There are lots of textbooks discussing symmetry analysis on differential equations such as Applications of Lie group to differential equations. All of the methods (at least all of them I have seen) focus on high order differential equations. But is there anything that focuses on dynamic systems? Or is there any proposition stating equivalence of high order differential equations and dynamic systems?
Yes, the most important to me is the book of Cicogna and Gaeta entitled:
Symmetry and Perturbation Theory in Nonlinear Dynamics
1. Bluman, Kumei: Symmetries and DEs
Ovsyannikov: Group anal of DEs
Olver: Applications of Lie groups to DEs
Bluman Cole Similarity methods for DEs
2. If you have, say, a second order ODE for y, then you can turn it into an equivalent system of first order ODEs by introducing y and y' as new variables.
I hope this helps, János
Thanks, János Tóth. I know that high-order ODEs can be translated to dynamical system by introducing high-order derivative as new variable. But how about the inverse process? Given a dynamical system, how to convert it to a single high-order ODE?
Garcia, you mean one can translate a dynamical system to a high-order ODE? How?
From a dynamical system to a higher order ODE: this is what you cannot always carry out because of computational difficulties. It depends on your problem. If the dynamical system is given by a linear DE, then (under some conditions) it is possible.
Is there any theoretical result proving the existence of under what circumstances such a conversion could be done? I wanna know the constructive method. What references can I refer?
As far as I know, it is not always possible to translate a dynamical system to a high-order ODE except in some excepcional cases. For example in you think only in that equivalence for linear system the answer is clear; see any textbook on linear control theory.
To study stability I did (in my paper accepted for publication) the construction of second order linear equation equivalent to 2 dimensional linear system u’(t)=A(t)u(t) under assumption that A is not diagonal. The similar construction should be true for higher dimensional first order system, but I don’t know if it was done before.
Hello Shu-Ch'ang Chang,
The paper is
"Stability of planar nonautonomous dynamic system"
Advances in difference equations. 2013
It is accepted, but not published yet. I could sent you pdf file if you are interested.
Dear Shu-Ch'ang Chang and Isaac Garcia ,
I uploaded my paper
"Stability of Planar Non-autonomous Dynamic Systems."
you are interested in under my publications as dataset.
Let me know if you can't see it.
Formulas (2.1)-(2.3) in the paper are giving the construction of the second order equation that is equivalent to a system (1.1).
As for dynamical systems, see also the book
A. Goriely, Integrability and Nonintegrability of Dynamical Systems,
https://www.researchgate.net/publication/31712346_Integrability_and_Nonintegrability_of_Dynamical_Systems__A._Goriely
Article Integrability and Nonintegrability of Dynamical Systems / A. Goriely.
I wrote a short note on the inversion of the transfer principle (on how to get a second order equaiton from a system of first order equations) in the nonlinear case. It is far from being precise. János
Now the attachment. If you wish to improve and extend it I can send you the TeX file.
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