`DynamicalSystems.DynamicalSystems`

— Module**DynamicalSystems.jl** is an award-winning Julia software library for nonlinear dynamics and nonlinear timeseries analysis.

To install **DynamicalSystems.jl**, run `import Pkg; Pkg.add("DynamicalSystems")`

as a Julia language command. To learn how to use it and see its contents visit the documentation, which you can either find online or build locally by running the `docs/make.jl`

file.

**DynamicalSystems.jl** is part of JuliaDynamics, an organization dedicated to creating high quality scientific software.

**Highlights**

Aspects of **DynamicalSystems.jl** that make it stand out among other codebases for nonlinear dynamics or nonlinear timeseries analysis are:

**Exceptional documentation**. All implemented algorithms provide a high-level scientific description of their functionality in their documentation string as well as references to scientific papers. The documentation features hundreds of tutorials and examples ranging from introductory to expert usage.**Accessible source code**. One of the main priorities of the library is that the source code of (almost) all implementations is small, simple, easy to understand and modify. This increases confidence, reduces bugs, and allows users to become developers without unnecessary effort.**Open source community project**. Built from the ground up entirely on GitHub,**DynamicalSystems.jl**is 100% open source and based on community contributions. Anyone can be a developer of the library. Everyone is welcomed.**Extensive content**. It aims to cover the entire field of nonlinear dynamics and nonlinear timeseries analysis. It has functionality for complexity measures, delay embeddings, periodic orbits, nonlocal stability analysis, continuation, chaos, fractal dimensions, surrogate testing, recurrence quantification analysis, and much more. Furthermore, all algorithms are "general" and work for any dynamical system applicable. Missing functionality that falls under this wide category of content is welcomed to be part of the library!**Well tested**. All implemented functionality is extensively tested. Each time any change in the code base is done, the extensive test suite is run and checked before merging the change in.**Extendable**.**DynamicalSystems.jl**is a living, evolving project. New contributions can become part of the library and be accessed by all users in the next release. Most importantly, all parts of the library follow professional standards in software design and implement extendable interfaces so that it is easy to contribute new functionality.**Active development**. Since the start of the project (May 2017) there has been activity every month: new features, bugfixes, and the developer team answers users questions on official Julia language forums.**Performant**. Written entirely in Julia, heavily optimized and parallelized, and taking advantage of some of the best packages within the language,**DynamicalSystems.jl**is*really fast*.

**Goals**

The **DynamicalSystems.jl** library started as a vision with three main goals; These same goals now are the core pillars guiding development, and are largely the source of where the aforementioned unique highlights stem from.

**Goal 1: Accessible and reproducible nonlinear dynamics**

The first goal of the library is to make this beautiful field **accessible and reproducible**.

**Accessible** means that if you read on some sorts of fancy algorithm online in a scientific article, you should be able to use it instantly. You shouldn't have to put in the work to code it yourself. The authors of the paper already did that. *So why should you do it again?!* To resolve this problem we developed, and continue to develop, a library that has an incredibly low threshold of entry: contributing to **DynamicalSystems.jl** and making your code available to all is truly *easier* than coding your own algorithms from scratch, due to the well thought out and generic interfaces it provides for dynamical systems.

**Reproducible** means that given some sorts of dynamical systems analysis in a scientific article, you should be able to do *exactly the same analysis* and get *exactly the same results* (within some numeric precision) as the article. After all, computers are deterministic constructs. **DynamicalSystems.jl** allows this by (1) being written in a modern programming language with incredible environment and reproducibility support, (2) being well tested, and (3) by providing thousands of algorithms out of the box, allowing most dynamical systems analysis to be done instantly while implementing only as little new stuff as necessary.

**Goal 2: Library in the literal sense**

**DynamicalSystems.jl** is not just a software library. It is also a library in the literal sense: *where people go to learn something new* (here in particular for nonlinear dynamics). That is why the documentation is of exceptionally high quality: detailed descriptions and explanations of algorithms, with references to the scientific articles articles. It is also partly a reason for the source code to be written as clearly as possible, so that it is examinable by any user.

**Goal 3: A general purpose software**

The third goal is to fill the missing gap of a high quality *general purpose software* for nonlinear dynamics which can be easily extended with new functionality. This can be particularly impactful in teaching. You see, it is unfortunately rarely the case that real, *runnable* code is shown in the classroom, because it is often long and messy. This is especially hurtful for nonlinear dynamics, a field where computer-assisted exploration is critical.

**DynamicalSystems.jl** provides teachers with a framework capable of demonstrating actual, real-world nonlinear dynamics code and its output, without having to invest the weeks to code the internal infrastructure themselves. Its high level syntax requires writing little code to get lots of meaningful analysis done, while its extensive functionality covers most typical classroom applications.

