توصيفگر ها :
فرم نرمال مداري , تجزيه ميدان هاي برداري سه بعدي , انشعاب هاپف-صفر , سيستم فتزوگ-نگومو
چكيده فارسي :
روش فرم نرمال يكي از روش هاي پركاربرد به منظور حذف (كاهش) عبارت هاي غيرخطي در مطالعه سيستم هاي ديناميكي به شمار مي رود. در اين پايان نامه كه براساس مقاله [ 10 ] مي باشد، با معرفي يك تجزيه برداري جديد براي خانواده اي از سيستم هاي سه بعدي كه قسمت اصلي آن ها داراي ديورژانس صفر و تنها وابسته به دو متغيير x,y (مستقل از z) است، به محاسبه فرم نرمال مداري مي پردازيم. خانواده سيستم هاي منفرد با تكينگي هاپف-صفر نمونه اي از اين دسته از سيستم ها هستند. ما با معرفي يك تجزيه جديد براي ميدان هاي برداري سه بعدي شبه همگن، فرم نرمال مداري شبه همگن را براي سيستم هاي مذكور تجزيه و تحليل مي كنيم و نتايج به دست آمده براي تكينگي هاپف-صفر غيرتباهيده كه در اين نوع سيستم ها قرار مي گيرد، اعمال مي شود. علاوه بر فرم نرمال هاپف-صفر، يك فرم نرمال پارامتريك به دست مي آيد و عبارات تحليلي براي ضرايب فرم نرمال ارائه مي شود. فرمول هاي ضريب فرم نرمال اساساً براي كاربردهاي عملي آن ها در تجزيه و تحليل و كنترل انشعاب مفيد هستند. در نهايت نتايج به دست آمده را بر روي يك سيستم سه بعدي فتزوگ-نگومو اعمال مي كنيم.
چكيده انگليسي :
This M.Sc. thesis is based on the following papers
• A. A, E. F, E. G, C. G., Orbital normal forms for a class of three-dimensional systems
with an application to Hopf-zero bifurcation analysis of Fitzhugh–Nagumo system, Applied Mathematics and
Computation 369 (2020) 1-21
The normal form method is one of the most important mathematical techniques that is commonly used to eliminate
(reduce) nonlinear expressions in the local theory of dynamic systems. In other words, by using the normal form
method for nonlinear systems that have complex local branching, their dynamics can be more easily analyzed near
critical points.
Normal forms of singular differential systems has had a long and extensive literature of more than a century. In fact,
normal form theory dates back to the original ideas of Poincare in his Phd thesis, where he tried to linearize nonlinear
systems using nonlinear changes of coordinates and solve the linear system. This would give rise to a solution for the
original system. However, the linearization is not typically possible for singular systems, yet the idea to simplify the
system remains fruitful.
In this method we seek to find a coordinate system in which the dynamic system becomes as simple as possible. In
general, coordinate changes are nonlinear functions of dependent variables and are obtained by solving a sequence of
linear problems.
We analyze the quasi-homogeneous orbital normal forms for this kind of systems. We introduce a new splitting for quasihomogeneous
three-dimensional vector fields. This spitting is discussed through its relations with the first two state
variables.
The above structure appears in a wide class of systems. The degeneracy corresponding to the interaction of Takens-
Bogdanov and fold bifurcations falls in this case. This is associated with a triple zero eigen-value with geometric
multiplicity two. A family of Hopf-zero singularities is also among large and important family of systems that they
falls into this kind of systems. The obtained results are applied to the nondegenerate Hopf-zero singularity. we obtain
normal forms under conjugation for this family. Furthermore, orbital normal forms for this family of Hopf-zero
systems are obtained.
We derive a parametric normal form for a versal unfolding system of the Hopf-zero singularity. Symbolic expressions
are derived and presented for the leading normal form coefficients. These normal form coefficient form formulas are
fundamentally useful for their practical applications in bifurcation analysis and control. Finally, the results are applied
to a case of the three-dimensional Fitzhugh–Nagumo system.
