look at the Toda flow and the physical meaning of it

There are many real live examples of natural processes that follow your equation for instance, the price X (t, w) of a stock at time t and space w in a stochastic market and population of samples in a dynamic population. The two are known to depend on time and the present value of the random variable X.

Is F a matrix? So, are you looking for examples of kind X'(t)=F(t).X? But It is linear...

May be you meant X and F(t,X) are matrices of the same order.

Whatever, a matrix can be seem as a vector and a vector can be seem as matrix, thus any interisting problem (finite dimension), of real life, that you know, can be seem as matrix equation.

However, if you are looking for examples of square matrix, than you can think, for instance, in 4-Dimensional (n^2-Dimensioanl) systems, since a matrix {{a11,a12},{a21,a22}} is (isomorphic sense) a vector (a11,a12,a21,a22).

The double pendulum is a example of 4 dimensional system. Thus, if you consider the matrix

X=|x1 y1|

|x2 y2|

where x1 and x2 are the positions of the particles and y1 and y2 are their velocity, then you obtain a non-linear equation X'(t)=F(X).

Hi Douglas, Yes, I mean X and F(t,X) are sq. matrices of the same order. Your example sounds interesting. I actually developed a case of 4-dim in the time scale setting and would like to know more about the double pendulum example. Can you cite the book or paper where it is? Thanks :)

Hi Sulaiman, can you send me a concrete example with citation at [email protected]? I will be interested more in population dynamics. Thanks:)

Hi Atiya.

There are many studies about double pendulum model. For instance, you can find this model in both of my two papers 1 and 2:

1 - On the periodic solutions of a perturbed double pendulum, J Llibre, DD Novaes, MA Teixeira. Journal of São Paulo Mathematical Sciences 5 (2), 317 - 330.

2 - On the periodic solutions of a generalized smooth and non–smooth perturbed planar double pendulum with small oscillations, J Llibre, DD Novaes, MA Teixeira, arXiv preprint arXiv:1203.0498.

2 is just a preprint an it is still beeing revised.

Hellow

The most simple example in real life is the harmonic oscillator. It's a second order linear system. The matrices relate its position with its velocity. When a non linear drag is put the complexity grows if I think what integrator I'm going to use. Some complex cases are impossibles to solve for a classic integrator, so we use the electronic analogy.

Other typical examples, that like me, are Mathieu and Van der Pol's oscillators, both build a 2° order flux in R2 and a lot of controlled electronic systems.

Thank you Douglas and Boris! I think I can get something out of your suggestions.

Douglas, please let me know when you upload your second paper on Arxive, if you do:).

Your equation is a compact way to express systems of differential equations. In fact, any such system can be expressed as X'=F(X,t). A great deal of physics, biology, chemistry etc. can be expressed in terms of systems of differential equations, so pick up any calculus-level textbook in the physical sciences and you will find many examples!

Dear Atiya. It is already there.

http://arxiv.org/abs/1203.0498

Best regards.

Dear Atiya, are you looking for a linear or non linear type of ODEs? I work on the heat transfer boundary layer. The governing PDEs are reduced to a set of ODEs using similarity transforms. The resulting non-linear high order differential equations can be written in the proposed form.

Hi Mohammad, I am looking at the nonlinear equations and it will be interesting to lookat your work on heat transfer. Please forward through and thank you very much!

Dear Atiya, i have uploaded three of my recent papers in my profile. The governing ODEs can be written in the proposed form. Please check them.

You may wish to look at the matrix Riccatti equations, see e.g. the papers

http://link.springer.com/chapter/10.1007%2F3-540-17171-1_33

http://scitation.aip.org/content/aip/journal/jmp/37/3/10.1063/1.531448

and references therein.

Another interesting class of examples is provided by integrable ODEs on associative algebras, see this paper by Mikhailov and Sokolov (the second author is on ResearchGate: https://www.researchgate.net/profile/Vladimir_Sokolov3/ ):

http://link.springer.com/article/10.1007/s002200050810

Your case is obtained when one chooses the associative algebra to be the algebra of N x N matrices.

For a related development, see also this paper by Odesskii and Sokolov https://www.researchgate.net/publication/2127554_Integrable_matrix_equations_related_to_pairs_of_compatible_associative_algebras

Article Integrable matrix equations related to pairs of compatible a...