`DynamicalSystems.interactive_cobweb`

— Function`interactive_cobweb(ds::DiscreteDynamicalSystem, prange, O::Int = 3; kwargs...)`

Launch an interactive application for exploring cobweb diagrams of 1D discrete dynamical systems. Two slides control the length of the plotted trajectory and the current parameter value. The parameter values are obtained from the given `prange`

.

In the cobweb plot, higher order iterates of the dynamic rule `f`

are plotted as well, starting from order 1 all the way to the given order `O`

. Both the trajectory in the cobweb, as well as any iterate `f`

can be turned off by using some of the buttons.

**Keywords**

`fkwargs = [(linewidth = 4.0, color = randomcolor()) for i in 1:O]`

: plotting keywords for each of the plotted iterates of`f`

`trajcolor = :black`

: color of the trajectory`pname = "p"`

: name of the parameter slider`pindex = 1`

: parameter index`xmin = 0, xmax = 1`

: limits the state of the dynamical system can take`Tmax = 1000`

: maximum trajectory length`x0s = range(xmin, xmax; length = 101)`

: Possible values for the x0 slider.

`DynamicalSystems.interactive_orbitdiagram`

— Function```
interactive_orbitdiagram(
ds::DynamicalSystem, p_index, pmin, pmax, i::Int = 1;
u0 = nothing, parname = "p", title = ""
)
```

Open an interactive application for exploring orbit diagrams (ODs) of discrete time dynamical systems. Requires `DynamicalSystems`

.

In essense, the function presents the output of `orbitdiagram`

of the `i`

th variable of the `ds`

, and allows interactively zooming into it.

Keywords control the name of the parameter, the initial state (used for *any* parameter) or whether to add a title above the orbit diagram.

**Interaction**

The application is separated in the "OD plot" (left) and the "control panel" (right). On the OD plot you can interactively click and drag with the left mouse button to select a region in the OD. This region is then **re-computed** at a higher resolution.

The options at the control panel are straight-forward, with

`n`

amount of steps recorded for the orbit diagram (not all are in the zoomed region!)`t`

transient steps before starting to record steps`d`

density of x-axis (the parameter axis)`α`

alpha value for the plotted points.

Notice that at each update `n*t*d`

steps are taken. You have to press `update`

after changing these parameters. Press `reset`

to bring the OD in the original state (and variable). Pressing `back`

will go back through the history of your exploration History is stored when the "update" button is pressed or a region is zoomed in.

You can even decide which variable to get the OD for by choosing one of the variables from the wheel! Because the y-axis limits can't be known when changing variable, they reset to the size of the selected variable.

**Accessing the data**

What is plotted on the application window is a *true* orbit diagram, not a plotting shorthand. This means that all data are obtainable and usable directly. Internally we always scale the orbit diagram to [0,1]² (to allow `Float64`

precision even though plotting is `Float32`

-based). This however means that it is necessary to transform the data in real scale. This is done through the function `scaleod`

which accepts the 5 arguments returned from the current function:

```
figure, oddata = interactive_orbitdiagram(...)
ps, us = scaleod(oddata)
```

`DynamicalSystems.interactive_poincaresos`

— Function`interactive_poincaresos(cds, plane, idxs, complete; kwargs...)`

Launch an interactive application for exploring a Poincaré surface of section (PSOS) of the continuous dynamical system `cds`

. Requires `DynamicalSystems`

.

The `plane`

can only be the `Tuple`

type accepted by `DynamicalSystems.poincaresos`

, i.e. `(i, r)`

for the `i`

th variable crossing the value `r`

. `idxs`

gives the two indices of the variables to be displayed, since the PSOS plot is always a 2D scatterplot. I.e. `idxs = (1, 2)`

will plot the 1st versus 2nd variable of the PSOS. It follows that `plane[1] ∉ idxs`

must be true.

`complete`

is a three-argument **function** that completes the new initial state during interactive use, see below.

The function returns: `figure, laststate`

with the latter being an observable containing the latest initial `state`

.

**Keyword Arguments**

`direction, rootkw`

: Same use as in`DynamicalSystems.poincaresos`

.`tfinal = (1000.0, 10.0^4)`

: A 2-element tuple for the range of values for the total integration time (chosen interactively).`color`

: A**function**of the system's initial condition, that returns a color to plot the new points with. The color must be`RGBf/RGBAf`

. A random color is chosen by default.`labels = ("u₁" , "u₂")`

: Scatter plot labels.`scatterkwargs = ()`

: Named tuple of keywords passed to`scatter`

.`diffeq = NamedTuple()`

: Any extra keyword arguments are passed into`init`

of DiffEq.

**Interaction**

The application is a standard scatterplot, which shows the PSOS of the system, initially using the system's `u0`

. Two sliders control the total evolution time and the size of the marker points (which is always in pixels).

Upon clicking within the bounds of the scatter plot your click is transformed into a new initial condition, which is further evolved and its PSOS is computed and then plotted into the scatter plot.

Your click is transformed into a full `D`

-dimensional initial condition through the function `complete`

. The first two arguments of the function are the positions of the click on the PSOS. The third argument is the value of the variable the PSOS is defined on. To be more exact, this is how the function is called:

