"...there are some variables (calculated inside the function @ODE_System) and i would like to see and save their values along with the solutions ; is there any way to do this"
There are ways to do this, but they are unlikely to be very helpful for you because only some calls of the function correspond to solutions returned by the ODE solver. This is because ODE solvers can change the step size arbitrarily, can even make steps backwards, and not all function calls correspond to output values. So if you simply save every value calculated on every function call, it is not a trivial task to work out which of them actually correspond to valid solutions values returned by the ODE solver. I would not recommend that approach.
Unless solving your ODE function is extremely slow (on the order of hours/days) then by far the simplest approach is to first solve the ODE and then call the function with the solution values to get the intermediate values.
Here is a working example:
[t,y] = ode45(@fun,0:9,[2;0])
[~,tmp] = cellfun(@fun,num2cell(t),num2cell(y,2),'uni',0);
function [dydt,intv] = fun(t,y)
intv = (1-y(1)^2)*y(2)-y(1);