I don't know about analytic solutions but this is the equation for a harmonic oscillator in a constant force field with nonlinear damping. It is a like a mass suspended by a nonlinear spring in a gravitational field. There is a single equilibrium point at x=V and x'=0 whose stability depends on the value of a. Solutions will either be unbounded (go to infinity), which is probably unphysical for your electrical circuit, or they will decay to the equilibrium point, following the usual exponential law for a linearly damped oscillator as they approach it. Only the case a=0 has (neutrally) stable oscillations whose amplitude depend on the initial conditions just as you would expect for a harmonic oscillator without damping. There are no limit cycles and no chaos.

Is there any evidence that an analytic solution does exist?

Please kindly use appropriate/standard linearization technique, I hope you will get to controllable solution. Another good tool is Lie-Algebra for nonlinear differential equations. In the worse case we recourse to some approximation technique.

Igor Ravve, Thank you for your answer.

I don't really know if there exists an analytical solution for my equation. However, numerical simulation does provide a solution. So my guess is, there should exist some kind of solution, if not exact then some approximate solution.

Muhammad Ali , Thanks for the suggestion.

However, I am not looking to linearise the system to obtain the solution as it may contain significant difference between the actual and the linearised solution. I am trying to obtain exact solution, if possible.

Hi

I guess you could easily apply First Integral Method to solve your problem.

Is V a constant or a function? In the first case you can try x = b1*tanh(b2*t) + b3.

you can get an analytical approximation with better convergence than Taylor series by using Adomian Decomposition Method

I agree with Fabio's idea. You can use the semi analytical methods as finite deferential method, Pade method, first integral method, G'/G-method, Adomian decomposition method, HAM, HPM, Tanh-method and so on, also its modification is applied.

I don't know about analytic solutions but this is the equation for a harmonic oscillator in a constant force field with nonlinear damping. It is a like a mass suspended by a nonlinear spring in a gravitational field. There is a single equilibrium point at x=V and x'=0 whose stability depends on the value of a. Solutions will either be unbounded (go to infinity), which is probably unphysical for your electrical circuit, or they will decay to the equilibrium point, following the usual exponential law for a linearly damped oscillator as they approach it. Only the case a=0 has (neutrally) stable oscillations whose amplitude depend on the initial conditions just as you would expect for a harmonic oscillator without damping. There are no limit cycles and no chaos.

Following

Othere then my subjecy sir , thanks

Interesting discussion

The best method depends upon the parameter regime. If your regime of interest is that of oscillations modified by the nonlinear term a*(x-1)*x' (includes but is not necessarily limited to small a), then an excellent approach is to use one of the WKB-like asymptotic methods. The best view of these methods is that they are a nonlinear time-dependent coordinate transform. See Chapter 3 of my magnum opus "Model emergent dynamics in complex systems" http://bookstore.siam.org/mm20

But also you can substitute such nonlinear systems to my web service that does the analysis for you.

The web service can do analysis to a couple more orders, and if you ask me, I can execute the computer algebra to much high-order.

However, if your parameter regime is strongly nolinear, large a or large x, then cognate analysis is less routine and needs more human effort.

Nice discussions sir

You apply He' s method . It is easy to get an analytical solution .You try for 2 days if wont' get write to me .I will find some time to give reply .

Hello,

You can also try the general solution set to x=h.exp(i.omega.t) using the new variables.

But the easiest way is to solve it numerically using the FTCS scheme.

Good question

Dear Manish Mandal

((d²x)/(dt²))+a(1-x)((dx)/(dt))+x-V = 0

Let u(t) = dx /dt, then

du/dt = d²x/dt² = -a(1-x)(dx/dt) - x + V

We obtan the system of differential equations in u and x.

dx/dt = u

du /dt = -a(1-x)u - x + V

All that you have to do is to study this simple autonomous dynamical system.

Observe that (v, 0) is an equilibrium state.

For linearization the system, find the Jacobian matrix

( 0 ................... 1 )

( au -1................... ax )

you can study the stability of the equilibrium state at (v, 0) based on the sign of a and V.

Find the eigenvalues and the eigenvectors of the Jacobian at (v, 0). The rest of the solution is straightforward.

