generates the Liouville transformed set of ODEs, which is the original ODE system with a new variable trJ appended, corresponding to the -tr(Jacobian). This variable is used for properties like uncertainty propagation from a given initial distribution density.

For example, if $u'=p*u$ and p follows a probability distribution $f(p)$, then the probability density of a future value with a given choice of $p$ is computed by setting the inital trJ = f(p), and the final value of trJ is the probability of $u(t)$.

Example:

using ModelingToolkit, OrdinaryDiffEq, Test

@parameters t α β γ δ
@variables x(t) y(t)
D = Differential(t)

eqs = [D(x) ~ α*x - β*x*y,
       D(y) ~ -δ*y + γ*x*y]

sys = ODESystem(eqs)
sys2 = liouville_transform(sys)
@variables trJ

u0 = [x => 1.0,
      y => 1.0,
      trJ => 1.0]

prob = ODEProblem(sys2,u0,tspan,p)
sol = solve(prob,Tsit5())

Where sol[3,:] is the evolution of trJ over time.

Sources:

Probabilistic Robustness Analysis of F-16 Controller Performance: An Optimal Transport Approach

Abhishek Halder, Kooktae Lee, and Raktim Bhattacharya https://abhishekhalder.bitbucket.io/F16ACC2013Final.pdf

function DiffEqBase.ODEFunction{iip}(sys::AbstractODESystem, dvs = states(sys),
                                     ps = parameters(sys);
                                     version = nothing, tgrad=false,
                                     jac = false,
                                     sparse = false,
                                     kwargs...) where {iip}

Create an ODEFunction from the ODESystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

function DiffEqBase.ODEProblem{iip}(sys::AbstractODESystem,u0map,tspan,
                                    parammap=DiffEqBase.NullParameters();
                                    version = nothing, tgrad=false,
                                    jac = false,
                                    checkbounds = false, sparse = false,
                                    simplify=false,
                                    linenumbers = true, parallel=SerialForm(),
                                    kwargs...) where iip

Generates an ODEProblem from an ODESystem and allows for automatically symbolically calculating numerical enhancements.

function SciMLBase.SteadyStateProblem(sys::AbstractODESystem,u0map,
                                    parammap=DiffEqBase.NullParameters();
                                    version = nothing, tgrad=false,
                                    jac = false,
                                    checkbounds = false, sparse = false,
                                    linenumbers = true, parallel=SerialForm(),
                                    kwargs...) where iip

Generates an SteadyStateProblem from an ODESystem and allows for automatically symbolically calculating numerical enhancements.

function ODEFunctionExpr{iip}(sys::AbstractODESystem, dvs = states(sys),
                                     ps = parameters(sys);
                                     version = nothing, tgrad=false,
                                     jac = false,
                                     sparse = false,
                                     kwargs...) where {iip}

Create a Julia expression for an ODEFunction from the ODESystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

function DAEFunctionExpr{iip}(sys::AbstractODESystem, dvs = states(sys),
                                     ps = parameters(sys);
                                     version = nothing, tgrad=false,
                                     jac = false,
                                     sparse = false,
                                     kwargs...) where {iip}

Create a Julia expression for an ODEFunction from the ODESystem. The arguments dvs and ps are used to set the order of the dependent variable and parameter vectors, respectively.

function SciMLBase.SteadyStateProblemExpr(sys::AbstractODESystem,u0map,
                                    parammap=DiffEqBase.NullParameters();
                                    version = nothing, tgrad=false,
                                    jac = false,
                                    checkbounds = false, sparse = false,
                                    skipzeros=true, fillzeros=true,
                                    linenumbers = true, parallel=SerialForm(),
                                    kwargs...) where iip

Generates a Julia expression for building a SteadyStateProblem from an ODESystem and allows for automatically symbolically calculating numerical enhancements.