# ODEResults

## Description

`ODEResults`

objects store the solution information from the
integration of an `ode`

object. The `solve`

and
`solutionFcn`

object functions of `ode`

can return
`ODEResults`

objects.

You can use the `Time`

and `Solution`

properties of
an `ODEResults`

object to extract or plot the integration results.

If events are being tracked (that is, the

`ode`

object has a nonempty`EventDefinition`

property), then the`EventTime`

,`EventSolution`

, and`EventIndex`

properties contain information related to events that triggered during the integration. See`odeEvent`

for more information on detecting events during the solution process.If sensitivity analysis is performed (that is, the

`ode`

object has a nonempty`Sensitivity`

property), then the`Sensitivity`

property contains partial derivative values of the equations taken with respect to parameters. If events are also being tracked, then the`EventSensitivity`

property contains sensitivity values at the time of each event. See`odeSensitivity`

for more information on performing sensitivity analysis.

All properties of `ODEResults`

objects are read-only. You can index into
the properties to extract their values with the syntax ```
data =
obj.property
```

.

## Creation

Use `solve`

or
`solutionFcn`

to
integrate an `ode`

object and
create an `ODEResults`

object. For example, ```
S =
solve(F,0,10)
```

creates an `ODEResults`

object
`S`

.

## Properties

### Standard Results

`Time`

— Evaluation points

row vector

This property is read-only.

Evaluation points, returned as a row vector. Typically, the independent variable in an ODE problem is time, but the independent variable can represent other quantities as well.

If you solve the ODE problem using a time range with two elements

`[t0 tf]`

, then`Time`

contains the internal step values chosen by the solver.If you solve the ODE problem using a vector of times

`[t0 t1 t2 ... tf]`

, then`Time`

contains the same values.

`Solution`

— Solution at evaluation points

matrix

This property is read-only.

Solution at evaluation points, returned as a matrix. Each row
`Solution(i,:)`

is the complete solution for one component, and
each column `Solution(:,j)`

is the solution of all components at the
corresponding evaluation point `Time(j)`

.

### Event Results

`EventTime`

— Time when events triggered

row vector

This property is read-only.

Time when events triggered, returned as a row vector.
`EventSolution(:,j)`

contains the solution of all variables at time
`EventTime(j)`

, and `EventIndex(j)`

indicates
which event triggered.

`EventSolution`

— Solution at time of events

matrix

This property is read-only.

Solution at time of events, returned as a matrix.
`EventSolution`

has a number of columns equal to
`numel(EventTime)`

, and the same number of rows as
`Solution`

. `EventSolution(:,j)`

contains the
solution of all variables at time `EventTime(j)`

, and
`EventIndex(j)`

indicates which event triggered.

`EventIndex`

— Index of triggered events

row vector

This property is read-only.

Index of triggered events, returned as a row vector.
`EventSolution(:,j)`

contains the solution of all variables at time
`EventTime(j)`

, and `EventIndex(j)`

indicates
which event triggered.

### Sensitivity Analysis

`Sensitivity`

— Sensitivity analysis results

array

This property is read-only.

Sensitivity analysis results, returned as an array.
`Sensitivity`

has a number of columns equal to the number of
parameters being analyzed, and the same number of rows as
`Solution`

. The number of pages in the array is equal to the
number of time steps.

Consider an ODE system of the form

$$\begin{array}{cc}\frac{d{y}_{i}}{dt}={f}_{i}\left(t,y,p\right),& i=1,2,\mathrm{...},n\end{array}\text{}\text{}\text{}$$

The parameters `p`

can be treated as independent variables to
obtain the sensitivity functions

$$\begin{array}{ccc}{u}_{ij}=\frac{\partial {y}_{i}}{\partial {p}_{j}},& i=1,2,\mathrm{...},n,& j=1,2,\mathrm{...},m\end{array}$$

The sensitivity analysis results contain these partial derivative values of the
equations taken with respect to the parameters. You can control aspects of the
sensitivity analysis by setting properties of the `odeSensitivity`

object assigned to the `Senstivity`

property of the `ode`

object.