```
x, y = mouseclick; z = plane[2]
newstate = complete(x, y, z)
```

The `complete`

function can throw an error for ill-conditioned `x, y, z`

. This will be properly handled instead of breaking the application. This `newstate`

is also given to the function `color`

that gets a new color for the new points.

`DynamicalSystems.interactive_poincaresos_scan`

— Function```
interactive_poincaresos_scan(A::StateSpaceSet, j::Int; kwargs...)
interactive_poincaresos_scan(As::Vector{StateSpaceSet}, j::Int; kwargs...)
```

Launch an interactive application for scanning a Poincare surface of section of `A`

like a "brain scan", where the plane that defines the section can be arbitrarily moved around via a slider. Return `figure, ax3D, ax2D`

.

The input dataset must be 3 dimensional, and here the crossing plane is always chosen to be when the `j`

-th variable of the dataset crosses a predefined value. The slider automatically gets all possible values the `j`

-th variable can obtain.

If given multiple datasets, the keyword `colors`

attributes a color to each one, e.g. `colors = [JULIADYNAMICS_COLORS[mod1(i, 6)] for i in 1:length(As)]`

.

The keywords `linekw, scatterkw`

are named tuples that are propagated as keyword arguments to the line and scatter plot respectively, while the keyword `direction = -1`

is propagated to the function `DyamicalSystems.poincaresos`

.

`DynamicalSystems.interactive_trajectory`

— Function`interactive_trajectory(ds::DynamicalSystem [, u0s]; kwargs...) → fig, dsobs`

Same as `interactive_trajectory_timeseries`

, but does not plot any timeseries only the trajectory in a (projected) state space.

`DynamicalSystems.interactive_trajectory_timeseries`

— Function`interactive_trajectory_timeseries(ds::DynamicalSystem, fs, [, u0s]; kwargs...) → fig, dsobs`

Create a Makie `Figure`

to visualize trajectories and timeseries of observables of `ds`

. This `Figure`

can also be used as an interactive GUI to enable interactive control over parameters and time evolution. It can also be used to create videos, as well as customized animations, see below.

`fs`

is a `Vector`

of "indices to observe", i.e., anything that can be given to `observe_state`

. Each observation index will make a timeseries plot. `u0s`

is a `Vector`

of initial conditions. Each is evolved with a unique color and displayed both as a trajectory in state space and as an observed timeseries. Elements of `u0`

can be either `Vector{Real}`

encoding a full state or `Dict`

to partially set a state from current state of `ds`

(same as in `set_state!`

).

The trajectories from the initial conditions in `u0s`

are all evolved and visualized in parallel. By default only the current state of the system is used. `u0s`

can be anything accepted by a `ParallelDynamicalSystem`

.

**Return**

Return `fig, dsobs::DynamicalSystemObservable`

. `fig`

is the created `Figure`

. `dsobs`

facilities the creation of custom animations and/or interactive applications, see the custom animations section below.

See also `interactive_trajectory`

.

**Interactivity and time stepping keywords**

*GUI functionality is possible when the plotting backend is GLMakie. Do using GLMakie; GLMakie.activate!() to ensure this is the chosen backend.*

`add_controls = true`

: If`true`

, below the state space axis some buttons for animating the trajectories live are added:`reset`

: results the parallel trajectories to their initial conditions`run`

: when clicked it evolves the trajectories forwards in time indefinitely. click again to stop the evolution.`step`

: when clicked it evolves the trajectories forwards in time for the amount of steps chosen by the slider to its right.

