@aroche 

Thank you for the help!

@dharr 

Thanks a lot for providing the alternative method. It is very helpful!

Dmitry

@ecterrab 

Thank you very much for the fast response. 

Dmitry

@mmcdara 

I have two more questions:

1. I have substituted some data for the final solution. But for each data set, I have received that the value of the integral is zero (please find the attached file). It looks a little bit strange.

2. If I had tried to solve the presented above integral in a common way, i.e., by applying the command Int in Maple, I suppose that I would have received a different solution than you presented. Am I right?

Final_solution.mw

Thanks in advance

Dmitry 

@mmcdara 

Thanks a lot for the help!

@ecterrab 

Hi,

First of all, thank you very much for your intention to help!

I will start with my purpose, then I will give the definitions of normal and hypernormal forms in my understanding. Finally, I will mention works (some of them with Maple code) that deal with this issue.

1. My purpose - to investigate the behavior of the non-linear dynamic system near the fixed points (including bifurcation analysis).

Definitions:

2. Normal Form - Under this term I assume that the differential equation has two parts: (a) linear (defined in the terms of Jordan normal form) + (b) non-linear part. Usually, this form is received after the Taylor expansion,  

3. Hypernormal form - sometimes I see in the literature the other name - "simplest form". In my understanding, this is a continuation of the reduction of the non-linear term in normal form as much as it is possible.

4. My expectation - (a) given any system of non-linear differential equations, I would like to receive normal and hypernormal (simplest) form. Today in the literature exists a number of algorithms for such a reduction but for very specific differential equations. In particular, for the differential equations which have zero eigenvalues. (b) since the reduction requires changing the coordinates, I would like to know, how  I can get back from the reduced form to the origin system.

Literature:

a. Edneral, V. F. (2007, September). An algorithm for construction of normal forms. In International Workshop on Computer Algebra in Scientific Computing (pp. 134-142). Springer, Berlin, Heidelberg - describes the algorithm for the normal form reduction. 

b. Yuan, Y., & Yu, P. (2001). Computation of simplest normal forms of differential equations associated with a double-zero eigenvalue. International Journal of Bifurcation and Chaos, 11(05), 1307-1330 - including Maple code.

c. Murdock, J. (2003). Hypernormal form theory: foundations and algorithms. Mathematics Preprint Archive, 2003(12), 225-271 - gives a comprehensive review of the existing reduction methods to the hypernormal form.

d. Gamero, E., Freire, E., Rodríguez-Luis, A. J., Ponce, E., & Algaba, A. (1999). Hypernormal form calculation for triple-zero degeneracies. Bulletin of the Belgian Mathematical Society-Simon Stevin, 6(3), 357-368 - shows the reduction to the hypernormal form for the system of differential equations with triple zero eigenvalues.

Please note, I saw that there exists add-in to Maple: BifTools. But, it's also limited to some specific types of differential equations.

If you need any additional information, please let me know.

I did not attach my problem because we still working on it and the problem may be two or three-dimensional where the eigenvalues are not zero (at least not all of them). 

Thanks in advance,

Dmitry

@Carl Love 

Thanks a lot!

@Preben Alsholm 

One more question: If the diff systems are really different, not only by uspilon, how is it possible to code such system in Maple to find numerical solution?

Thanks,

Dmitry

@Preben Alsholm 

First of all, thanks a lot. But please pay attention, that the initial values for the state variables are given for the first regime, whereas terminal conditions for the co-state variables are given for the second regime. This means, that I first have to solve the system of the state differential equations for the first regime, then the obtained solutions at t=t1 (where t1 is uknown) I have to used as a initial conditions for the state differential equations of the second regime. Finally, I have to solve the diff system (including both state and co-state equations) of the second regime to find psi_S, which when substituting into the given algebraic equation at t=t1, I will be able to determine unknown t1. I don't know how to do that in Maple, especially when the system cannot be solved explicitly and only numerical solution is possible. 

@elsa21k 

Hi,

First of all thanks a lot for the help. My question is why you left first epsilon in the equation as it is, that is: N*exp(-(1/2)*eta*epsilon. Why?

Thanks,

Dmitry

@Preben Alsholm 

Thank you very much for so helpful answers.

Dmitry

@Preben Alsholm 

First of all thank you very much for your help. I would like to ask additional question. When you solved this equation you assumed that all parameters are equal (specifically, equal to 7). This equation reflects real case where all parameters couldn't be equal, the solution also couldn't be complex and t1,t3>0. In the light of above, how can I know that this condition has additional solutions rather than t1=t3?

Thanks in advance,

Dmitry

@Preben Alsholm 

Hi,

If you can please clarify your answer. Do you mean that this condition holds only when t1=t3?

Thanks,

Dmitry

@Markiyan Hirnyk 

Hi,

Thanks for the answer. 

Dmitry

@Markiyan Hirnyk 

Hi,

Thanks for the response, but actually, it's not the same question. Now, I put attention more on the formal description of the problem (mathematical formulation), rather than on the solution technique. The main question: is it correct way (in mathematical sense) to introduce the reverse system (backward) of the diff. equations by changing the signs of the right hand sides of the original system (of course, the boundary conditions are converted to the the initial conditions). As I saw from the link above, if the system is autonomous (like mine) this is a right way to do it.

Thanks,

Dmitry