`EventSensitivity`

— Sensitivity values at time of events

array

This property is read-only.

Sensitivity values at time of events, returned as an array.
`EventSensitivity`

has a number of columns equal to the number of
parameters being analyzed, and the same number of rows as
`Solution`

. The number of pages in the array is equal to the
number of events, `numel(EventTime)`

.

## Examples

### Solve Single ODE

Create an empty `ode`

object, and then specify values for the `ODEFcn`

and `InitialValue`

properties.

F = ode; F.ODEFcn = @(t,y) 2*t; F.InitialValue = 0;

Check which solver is selected for the problem, and then solve the equation over the time range `[0 10]`

.

F.SelectedSolver

ans = SolverID enumeration ode45

sol = solve(F,0,10)

sol = ODEResults with properties: Time: [0 0.2500 0.5000 0.7500 1 1.2500 1.5000 1.7500 2 2.2500 2.5000 2.7500 3 3.2500 3.5000 3.7500 4 4.2500 4.5000 4.7500 5 5.2500 5.5000 5.7500 6 6.2500 6.5000 6.7500 7 7.2500 7.5000 7.7500 8 8.2500 8.5000 8.7500 9 9.2500 9.5000 9.7500 10] Solution: [0 0.0625 0.2500 0.5625 1.0000 1.5625 2.2500 3.0625 4 5.0625 6.2500 7.5625 9 10.5625 12.2500 14.0625 16 18.0625 20.2500 22.5625 25 27.5625 30.2500 33.0625 36 39.0625 42.2500 45.5625 49 52.5625 56.2500 60.0625 64 68.0625 ... ] (1x41 double)

Plot the results.

`plot(sol.Time,sol.Solution,"-o")`

### Interpolate ODE Solution Outside of Original Interval

Create an `ode`

object for the equations `@(t,y) [y(2); 1000*(1-y(1)^2)*y(2)-y(1)]`

with the initial conditions `[2 0]`

. Specify the solver as `"stiff"`

.

`F = ode(ODEFcn=@(t,y) [y(2); 1000*(1-y(1)^2)*y(2)-y(1)],InitialValue=[2 0],Solver="stiff");`

Create a function handle that can evaluate the solution of the problem in the interval `[0 1000]`

by using the `solutionFcn`

method.

Specify

`Extension="on"`

to enable the function handle to evaluate the solution at times outside of the original interval`[0 1000]`

.Specify

`OutputVariables=1`

so that the function handle interpolates only the first solution variable.Specify two outputs to also return the integration results in the original interval.

`[fh,S] = solutionFcn(F,0,1000,Extension="on",OutputVariables=1)`

`fh = `*function_handle with value:*
@ode.solutionFcn/interpolate

S = ODEResults with properties: Time: [0 1.4606e-05 2.9212e-05 4.3818e-05 1.1010e-04 1.7639e-04 2.4267e-04 3.0896e-04 4.5006e-04 5.9116e-04 7.3226e-04 8.7336e-04 0.0010 0.0012 0.0013 0.0015 0.0017 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 0.0044 0.0055 ... ] (1x227 double) Solution: [2x227 double]

Extract and plot the integration results for the first solution variable in `S`

, and then use the function handle `fh`

to also evaluate the solution in the extended interval `[1000 3000]`

. Plot the extended solution in red.

plot(S.Time,S.Solution(1,:),"-o") t = linspace(1000,3000); ys = fh(t); hold on plot(t,ys,"r-o") hold off

### Examine Parameter Sensitivity

Solve an ODE system with two equations and two parameters, and perform sensitivity analysis on the parameters.