Best wishes

Dear Issam,

It gives : int(du)=-int(a.u.dt)+int(x.u.dt)-int(x.dt)+int(V.dt)

if V=constant, f=x and g'=u

then : u2-u1=-(a/2).(u2^2-u1^2)+(x2/2).u2^2-(x1/2).u1^2-(1/3).u2^3+(1/3).u1^3-(1/2).(x2^2-x1^2)+V.(t2-t1)

Dear Mohammed Lamine Moussaoui

Sorry for the late reply. I am confused with your notations! In the case of a 2x2 nonlinear autonomous dynamical system, phase portrait is very useful to show the solution behavior. Also, we need to follow the matrix approach to show the solution form in the neighborhood of the equilibrium point. The values of a and V signs will show the nature of the Jacobian matrix if it is elliptic or hyperbolic. Each has a different interpretation. Best wishes

Hi Issam Kaddoura,

the symbol ^2 is used for power 2

int is for integral

the functions f and g=(1/2).u^2 are used to calculate int(x.u.dt)=[x.(1/2).u^2]-int(1.(1/2).u^2 dt)

f'=1 and g'=u are the corresponding derivatives

integration boundaries are (u1,u2) and (t1,t2)

I see here that there is two functions x(t) and u(t) and there is NOT a system of differential equations.

Dear Mohammed Lamine Moussaoui

No, it is an autonomous dynamical system in two nonlinear equations

dx/dt = f(x,u) = u

du/dt= g(x,u) = -a(1-x)u - x + V ,

direct integrals didn't work!!

To study the Jacobian matrix is a must.

I can show the complete answer. It is a simple exercise.

Any way the attached textbook is an excellent reference to show you the details.

Best wishes

i think this is nice update

Dear Mehdi Mesrizadeh

The method of discriminant doesn't work to solve the current second-order differential equation. It works when the equation has constant coefficients.

Best regards

Dear Issam Kaddoura

I think two parameters 'a' and 'V' are constant, else you can apply reproducing kernel Hilbert space method to compute its analytical solution.

In appreciation

Dear Mehdi Mesrizadeh

The coefficients of the original second-order equation are

1, a(x-1) and x - V which are variables !

The reproducing kernel Hilbert space method is practical for solving integrodifferential equations.

Regards

Dear Issam Kaddoura,

Thank you very much for the text book.

If you include the assumption (dx/dt)=u as another equation it is okay.

You know also that u=u(x(t)) which is u(x,t).

But from your cited assumptions and to avoid any confusion, how do you calculate your Jacobian ?

Dear Mohammed Lamine Moussaoui

The Jacobian 2x2 matrix is given by the general form as

((∂f)/(∂x)) ((∂f)/(∂u))

((∂g)/(∂x)) ((∂g)/(∂u))

= 0 1

au-1 ax

Next compute the Jacobian matrix at the equilibrium state (V, 0)

= 0 1

-1 aV

, eigenvalues: (1/2)√(V²a²-4)+(1/2)Va,(1/2)Va-(1/2)√(V²a²-4)

For the supposed values for V=3.887 and a = 65

The eigenvalues are positive, and we deduce that the equilibrium state is not stable.

To find the analytic solution, you can replace the nonlinear system by the linear one and apply the Hartman - Groban theorem.

It is elementary and direct.

Best regards

Dear Issam Kaddoura and Manish Mandal,

I think that the signs in the first post have changed (updated) !

I have almost found the same elements like you in the J matrix where J(1,1)=0 J(1,2)=u J(2,1)=-a.u-1 J(2,2)=a.(1-x)

I am not familiar with the given solving methods names because I am a Mathematical Mechanician not a Specialysed Mathematician but I have studied in the past some solving methods with Russian Professors see Piskounov's notations. Good luck for the continuation.

I am sure that the first solution that I have given is mathematically correct. You can also check if there is a correspondance.

Manish Mandal :

I think the (reformulated) question is remarkably unambitious! Don't you want to learn in more detail how your circuit will behave under various conditions? With the given parameters the system will undergo a rapid, extremely dull, decay to its stable minimum at x=V. Not much to do research on!

The attached pdf contains some general analysis, not yet complete, and not yet illustrated and quality controlled with numerical solutions.

Kåre Olaussen

Thank You for your insightful answer. I will surely look into the dynamics in general as it is really very interesting problem.

I have one question. What is your motivation behind your change of variables/ substitutions?

Mohammed Lamine Moussaoui

Thank you for the suggestion. While trying to set the general solution to x=h*exp(i.omega.t), is h is taken as constant ? and also is 'i' here is the complex number iota?

Dear Manish Mandal

You look its solution having the following

form, possiblity.

Thanks

Manish Mandal: "What is your motivation behind your change of variables/ substitutions?"

