Kamal. H. Yasir1 , Hassan. Sh.Kadem2

1 Department of Mathematics, College of Computer Sciences and Mathematics,Thi-Qar University, Thi-Qar, Iraq.

2 Department of Mathematics, College of Education for Pure Sciences, Thi-Qar University, Thi-Qar, Iraq

**Abstract**

Motivated by a great useful of some types of non autonomous differential algebraic

equation systems ( so called strangeness free ) and its applied in different scientific

fields, we present several new results for studying such systems by classical Floquet

Theory, which we extended from linear periodic ordinary differential equation systems

into linear periodic differential algebraic equation systems. For both systems we

investigate that they have the same Floquet exponents. The relation between monodromy

matrices of both systems is also presented. Classification of solution according to the

nature of Floquet exponent is established. Then according to these results, we study the

stability and bifurcation phenomenon of our differential algebraic equation systems.

Key Words: Differential lgebraic equation Floquet theory stability bifurcation

**References**

[1] Cutsem.T.V.,”Voltage instability: Phenomena, countermeasures, and analysis methods”

Proceedings of the IEEE, vol. 88, Issue 2, pp. 208-227, Feb. (2000.)

[2] Lawrence P.“Differential Equations and Dynamical Systems”,Third addition, Springerverlag, USA,

(2001.)

[3] Jianjun P. “New Results in Floquet theory”, Department of Mathematics. The College of William

and Mary Williamsburg, VA 23187, USA, (2003).

[4]Vorgelegt V., “ Linear Differential-Algebraic Equations of Higher-Order and the Regularity

orSingularity of Matrix Polynomials”, Dipl.-Math. Chunchao Shi von der Fakultat II –Mathematikund

Naturwissenschaften der TechnischenUniversitat Berlin zurErlangung des akademischenGrades,Berlin,

( 2004).

[5] Alexandra D. and S.c. Sinha.., “Bifurcation Analysis of Nonlinear Dynamic Systems withTime-

Periodic Coefficients”,Nonlinear Systems Research Laboratory,Department of MechanicalEngineering,

Auburn University,202 Ross Hall, Auburn, AL 36849, USA, (2006).

[6] 1M. Sahadet H. and 2M. Mosta_zur R.,”linear differential algebraic equations withconstant

coefficients” 1Deparment of Mathematics, Chemnitz University of Technology, Germany and

2Department of Information Engineering and Science, University of Trento, Trento,Italy.July, (2009.)

[7]Thomas B.“On stability of time-varying linear differential-algebraic equations”,Technische

UniversitatIlmenau, 22.02, (2010)

[8] Vu Hoang L.,“spectral analysis for linear differential algebraic equations”, Faculty of Mathematics,

Mechanics and Informatics Vietnam National University 334, Nguyen Trai Str., Thanh Xuan,

Hanoi,Vietnam Volker MehrmannInstitut fur Mathematik, MA 4-5 Technische Universitat Berlin D-

10623 Berlin, Fed. Rep. Germany, (2011.)

[9] Ernst H.,“Solving Differential Equations on Manifolds”, University of Reodiganero, June, (2011.)

[10]Thomas B., and Achim Ilchman.“On stability of time-varying linear differential algebraic

equations”,Institute for mathematics, Ilmenau, University of technology, Germany, 31 January, (2013.)

[11] Eich-Soellner, E., and F ̈uhrer, C.,”Numerical methods in multibody dynamics” Stuttgart:

Teubner,(1998).

[12] Riaza, R.,”Differential-algebraic systems. Analytical aspects and circuit applications” World

Scientific Basel: Publishing, (2008).

[13] Kumar, A., and Daoutidis, P.,” Control of nonlinear differential algebraic equation systems with

applications to chemical processes” volume 397 of Chapman and Hall/CRC research notes in

mathematics. Boca Raton: Chapman and Hall, (1999).

[14]Ilchmann, A., and Mehrmann, V., “A behavioral approach to time-varying linear systems, Part 1:

General theory”SIAM Journal on Control and Optimization, 44(5), 1725–1747. (2005a).