Create an `ode`

object to represent this system of equations.

$$\begin{array}{l}\frac{{\mathrm{dy}}_{1}}{\mathrm{dt}}={\mathit{p}}_{1}{\mathit{y}}_{1}-{\mathit{y}}_{2}\\ \frac{{\mathrm{dy}}_{2}}{\mathrm{dt}}={-\mathit{p}}_{2}{\mathit{y}}_{2}\end{array}$$

Specify the initial conditions as ${\mathit{y}}_{1}\left(0\right)=2$ and ${\mathit{y}}_{2}\left(0\right)=3$, and parameter values of ${\mathit{p}}_{1}=0.05$ and ${\mathit{p}}_{2}=1.5$. To enable sensitivity analysis of the parameters, set the `Sensitivity`

property of the `ode`

object to an `odeSensitivity`

object.

p = [0.05 1.5]; F = ode(ODEFcn=@(t,y,p) [p(1)*y(1)-y(2); -p(2)*y(2)], ... InitialValue=[2 3], ... Parameters=p, ... Sensitivity=odeSensitivity)

F = ode with properties: Problem definition ODEFcn: @(t,y,p)[p(1)*y(1)-y(2);-p(2)*y(2)] Parameters: [0.0500 1.5000] InitialTime: 0 InitialValue: [2 3] Sensitivity: [1x1 odeSensitivity] EquationType: standard Solver properties AbsoluteTolerance: 1.0000e-06 RelativeTolerance: 1.0000e-03 Solver: auto SelectedSolver: cvodesnonstiff Show all properties

Because the equations are nonstiff and sensitivity analysis is enabled, the `ode`

object automatically chooses the `cvodesnonstiff`

solver for this problem.

Solve the ODE over the time interval `[0 5]`

, and plot the solution for each component.

S = solve(F,0,5)

S = ODEResults with properties: Time: [0 2.9540e-09 2.9543e-05 2.2465e-04 4.1976e-04 0.0024 0.0080 0.0137 0.0245 0.0353 0.0611 0.0869 0.1499 0.2129 0.3169 0.4208 0.5248 0.7204 0.9161 1.1118 1.3075 1.5031 1.6988 1.8945 2.0901 2.2858 2.4815 2.6772 2.8728 ... ] (1x40 double) Solution: [2x40 double] Sensitivity: [2x2x40 double]

plot(S.Time,S.Solution(1,:),"-o",S.Time,S.Solution(2,:),"-o") legend("y1","y2")

The values in `S.Sensitivity`

are partial derivatives of the equations with respect to the parameters. To examine the effects of the parameter values during the integration, plot the sensitivity values.

figure hold on plot(S.Time,squeeze(S.Sensitivity(1,1,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(1,2,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(2,1,:)),"-o") plot(S.Time,squeeze(S.Sensitivity(2,2,:)),"-o") legend("p1,eq1","p2,eq1","p1,eq2","p2,eq2") hold off

### Use Callback Function to Restart Integration of Bouncing Ball

Consider a ball thrown upward with an initial velocity $\mathrm{dy}/\mathrm{dt}$. The ball is subject to acceleration due to gravity aimed downward, so its acceleration is

$$\frac{{d}^{2}y}{d{t}^{2}}=-g.$$

Rewriting the equation as a first-order system of equations with the substitutions ${\mathit{y}}_{1}=\mathit{y}$ and ${\mathit{y}}_{2}=\frac{\mathrm{dy}}{\mathrm{dt}}$ yields

$$\begin{array}{l}{y}_{1}^{\prime}={y}_{2}\\ {y}_{2}^{\prime}=-g.\end{array}$$

Solve the equations for the position ${\mathit{y}}_{1}$ and velocity ${\mathit{y}}_{2}$ of the ball over time.

