The latest version of Maple shows the behaviour you desire (default folder is set to the location where a double-click is performed). Maybe these settings influence the selection of the default folder

@Rouben Rostamian  

I am aware of the advantages of quaternions but lack experience. Very little to find about them in combination with Maple.

For this reason I was looking for supporting functionality from Maple for something I know (Euler angles).

For one static rotation plottools:-rotate is a step in the right direction. Looks like that a bunch of vectors can be converted with one statement.

For dynamic problems: I have seen that you use quaternions for systems ode's. Very interesting, but I need to get hands on (one day...). 

@Christopher2222 

It was possible to add the contact force to the simulation for this simple case (see attached). Now it is easier to observe the force.

simplified_disk_pendulum_with_force.msim

Note:
If the mass m in the center is set to zero (try the new rerun feature for this. decend to 0.001 and then to zero), the excentric mass is not falling in a straight line. The problem has become unphysical. Change the solver to variable. Then you can approach zero (but not run at zero). All this are signs for an ill-conditionned case.

@Christopher2222 

The fix does not change the behaviour. The behaviour stems from a combination of the "extreme parameters" (i.e. an excentric mass inside a massless contact body without friction but with elasticity in the contact).

Make a thought experiment and try to model this in Maple: It is basically a free falling mass. In the animation of Orang ("hits the wall" here) it looks like that a contact force is only generated when the contact point is between the center of the contact element and the excentric mass. I do not know if this a correct observation and if it makes sense. I have to think about it. (Edit: striked through since the observation was wrong) A contact point and force visualization in MapleSim would be helpfull for an analysis.

The rolling case is much easier to model n Maple

@mmcdara

Thank you so much for all the references. I made this post one of my favourites and added new tags in the hope that others can find it and profit from it as I did.

I have to say it again: The content is worth a post. Proper classification of periodic, quasi-periodic, psuedo-chaotic and chaotic behaviour is less trivial than I thougth.

@nm 

I could not reproduce this pseudo-ramdom error with former versions of Maple.

If you want, I can try it for the new error

@Rouben Rostamian  

Correct: eta is not periodic (at least for small timescales). It could be that after many cycles (stop-and-go's of the big circle) eta has the same value as at the beginning of the simulation. For that a certain ratio of radii and masses is required. Since this is not a celestial motion (or a gear drive), its difficult to tell what rational ratio of numbers is required. I would be surprised if there is no periodic solution at all.

Chaos: The type of chaos I had in mind is an unexpected behaviour. Like here for the double pendulum where the small link changes the sense of rotation while the big link swings back and forth.

@mmcdara 

This is an excellent lecture! I do not remember having it seen this way. The Maple code is simple and elegant. Plots are nice.

Something like this should be part of Maples educational documentation (maybe there is already something). I hope someone from Maplesoft is reading this. I have doubts that a better presentation is possible with other math tools.

TimeSeriesAnalysis: New to me.

Chaos: Regardless of the mathematical features used, the detection of chaos depends on the sampling interval. With some signals (timeseries) it can manifest “immediately”, with others the interval must be extended. In any case, a statement about chaos of a timeseries will be a probalistic statement - IMO.
If demonstration of analytical periodicy with (one) timeseries is theoretically impossible, how should it be possible to proof chaos.

I am sure we will come back to this topic one day.

@Rouben Rostamian  

Looking at the trace of the marker of the small circle, I would agree. However, plotting psi does not give the impression

plot(psi(Theta(t),Phi(t))-t/4.4, t=0..100, color=red);

Reminds me of the stop-and-go motion of a sine force on a mass

plots:-odeplot(dsolve({diff(x(t),t$2)=sin(t),x(0)=0,D(x)(0)=0},numeric))

There seems to be no special command to test for periodicy or chaos.

Exact periodic solutions are important bench tests for numerical methods and results. Finding them is an art.

@mmcdara

Great answer, worth a post on the topic because of the theoretical background and the inclusion of almost periodic cases.

I have to give some backgound why this question came up. In a post a user complained that Maple could not evaluate certain definite integrals of periodic functions. I was wondering whether there are methods that can be used to detect periodicy and then provide solutions for multiples of integration ranges of known solutions.

In another post solutions are discussed that could be chaotic or periodic. Verifying the two cases is of interest. For the later case I was looking for a software function similar to what other tools provide. When I use Maple's autocorrelation out of the box I get something like this for a sine function. I know why that is but a decreasing amplitude is not a good indicator for periodicy.  A value always displaying "1" at multiples of periods would be desirable. But this has to be programmed (period detection followed by cross-correlation of a period patter with the signal)

Can somethig be assumed for the 12 parameters Y... ? Are they real or positive?

@OrangVahid 

Yes, the physics are correct as I have tired to explain in my reply "Physics" to Rouben Rostamian. I see in your animation the contact force at the contact point animated. Visualising the contact point and contact forces (and slipage) is one of the features I would like to see one day.

In a planar case I could probably find a way to visualize such things but in 3d including multiple contacts I cannot see how this can be done with the standard library. 

Thank you for following up on this.

@OrangVahid 

Yes, it vanishes when the number of plot points in the simulation settings are increased or the simulation time is reduced.

@Rouben Rostamian  

Beautiful. It's kind of a double pendulum that is not chaotic. I have to think about it (and how to make it chaotic)

@mmcdara 

Thank you too. MapleSim is a good explorative tool but also kind of a grey box. The governing equations of motion are not intended to be easily understandable. MapleSim assembles them for efficient computation.
For example, I have not found a good explanation why the vector of agular acceleration (orange) is perpendicular to the initial axis of rotation. Note also the flip the red and green forces.

For a further analysis, I think, Maple has to be involved.