`add_controls`

only concerns the buttons and interactivity added to the created figure.`parameter_sliders = nothing`

: If given, it must be a dictionary, mapping parameter indices (any valid index that can be given to`set_parameter!`

) to ranges of parameter values. Each combination of index and range becomes a slider that can be interactively controlled to alter a system parameter on the fly during time evolution. Below the parameter sliders, three buttons are added for GUI usage:`update`

: when clicked the chosen parameter values are propagated into the system`u.r.s.`

: when clicked it is equivalent with clicking in order: "update", "reset", "step".`reset p`

: when clicked it resets

`parameter_sliders`

only conserns the buttons and interactivity added to the created figure.`parameter_names = Dict(keys(ps) .=> string.(keys(ps)))`

: Dictionary mapping parameter keys to labels. Only used if`parameter_sliders`

is given.`Δt`

: Time step of time evolution. Defaults to 1 for discrete time, 0.01 for continuous time systems. For internal simplicity, continuous time dynamical systems are evolved non-adaptively with constant step size equal to`Δt`

.`pause = nothing`

: If given, it must be a real number. This number is given to the`sleep`

function, which is called between each plot update. Useful when time integration is computationally inexpensive and animation proceeds too fast.`starting_step = 1`

: the starting value of the "step" slider.

**Visualization keywords**

`colors`

: The color for each initial condition (and resulting trajectory and timeseries). Needs to be a`Vector`

of equal length as`u0s`

.`tail = 1000`

: Length of plotted trajectory (in units of`Δt`

).`fade = 0.5`

: The trajectories in state space are faded towards full transparency. The alpha channel (transparency) scales as`t^fade`

with`t`

ranging from 0 to 1 (1 being the end of the trajectory). Use`fade = 1.0`

for linear fading or`fade = 0`

for no fading. Current default makes fading progress faster at trajectory start and slower at trajectory end.`markersize = 15`

: Size of markers of trajectory endpoints. For discrete systems half of that is used for the trajectory tail.`plotkwargs = NamedTuple()`

: A named tuple of keyword arguments propagated to the state space plot (`lines`

for continuous,`scatter`

for discrete systems).`plotkwargs`

can also be a vector of named tuples, in which case each initial condition gets different arguments.

**Statespace trajectory keywords**

`idxs = 1:min(length(u0s[1]), 3)`

: Which variables to plot in a state space trajectory. Any index that can be given to`observe_state`

can be given here.`statespace_axis = true`

: Whether to create and display an axis for the trajectory plot.`idxs = 1:min(length(u0s[1]), 3)`

: Which variables to plot in a state space trajectory. Any index that can be given to`observe_state`

can be given here. If three indices are given, the trajectory plot is also 3D, otherwise 2D.`lims`

: A tuple of tuples (min, max) for the axis limits. If not given, they are automatically deduced by evolving each of`u0s`

1000`Δt`

units and picking most extreme values (limits are*not*adjusted by default during the live animations).`figure, axis`

: both can be named tuples with arbitrary keywords propagated to the generation of the`Figure`

and state space`Axis`

instances.

**Timeseries keywords**

`linekwargs = NamedTuple()`

: Extra keywords propagated to the timeseries plots. Can also be a vector of named tuples, each one for each unique initial condition.`timeseries_names`

: A vector of strings with length equal to`fs`

giving names to the y-labels of the timeseries plots.`timeseries_ylims`

: A vector of 2-tuples for the lower and upper limits of the y-axis of each timeseries plot. If not given it is deduced automatically similarly to`lims`

.`timeunit = 1`

: the units of time, if any. Sets the units of the timeseries x-axis.`timelabel = "time"`

: label of the x-axis of the timeseries plots.

**Custom animations**

The second return argument `dsobs`

is a `DynamicalSystemObservable`

. The trajectories plotted in the main panel are linked to observables that are fields of the `dsobs`

. Specifically, the field `dsobs.state_obserable`

is an observable containing the final state of each of the trajectories, i.e., a vector of vectors like `u0s`

. `dsobs.param_observable`

is an observable of the system parameters. These observables are triggered by the interactive GUI buttons (the first two when the system is stepped in time, the last one when the parameters are updated). However, these observables, and hence the corresponding plotted trajectories that are `map`

ed from these observables, can be updated via the formal API of `DynamicalSystem`

:

`step!(dsobs, n::Int = 1)`

will step the system for `n`

steps of `Δt`

time, and only update the plot on the last step. `set_parameter!(dsobs, index, value)`

will update the system parameter and then trigger the parameter observable. Lastly, `set_state!(dsobs, new_u [, i])`

will set the `i`

-th system state and clear the trajectory plot to the new initial condition.

This information can be used to create custom animations and/or interactive apps. In principle, the only thing a user has to do is create new observables from the existing ones using e.g. the `on`

function and plot these new observables. Various examples are provided in the online documentation.

`DynamicalSystems.scaleod`

— Function`scaleod(oddata) -> ps, us`

Given the return values of `interactive_orbitdiagram`

, produce orbit diagram data scaled correctly in data units. Return the data as a vector of parameter values and a vector of corresponding variable values.