**Define Equations and Initial Conditions**

Create a function handle for the first-order system of equations that accepts two inputs for `(t,y)`

. Use the value $\mathit{g}=9.8$ for the acceleration due to gravity.

dydt = @(t,y) [y(2); -9.8];

Next, create a vector with the initial conditions. The ball starts at position ${\mathit{y}}_{1}=3$ at $\mathit{t}=0$ as it is thrown upward with initial velocity ${\mathit{y}}_{2}=20$.

y0 = [3 20];

**Model Ball Bounces as Events**

The ball initially travels upward until the force due to gravity causes it to change direction and head back down to the ground. If you solve the equations without more consideration, then the ball falls back downward forever without striking the ground. Instead, you can use an event function to detect when the position of the ball goes to zero where the ground is located. Because the solution component ${\mathit{y}}_{1}=\mathit{y}$ is the position of the ball, the event function tracks the value of ${\mathit{y}}_{1}$ so that an event triggers whenever ${\mathit{y}}_{1}=0$.

Create a function handle for the event function that accepts two inputs for `(t,y)`

.

bounceEvent = @(t,y) y(1);

When the ball strikes the ground, its direction changes again as it heads back upwards with a new (smaller) initial velocity. To model this situation, use a callback function along with the event function. When an event triggers, the ODE solver invokes the callback function. The callback function resets the position and initial velocity of the ball so that the integration can restart with the correct initial conditions. When an event occurs, the callback function sets the position ${\mathit{y}}_{1}=0$ and attenuates the velocity by a factor of $0.9$ while reversing its direction back upward. Define a callback function that performs these actions.

function [stop,y] = bounceResponse(t,y) stop = false; y(1) = 0; y(2) = -0.9*y(2); end

(The callback function is included as a local function at the end of the example.)

Create an `odeEvent`

object to represent the bouncing ball events. Specify `Direction="descending"`

so that only events where the position is decreasing are detected. Also, specify `Response="callback"`

so that the solver invokes the callback function when an event occurs.

E = odeEvent(EventFcn=bounceEvent, ... Direction="descending", ... Response="callback", ... CallbackFcn=@bounceResponse)

E = odeEvent with properties: EventFcn: @(t,y)y(1) Direction: descending Response: callback CallbackFcn: @bounceResponse

**Solve Equations**

Create an `ode`

object for the problem, specifying the equations `dydt`

, initial conditions `y0`

, and events `E`

as property values.

F = ode(ODEFcn=dydt,InitialValue=y0,EventDefinition=E);

Integrate the equations over the time interval `[0 30]`

by using the `solve`

method. Specify `Refine=8`

to generate 8 points per step. The resulting object has properties for the time and solution, and because events are being tracked, the object also displays properties related to the events that triggered during the integration.

S = solve(F,0,30,Refine=8)

S = ODEResults with properties: Time: [0 0.0038 0.0075 0.0113 0.0151 0.0188 0.0226 0.0264 0.0301 0.0490 0.0678 0.0867 0.1055 0.1243 0.1432 0.1620 0.1809 0.2751 0.3692 0.4634 0.5576 0.6518 0.7460 0.8402 0.9344 1.3094 1.6844 2.0594 2.4344 2.8094 3.1844 ... ] (1x468 double) Solution: [2x468 double] EventTime: [4.2265 8.1607 11.7015 14.8882 17.7563 20.3375 22.6606 24.7514 26.6331 28.3267 29.8509] EventSolution: [2x11 double] EventIndex: [1 1 1 1 1 1 1 1 1 1 1]

**Plot Results**

Plot the position ${\mathit{y}}_{1}$ of the ball over time, marking the initial position with a green circle and events with red circles.

plot(S.Time,S.Solution(1,:),"--") hold on plot(S.EventTime,S.EventSolution(1,:),"ro") plot(0,y0(1),"go") hold off ylim([0 25]) xlabel("Time") ylabel("Position y_1")

**Local Functions**

function [stop,y] = bounceResponse(t,y) stop = false; y(1) = 0; y(2) = -0.9*y(2); end

## Version History

**Introduced in R2023b**

## See Also

`ode`

| `odeEvent`

| `odeSensitivity`

| `solve`

| `solutionFcn`

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