The motivation is to transform the equation to a form where the behaviour of the solution can be understood by just looking. Which is not always possible, but in this case it is. For the integrable case, V=1, the equations take Hamiltonian form

q' = ∂H/∂p, p' = -∂H/∂q

so they can be interpreted as the equation of motion of a particle that oscillates back and forth in a potential well. For the general case, with V > 1, there is some mechanical friction in addition. This gradually reduces the energy of the particle, and brings it to halt at the bottom of the potential well.

It is not mathematically necessary to make the transformations, but then one (at least I) has no way of understanding the solution by in-your-head visual calculations.

I don't think it generally is much gain in searching for `analytic solutions'. Most of the time they don't exist, and even when they do they may not provide much insight, if presented as a complicated algebraic formula involving obscure functions. It is better to interpret the equations as describing an already familiar phenomenon, if possible.

Dear Kåre Olaussen

Thank you for the careful interpretation to study the current second-order DE.

I have the following notes on your approach:

(1) Your autonomous system is equivalent to the one that I have shown in my previous answer.

(2)The difference between the two systems is that it transforms the equilibrium point from ( x, u) = (V, 0) into (ξ, η)= (0, 0).

(3) The restriction adx/dt > -1 is unnecessary if we use the first system.

(4) In contrary to your conclusion, the equilibrium point at (V, 0) in my notations is not stable. So, the equilibrium at (0,0) should be not stable too!

To check the stability at the origin, first you need to compute the Jacobian at (0,0)

which is

((∂f)/(dξ)) ((∂f)/(dη))

((∂g)/(dξ)) ((∂g)/(dη))

=

e^{aη} 0

-1 -a(V-1)e^{-aη}

=

1 0

-1 -a(V-1)

it has the eigenvalues: 1 and -a(1-V) that confirm the instability of the equilibrium state at (0,0). It is not stable for any other choices for a and V

since 1 is a positive eigenvalue.

Best regards

Issam Kaddoura :"Your autonomous system is equivalent to the one that I have shown in my previous answer"

It better be, if we do our algebra right. Because we are studying the same system.

Issam Kaddoura : "In contrary to your conclusion, the equilibrium point at (V, 0) in my notations is not stable. So, the equilibrium at (0,0) should be not stable too!"

Consider small deviations from the the point (V, 0), i.e. x = V + ξ, with ξ small. The linear equation for ξ becomes, when (safest) inserted into the original equation:

ξ'' + a(V-1)ξ' + ξ = 0

This obviously has positive damping when 2α ≡ a(V-1) is positive, in which case the point (V, 0) is stable. Please recheck your analysis. The ansatz ξ = exp(λt) leads to the eigenvalues

λ = -α ± √(α2-1)

So the critical point (V, 0) is attractive for α ≥ 1, inward spiral for 0

There is one simple exact solution

x=V,

a constant independent of t

Then the derivatives are zero of it, and voila.

Juan Weisz : "There is one simple exact solution"

Well, if you look at the previous discussions, that is sort of known...

Finding, and classifying, the critical points is the mandatory first step when analysing autonomous systems.

Dear Kåre Olaussen

I think we are on the same track, and I agree that approach will provide an exciting choice of the Hamiltonian, which you may sure it acts as Lyapunov function. It means H should be positive, and dH/dt is negative. ( Both zero at origin). I think you are interested in analyzing some particular cases. After checking the system, the eigenvalues are complex + i, -i and the real parts are zeros. So the Lyapunov function or( Hamiltonian) is necessary to decide the stability of the origin. You have proved that dH/dt is strictly negative ( besideH(0,0)= 0) We need to set the required conditions for H positiveness( I think you did). For the case V =1, I think we have dH/dt =0, which means the solution trajectory is a subset of H = constant. PS. For a complete analysis, I think we have limit cycles. Best regard

You can certainly proceed by assuming the solution is nearly this (x=V)

First y=x-V

DDy +a(y+V-1)Dy +y=0

Next neglect y compared to V-1

You just get a damped harmonic oscilator which you can solve.

Juan Weisz

Your observation is already done.

We are in the process of studying the generated autonomous dynamical system.

(V, 0) is called the equilibrium state of the system.

Regards

Manish Mandal

Thank you for your interest.

The suggestion is for the post of {6 days ago} with the given solution.

Your question is for the post of {5 days ago} : in this case I haven't detailed but I have told you {using the new variables} because it is difficult to solve it with x multiplied by x' so you have to use w=h.exp(i.omega.t) instead of x where h is the constant amplitude and i the complex number.

