# Event Handling

`BifurcationKit.jl`

allows the detection of events along the branch of solutions. Its main use consists in detecting bifurcation points but they can be used and combined together by the user too.

The events are detected during a call to `br = continuation(prob, alg, contParams::ContinuationPar;kwargs...)`

by turning on the following flag(s):

`contParams.detect_event = 1`

The event points are located by looking at the function defining the event (see below). The located event points are then returned in `br.specialpoint`

.

## Precise detection of event points using Bisection

Note that the event points detected when `detect_event = 1`

are only approximate event points. Indeed, we only signal that, in between two continuation steps which can be large, a (several) event point has been detected. Hence, we only have a rough idea of where the event is located, unless your `ContinuationPar().dsmax`

is very small... This can be improved as follows.

If you choose `detect_event = 2`

, a bisection algorithm is used to locate the event points more precisely. It means that we recursively track down the event. Some options in `ContinuationPar`

control this behavior:

`n_inversion`

: number of sign inversions in the bisection algorithm`max_bisection_steps`

maximum number of bisection steps`tol_param_bisection_event`

tolerance on parameter to locate event

## Different event types

The set of possible events `DiscreteEvent, ContinuousEvent, SetOfEvents, PairOfEvents`

is detailed in the Library.

## Built-in events

`BifurcationKit.SaveAtEvent`

— Function``SaveAtEvent(positions::Tuple)``

This event implements the detection of when the parameter values, used during continuation, equals one of the values in `positions`

. This state is then saved in the branch.

For example, you can use it like `continuation(args...; event = SaveAtEvent((1., 2., -3.)))`

`BifurcationKit.FoldDetectEvent`

— Constant``FoldDetectEvent``

This event implements the detection of Fold points based on the p-component of the tangent vector to the continuation curve. It is designed to work with `PALC(tangent=Bordered())`

as continuation algorithm. To use it, pass `event = FoldDetectEvent`

to `continuation`

.

`BifurcationKit.BifDetectEvent`

— Constant``BifDetectEvent``

This event implements the detection of bifurcations points along a continuation curve. The detection is based on monitoring the number of unstable eigenvalues. More details are given at Detection of bifurcation points of Equilibria.

## Examples

We show how to use the different events. We first set up a problem as usual.

```
using Revise, BifurcationKit
using Plots
const BK = BifurcationKit
####################################################################################################
# test vector field for event detection
function Feve(X, p)
p1, p2, k = p
x, y = X
out = similar(X)
out[1] = p1 + x - y - x^k/k
out[2] = p1 + y + x - 2y^k/k
out
end
# parameters for the vector field
par = (p1 = -3., p2 = -3., k = 3)
# bifurcation problem
prob = BifurcationProblem(Feve, -2ones(2), par, (@lens _.p1);
record_from_solution = (x, p) -> x[1])
# parameters for the continuation
opts = ContinuationPar(max_steps = 150, p_min = -3., p_max = 4.0,
# we disable bifurcation detection to focus
# on the user passed events
detect_bifurcation = 0, detect_fold = false,
# parameters specific to event detection
detect_event = 2)
# arguments for continuation
args = (prob, PALC(), opts)
kwargs = (plot = false, verbosity = 0,)
```

### Example of continuous event

In this first example, we build an event to detect when the parameter value is `-2`

or when the first component of the solution is `1`

.

```
br = continuation(args...; kwargs...,
event = BK.ContinuousEvent(2,
(iter, state) -> (getp(state)+2, getx(state)[1]-1)),)
nothing #hide
```

