@nm ,  thanks , good idea to add small code directly

and c is wave speed 

Impressive ...
A good exercise to do (try ) this in geometric expression app

Lotka_Volterra system as start for experimentation with bifurcation 
Note: the spiral must be bigger ..
IterativeMaps:-Bifurcation ..use?
I do have extensive old  Maple studymateria from my education  for Lotka -Volterra , but it is not using ODE's, but works with difference equations, to study the phaseplot with equibrillium lines


lotka-volterra_bifurcatie_analyse_test_-25-8-2024.mw

Dynmod08_-_kopie.mws

@Paras31 
Getting a good plot in Maple is still a challence as a non-expert as i am.
You have done well.

@Paras31 
 

systems := [
    (x, y) -> y, 
    (x, y) -> -delta * y - alpha * x - beta * x^3 + gamma * cos(omega * t)
];

Conclusion:

For the given system with a time-dependent forcing term gamma * cos(omega * t), there are no true equilibrium points in the classical sense because the term gamma * cos(omega * t) is time-dependent and prevents the system from reaching a stationary point. 

Putting the system in the right input format of the Dsys procedure ,then you will probably see this phaseplot?
For t = 10 , this is a solution curve 

About Duffings eq : how about seeing it  in DSys procedure ?

Duffing_eq_as_system.mw

Possible to ad a list of ic , to show the solutioncurve better 
Adding population curves and add Explore for handling parameters?
Don't know if is possible to use a connected Phase plot with population curves ? 

faseplots_ics_not_ics24-8-2024Mprimes_evenwichtpunten_.mw

@Paras31 ,thanks
Another procedure , the same as yours example, need some adjustments 


faseplots_ics_not_ics23-8-2024Mprimes_evenwichtpunten.mw

Adding a procedure for plotting  phase plots 

22-8-2024_Mprimes_evenwichtpunten_Dsys

@Scot Gould , thanks for this exercise, get some help en ideas from it.  

@Paras31  i made it a little bit more userfriendly 

Mprimes21-8-20124sytem_ode_population_B.mw
 

with classification of equilibrium points
If the system is in equilibrium : the growth rates of both populations are then 0.

Mprimes21-8-20124sytem_ode_population_A_.mw

 

As a start..
Classification of equilibria for this purpose becomes a Jacobi matrix
used....
I see 6 classifications , let's see if the code can be modified to start classifying the type of equilibrium weight

Mprimes21-8-20124sytem_ode_populatin.mw

 

@salim-barzani :  userfriendly for experimentation

A adjustable integralcurve can be set in fieldplot, with initial value point for first order ode

module_20-8-2024-Odegeneratorplots_Mprimes_A.mw

@salim-barzani  make it more richer :-)

generatorfunctie_ode_Mprimes_postDEF_20-8-2024_var_methelp_dfieldplot_rownumber_.mw