How we can perturb an abstract model.?Like if we have a system of equations $dx/dt=f(x,y), dy/dt=g(x,y), x(0)=x_0, y(0)=y_0$. If we wish to perturb the given system, so in how many ways one can do that? Like what I saw that we can do it by adding perturbation term(s) to right side of the above system, varying the initial conditions, or adding some singularity terms e.g multiplying epsilon to left side of any of above equations.
We take multi epsilon to the consider system is the perturbation, which form a matrix and after that, we check the eigenvalues
I agree with the answer of Usman Riaz. You can also use a function of time as a perturbation.
A very nice approach is to make the first derivations of functions f and g with respect to the time variable. You get linearized set of questions. By inserting small perturbations to the linear equations you can find out whether the system is stable, unstable or mixed, in other words, whether contracts to the equilibrium, expand, or it is a saddle point. The fixed point, where dd/dt and dg/dt are both equal to zero, is achieved for the original nonlinear set of equations.
In other words, you make a tangential plane to the original set of equations and observe to the evolution of small perturbations from the points achieved for no evolving system.
This method was used in physics for centuries and there is a lot of literature about it. Try to find terms 'linear perturbation methods' and similar.
Consider A=(1,0;1,0)A=(1,0;1,0) and b=(1;1)b=(1;1). Let xx be any solution of this, so x=(1;t)x=(1;t) for some tt.
Now, perturbe this to (1,0;1−1/n,1/n2)(1,0;1−1/n,1/n2) and bb unchanged, so pertruebed by 00 (but one could also impose some nontrivial perturbation). The perturbation(s) tends to zero. But the (only) solution is (1;n)(1;n). Thus xnxn does not converge, and one seees that the xnxn and the inverse (in this case it is actually invertible, but one could expand the system to avoid this) of the perturbed system are not bounded.
Wajid Ali
well it depends on the PDE system you are considering and the type of perturbations you want to introduce. The answer is very case dependant. In general you add a small term (which could be a function of every variable) to the system inputs, and see what happens to the system outputs.
Wajid Ali
As I mentioned in my first answer, in general you add a perturbation to the system inputs, in order to study what happens to the system outputs when you introduce such perturbation.
As Jiří Kroc explained above, one practical reason is to study the stability of a system (in structural engineering for example). Another practical example it came to my mind where you want to introduce a perturbation to a PDE system is to study the turbolence of a fluid flow (in Fluid Dynamics). The reason and the way you introduce the perturbations are very case dependant, and in fact they are very different in the two examples I mentioned here.
1) dx/dt=f(x,y)
make the differential equation (by going to the definition of derivative) to a differences equation,
[] are considered as subscripts, here with [t] – [t-1] = h:
(x(t) – x(t-h))/h = (x[t] – x[t-1])/h = f(x[t-1],y[t-1])
x[t] – x[t-1] = h * f(x[t-1],y[t-1])
then 1) will be:
x[t] = x[t-1] + h * f(x[t-1],y[t-1])
then 1) will be a time series.
This may be perturbed to:
x[t] = x[t-1] + h * f(x[t-1],y[t-1]) + fperturb(x[t-1], …, x[t-n], y[t-1], …, y[t-n])
fperturb can be any (also nonlinear) function,
then 1) will be a (in general nonlinear) time series,
can also destroy linearity in x[t-1] if it is so in fperturb,
fperturb(x[t-1], …, x[t-n], y[t-1], …, y[t-n]) -> 0 if going back to top.
That is of the general form:
x[t] = fnew(x[t-1], …, x[t-n], y[t-1], …, y[t-n]),
on principal, fnew can be any (also nonlinear) function of the arguments,
Do the same for
2) dy/dt=g(x,y)
Then apply books about nonlinear time series.
If the system gets chaotic, you do not need any linearization.
But when fperturb gets small, you can linearize and
when fperturb gets 0, you should get the original results.
A typical requirement for a perturbation method to be useful is that it is a small perturbation from a problem that you already know how to solve. So the most useful perturbation depends not only on the original problem that you must solve, but also on the kinds of problems that you already know how to solve. This is not a simple thing. The idea is to start with a family of problems that you can solve, and then look for the one that is closest to the problem that you must solve. This is the difficult part and a lot of study might be needed just to recognize the solvable problem that is closest to the problem of interest. If you can get past that step, the perturbation becomes fairly obvious and applications of perturbation theory are the easy steps.
Wajid Ali
unfortunately I do not know any specific book on perturbation theory. Usually you learn perturbation theory when studying something else as a tool. Try the attched ".pdf" for a beginner introduction to the Maths behind it.
It all depends on the goals of your research.
If you want to perturb the system so that the shift operator along the trajectories of differential equations or the Poincaré map (and thus some qualitative properties of solutions) are preserved, then you can use the method of reflecting function.
To get started, look at the article https://www.researchgate.net/publication/220318502_How_to_construct_equivalent_differential_systems
and project
https://www.researchgate.net/project/ADMISSIBLE-PERTURBATIONS-OF-VARIOUS-MATHEMATICAL-MODELS
Good luck!
about your pdf:
It is ok, when a Taylor series expansion, here stopped after the square term, is a good approximation (e.g. in your pdf, when the ε is small).
Sometimes, it is sufficient, to stop the expansion after the linear term, keeping things linear (with linear differential equations, the superposition of the solutions is also a solution - fourier synthesis).
Sometimes you have to go further with the expansion, especially when the linear term is 0, or when the hoped effect is not showing up.
Sometimes, you can not make any aproximation at all. This is the case, just e.g., when you have a soliton solution (google) or when you expect chaotic behaviour. In that case, you need to solve the full differential equation (some use a computer); you can not make an approximation, starting with a simpler equation and perturb that one as an approximation, for the proper solution, showing the most wanted effect, gets lost.
What you can do is, going from the differencial equation to a differences equation, (as described above) and program the time series, to get the trajectory.
Following that, you can try to get an analytical solution. When you make an approximation, you can check, if you lose exactly what you want to get.
Starting with your dx/dt=f(x,y), dy/dt=g(x,y), that depends on the f(x,y) and g(x,y).
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