محاسبه فرم نرمال و كاربرد آن در آناليز ديناميك يك مدل درمان ايدز

چكيده انگليسي :

Abstract Many phenomena in chemistry physics and engineering can be mod eled by parametric nonlinear di erential systems These systems demonstratecomplicated dynamics when the parameters reach certain singular values There fore it is important to understand their dynamics near the critical values Normalform theory is one of the most e cient methods for the local bifurcation analysisof such systems The main idea is to use a nonlinear change of coordinates toconvert a given vector eld to its simplest form called the simplest normal form The resulting system shares certain qualitative properties of the original system Here we mean by qualitative properties as those properties invariant under ourpermissible changes of coordinates In this thesis we discuss normal forms for those systems whose linear part hasa pair of imaginary eigenvalues called Hopf singularity as well as those with azero eigenvalue along with a pairof imaginary eigenvalues so called Hopf zero singularity Derivation of the focusvalues are also considered as an application of our computer program implemen tation in Maple of the parametric Hopf singularity with symbolic coe cients Given our approach the rst well known and second order focus values arereadily derived and presented Recently the simplest normal form for Hopf zero singularity has been obtainedthrough a representation of sl 2 Lie algebra over the space of all classical Hopf zero normal forms This gives rise to a decomposition of the space into subspacesinvariant under the sl 2 action This way certain families of vector elds areintroduced and their dynamical interpretations are presented The normal formof this singular system is divided into three general cases In this thesis weobtain some new results that are the extension of the simplest normal forms intothe orbital normal forms Given the additional transformations associated withtime rescaling the orbital normal form is substantially more simple than thesimplest normal form system In fact the main idea in orbital normal form is touse a sequence of near identity polynomial time rescalings along with a sequenceof well coordinated changes of coordinates This has been rarely been performedin the existing literature This is a di cult task due to the di erent algebraicstructures associated with time rescaling and coordinate changes In the existing literature many mathematical models have been introduced inorder for the dynamics study of the HIV 1 virus In this thesis we analyze anordinary di erential equation system that models the ghting of the HIV 1 viruswith a genetically modi ed virus This is to continue a previous result on anHIV 1 therapy model by ghting the HIV 1 virus by injecting another virusinto the infected patient Here a modi cation of the model is proposed that is toadd a constant into the recombinant virus equation The associated dynamics isstudied in details It is showed that an increase in the constant greatly increasesthe Hopf critical value A numerical example is provided to demonstrate thebifurcation direction and stability Here a normal form computational approachis applied and an accurate estimates for the amplitudes and the periods of thebifurcated limit cycles is given Numerical simulations are performed in order tocon rm the theoretical results Finally it is concluded that any increase in isbene cial for controlling eliminating the HIV virus and in order to control thelimit cycles amplitudes Keywords Normal form Hopf Bifurcation Hopf zero Bifurcation HIV 1 ther apy model

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محاسبه فرم نرمال و كاربرد آن در آناليز ديناميك يك مدل درمان ايدز