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Definition of dynamical systems and integrators for DynamicalSystems.jl

Results 12 DynamicalSystemsBase.jl issues
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As discussed with @Datseris, we could add the possibility of defining an SDE system as a type of `DynamicalSystem`. This type would allow to add stochastic dynamics, and it would...

When using the _lyapunovspectrum_ function, it would be nice to use the Jacobian's structure (diagonal, banded, rank1, and so on) especially as long as one doesn't use a GPU for...

hard
performance

With the new introduction of the `observe_state` function, we can easily allow any index/observable/symbolic in the `save_idxs` argument of the `trajectory` function, allthewhile simplifying its source code as well!

good first issue
qol

## Describe the feature you'd like to have See discussion [here](https://github.com/JuliaDynamics/CausalityTools.jl/issues/237#issuecomment-1433254557). It would be nice to have a dedicated implementation of `DiscreteDynamicalSystemWithMemory`, which could iterate systems where the next state...

new-system

Right now, construction of a `ParallelDynamicalSystem` is constrained to these types only: ```julia methods(ParallelDynamicalSystem) # 2 methods for type constructor: [1] ParallelDynamicalSystem(ds::CoreDynamicalSystem, states) @ ~/.julia/packages/DynamicalSystemsBase/Dx8Dp/src/derived_systems/parallel_systems.jl:47 [2] ParallelDynamicalSystem(ds::DiscreteTimeDynamicalSystem, states) @ ~/.julia/packages/DynamicalSystemsBase/Dx8Dp/src/derived_systems/parallel_systems.jl:160...

enhancement

I notice that in the latest version of ```dynamicalsystems.jl``` ds becomes the integrator itself. It's a good improvement but I don't know how to get the derivative via get_du() now....

qol

I haven't had the time to extensively test performance of each individual function of the new DynamicalSystems.jl v3.0 API. I've checked the over-arching performance, e.g., `lyapunovspectrum` has approximately same speed...

help wanted
good first issue
performance

Just use by default `ds.t0` as a keyword. Pass directly to the generated `integrator` underlying the evolution.

documentation
easy

At the moment `parallel_integrator` uses a vector of vectors as a state. However, since we have anyways established that the `get_state` function is used to obtain one of the parallel...

performance

Hi, as shown here https://discourse.julialang.org/t/chaostools-tangent-integrator-stiff-solvers/26464, the tangent_integrator errors, if one provides an algorithm, which uses ForwardDiff.jl to calculate the Jacobian, etc. This is the case for all stiff solvers. One...

enhancement
help wanted