By ODE theory, we know that there is the uniqueness of solution. Also, for this problem, I obtain its solution by the first Integral method, as follows

If a=0

x(t)=c_1 sint+c_2cost+b

else if a\noneq 0

x(t)=constant

The problem with previous discussion is very little or no

theory for nonlinear ODE. If it were linear you are assured of

two independent solutions.

Here you are assured of nothing.

Dont think that to try to visualize this in phase space, with the extra variable u, adds much.

The large value for (a) suggests a strongly damped problem.

For those who are not known how to transform a second-order differential equation into an autonomous system, the attached article is useful. See the introduction and read the example 1.1.3.

Juan Weisz "The problem with previous discussion is very little or no

theory for nonlinear ODE. Here you are assured of nothing."

That is not true for this case, for initial conditions satisfying (1+ ax') > 0. As you can read from my pdf-note some posts ago. A lot is known and understood about mechanical problems with conservative forces, also when the dynamics is non-linear. The presence of friction complicates matter a bit, but one may still get a reasonably good qualitative understanding of the system by just looking at (or imagining) the potential function. In this case, extension to the region (1+ ax') < 0 leads to some complication, including runway solutions. So, the system may not be modelled physically realistic in that region.

Manish Mandal

The General Solution that I have Developed is in the Attached File.

It Complies with the Updated Main Post Informations.

You Can Remark that Your System Must Satisfy a New Condition on the Constant a.

Regards

Mohammed Lamine Moussaoui:

1. You have a sign error in your second equation (in the post above).

2. It is a very good idea to check your eigenvalues of the Jacobi matrix (at the critical point) against a direct linearisation of the second order equation (around the point x=V).

Manish Mandal

I have Updated the Attached File.

Dear Mohammed,

The ethics of research, you should mention that:

You followed my method which was repeated many times in this thread.

Otherwise, it is called "plagiarism"!!

Best regards

Kaddoura,

You are wrong.

The DETERMINANT’S METHOD is published in the book of N. PISKOUNOV, CALCUL DIFFERENTIAL ET INTEGRAL, EDITIONS OPU, ALGERIA as I have mentioned above.

The SOLVING PROCESS that I have given is MY DEVELOPMENT. These are MY RIGHTS.

I have not found the same eigenvalues like you.

Where are your details and where is the similarity ? GET OUT !

I have also found that my colleague M. MESRIZADEH has given a solution which does not correspond to the main differential equation. She must check his developments. I have not enough time to do this for her. It is also important to notice that the form of the assumed solution is a choice but it must correspond.

Another remark is that my other colleagues have given brief solutions which are not clear and detailed. I preferred to share the detailed method with my other colleagues.

Mr MANISH MANDAL has also to check the origin of his differential equation. This not the end of the world.

Mohammed Lamine Moussaoui

Go back to my answers before 5 and 6 days. It is the same answer.

It transformed into an autonomous system and using the Jacobian and then the rest is straight forward. This was my answer!!

The details are a direct consequence!!

All of us need to avoid " plagiarism" when show answers.

Or at least you need to mention the answers of the others.( Quoting)

My answer was ( 6 days ago )

((( Dear Mohammed Lamine Moussaoui

No, it is an autonomous dynamical system in two nonlinear equations

dx/dt = f(x,u) = u

du/dt= g(x,u) = -a(1-x)u - x + V ,

direct integrals didn't work!!

To study the Jacobian matrix is a must.

I can show the complete answer. It is a simple exercise.

Any way the attached textbook is an excellent reference to show you the details.

Best wishes )))

Do you think that your answer is a different one?

Regards

Issam Kaddoura : I don't think this is much to fuzz about. As I have pointed out above, Mohammed Lamine Moussaoui does not start with correct equations. It is silly to quarrel over priority to wrong results.

Kåre Olaussen

The quarrel about the priority of which results? The whole issue is elementary.

It is a matter of the approach on how to tackle the original problem!

Regardless of the wrong results that may be caused by a typing error, it is also followed by incorrect interpretations. The main point is to preserve the ethics of answers and show respect to all contributors.

Issam Kaddoura :

It is not a matter of typing error, since the mistake propagates further in the note.

Thank you everyone for your suggestions and answers. There is no point in quarrelling as it neither helps you nor me in actually trying to tackle the problem.

Thank you again.

Hi,

The following documents and links give details.

https://www.springer.com/journal/12591

Best Regards.

See the following link.

https://b-ok.africa/book/1243202/ba8fc5

There is a simple and effective classical method for solving autonomous ODEs. Let x'(t) = p(x). Then x'' = p' * p and your equation transforms as follows: p' * p +a * (x-1) * p + x = V. That is Chini's equation.

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