gives

`br`

```
┌─ Curve type: EquilibriumCont
├─ Number of points: 115
├─ Type of vectors: Vector{Float64}
├─ Parameter p1 starts at -3.0, ends at 4.0
├─ Algo: PALC
└─ Special points:
- # 1, userC-1 at p1 ≈ -1.98380428 ∈ (-2.00059057, -1.98380428), |δp|=2e-02, [converged], δ = ( 0, 0), step = 12
- # 2, userC-1 at p1 ≈ -2.00301167 ∈ (-2.00301167, -1.99953208), |δp|=3e-03, [converged], δ = ( 0, 0), step = 28
- # 3, userC-2 at p1 ≈ -2.30722875 ∈ (-2.30888689, -2.30722875), |δp|=2e-03, [converged], δ = ( 0, 0), step = 33
- # 4, userC-1 at p1 ≈ -1.99895595 ∈ (-2.01435831, -1.99895595), |δp|=2e-02, [converged], δ = ( 0, 0), step = 36
- # 5, userC-2 at p1 ≈ -0.83478733 ∈ (-0.83598611, -0.83478733), |δp|=1e-03, [converged], δ = ( 0, 0), step = 48
- # 6, userC-2 at p1 ≈ +1.15208549 ∈ (+1.14075593, +1.15208549), |δp|=1e-02, [converged], δ = ( 0, 0), step = 92
- # 7, endpoint at p1 ≈ +4.00000000, step = 114
```

This shows for example that the first component of the event was detected `userC-1`

first. This yields

`plot(br)`

You can also name the events as follows

```
br = continuation(args...; kwargs...,
event = BK.ContinuousEvent(2,
(iter, state) -> (getp(state)+2, getx(state)[1]-1),
("event1", "event2")))
nothing #hide
```

And get:

`br`

```
┌─ Curve type: EquilibriumCont
├─ Number of points: 115
├─ Type of vectors: Vector{Float64}
├─ Parameter p1 starts at -3.0, ends at 4.0
├─ Algo: PALC
└─ Special points:
- # 1, event1 at p1 ≈ -1.98380428 ∈ (-2.00059057, -1.98380428), |δp|=2e-02, [converged], δ = ( 0, 0), step = 12
- # 2, event1 at p1 ≈ -2.00301167 ∈ (-2.00301167, -1.99953208), |δp|=3e-03, [converged], δ = ( 0, 0), step = 28
- # 3, event2 at p1 ≈ -2.30722875 ∈ (-2.30888689, -2.30722875), |δp|=2e-03, [converged], δ = ( 0, 0), step = 33
- # 4, event1 at p1 ≈ -1.99895595 ∈ (-2.01435831, -1.99895595), |δp|=2e-02, [converged], δ = ( 0, 0), step = 36
- # 5, event2 at p1 ≈ -0.83478733 ∈ (-0.83598611, -0.83478733), |δp|=1e-03, [converged], δ = ( 0, 0), step = 48
- # 6, event2 at p1 ≈ +1.15208549 ∈ (+1.14075593, +1.15208549), |δp|=1e-02, [converged], δ = ( 0, 0), step = 92
- # 7, endpoint at p1 ≈ +4.00000000, step = 114
```

`plot(br)`

### Example of discrete event

You can also use discrete events to detect a change. For example, the following detect when the parameter value equals `-2`

:

```
br = continuation(args...; kwargs...,
event = BK.DiscreteEvent(1,
(iter, state) -> getp(state)>-2))
```

```
┌─ Curve type: EquilibriumCont
├─ Number of points: 114
├─ Type of vectors: Vector{Float64}
├─ Parameter p1 starts at -3.0, ends at 4.0
├─ Algo: PALC
└─ Special points:
- # 1, userD at p1 ≈ -1.98380428 ∈ (-2.00059057, -1.98380428), |δp|=2e-02, [converged], δ = ( 0, 0), step = 12
- # 2, userD at p1 ≈ -2.00301167 ∈ (-2.00301167, -1.99953208), |δp|=3e-03, [converged], δ = ( 0, 0), step = 28
- # 3, userD at p1 ≈ -1.99670519 ∈ (-2.00056478, -1.99670519), |δp|=4e-03, [converged], δ = ( 0, 0), step = 35
- # 4, endpoint at p1 ≈ +4.00000000, step = 113
```

### Example of PairOfEvents event

Let us be a bit more creative and combine a continuous event with a discrete one:

```
br = continuation(args...; kwargs...,
event = BK.PairOfEvents(
BK.ContinuousEvent(1, (iter, state) -> getp(state)),
BK.DiscreteEvent(1, (iter, state) -> getp(state)>-2)))
```

```
┌─ Curve type: EquilibriumCont
├─ Number of points: 114
├─ Type of vectors: Vector{Float64}
├─ Parameter p1 starts at -3.0, ends at 4.0
├─ Algo: PALC
└─ Special points:
- # 1, userD at p1 ≈ -1.98380428 ∈ (-2.00059057, -1.98380428), |δp|=2e-02, [converged], δ = ( 0, 0), step = 12
- # 2, userD at p1 ≈ -2.00301167 ∈ (-2.00301167, -1.99953208), |δp|=3e-03, [converged], δ = ( 0, 0), step = 28
- # 3, userD at p1 ≈ -1.99670519 ∈ (-2.00056478, -1.99670519), |δp|=4e-03, [converged], δ = ( 0, 0), step = 35
- # 4, userC at p1 ≈ +0.01029312 ∈ (-0.00414058, +0.01029312), |δp|=1e-02, [converged], δ = ( 0, 0), step = 55
- # 5, endpoint at p1 ≈ +4.00000000, step = 113
```

Here `userD-1`

means that the first component of the discrete event was detected. Of course, you can name the event like done above.

### Example of set of events

We can combine more events and chain them like we want using `SetOfEvents`

. In this example, we show how to do bifurcation detection and event location altogether:

```
ev1 = BK.ContinuousEvent(1, (iter, state) -> getp(state)-1)
ev2 = BK.ContinuousEvent(2, (iter, state) -> (getp(state)-2, getp(state)-2.5))
# event to detect bifurcation
ev3 = BK.BifDetectEvent
# we combine the events together
eve = BK.SetOfEvents(ev1, ev2, ev3)
br = continuation(args...; kwargs...,
event = eve)
```

```
┌─ Curve type: EquilibriumCont
├─ Number of points: 119
├─ Type of vectors: Vector{Float64}
├─ Parameter p1 starts at -3.0, ends at 4.0
├─ Algo: PALC
└─ Special points:
- # 1, bp at p1 ≈ -1.13331997 ∈ (-1.13331997, -1.13288137), |δp|=4e-04, [converged], δ = ( 1, 0), step = 20
- # 2, bp at p1 ≈ -2.32501040 ∈ (-2.32501894, -2.32501040), |δp|=9e-06, [converged], δ = (-1, 0), step = 33
- # 3, hopf at p1 ≈ -0.95023978 ∈ (-0.95537606, -0.95023978), |δp|=5e-03, [converged], δ = ( 2, 2), step = 47
- # 4, hopf at p1 ≈ +0.95861227 ∈ (+0.94836964, +0.95861227), |δp|=1e-02, [converged], δ = (-2, -2), step = 67
- # 5, userC1 at p1 ≈ +1.00022767 ∈ (+0.99917369, +1.00022767), |δp|=1e-03, [ guess], δ = ( 0, 0), step = 69
- # 6, userC2-1 at p1 ≈ +2.00505846 ∈ (+1.98961239, +2.00505846), |δp|=2e-02, [converged], δ = ( 0, 0), step = 78
- # 7, bp at p1 ≈ +2.32505861 ∈ (+2.32488513, +2.32505861), |δp|=2e-04, [converged], δ = ( 1, 0), step = 82
- # 8, userC2-1 at p1 ≈ +1.99013402 ∈ (+1.99013402, +2.00407138), |δp|=1e-02, [converged], δ = ( 0, 0), step = 86
- # 9, bp at p1 ≈ +1.13286568 ∈ (+1.13286568, +1.13289065), |δp|=2e-05, [converged], δ = (-1, 0), step = 95
- # 10, userC2-1 at p1 ≈ +2.01959644 ∈ (+1.98600267, +2.01959644), |δp|=3e-02, [converged], δ = ( 0, 0), step = 103
- # 11, userC2-2 at p1 ≈ +2.52953918 ∈ (+2.49528893, +2.52953918), |δp|=3e-02, [converged], δ = ( 0, 0), step = 107
- # 12, endpoint at p1 ≈ +4.00000000, step = 118
```

Which gives

`plot(br)`