توصيفگر ها :
مدهاي نرمال , انشعاب مضاعف سازي دوره تناوب , تقارن كاهشي , پايا , فرم هاي نرمال , اختلال
چكيده فارسي :
در اين پايان نامه به مطالعه ي خانواده اي از دستگاه هاي هميلتوني از دو درجه آزادي با يك تعادل در تشديد 1:2 مي پردازيم. فرم نرمال برشي دستگاه را معرفي كرده و ابتدا با تقارن S1، سپس با تقارن Z2 ديناميك ها را به يك درجه آزادي كاهش مي دهيم و نشان مي دهيم كه تقارن Z2*Z2باعث مي شود، تشديد 1:2 به تشديد 2:4 در شكل اصلي خود تبديل شود. سپس با يك روش هندسي نقاط تعادل دستگاه كاهش يافته را مطالعه كرده و دنباله ي انشعابات ممكن دستگاه را بررسي مي كنيم. در پايان نتايج به دست آمده را روي ديناميك هاي دستگاه اصلي پياده سازي مي كنيم و با استفاده از يك مثال نتايج به دست آمده را به كار مي گيريم.
چكيده انگليسي :
This thesis is defined by [23]. We deal with families of Hamiltonian systems in two degrees of freedom with an
equilibrium in 1 : 2 resonance which is equivariant with respect to the mirror symmetries. We consider a family of
Z2 * Z2 symmetric Hamiltonian systems in two degrees of freedom near an elliptic equilibrium and will show this
Z2 * Z2 symmetry causes the 1 : 2 resonance to become a higher-order resonance. Then we study 2 : 4 resonance
in its own right.
Normalization methods are one of the most important mathematical techniques for studying dynamical systems. The
normal form of Hamiltonian systems causes to remove some terms and then their dynamics can be analyzed more
easily. Symmetric mathematical models are very important in physical systems. We apply the truncated normal form
for the system and reduce the system to a system of one degree of freedom by the S1 symmetry and then by the
remaining Z2 symmetry. Using a geometric method, we investigate the equilibria of the reduced system and describe
the possible bifurcation sequences. Many scientists, such as Kozlov, Tuwankotta, Verhulst and Broer have researched
different topics related to 1 : 2 resonance and they have achieved interesting results. Following Contopoulos [12],
we denote the resulting problem as 2 : 4 resonance. It has several features of the lowest-order case and can be studied
with similar methods.
The elliptic equilibria are called banana orbits and hyperbolic equilibria are called anti-banana orbits. Also, in astronomical
systems, the stable orbits are usually called bananas and the unstable ones anti-bananas. In this thesis, we
apply more general systems and prefer to follow a different nomenclature. We show the short axial orbit becomes
dynamically stable everywhere except at a simultaneous bifurcation of banana and anti-banana orbits, while the long
axial orbit loses and regains stability through two successive period-doubling bifurcations.
By KAM theory, the persistence of invariant 2-tori can be understood when higher-order terms add. The main goal
is a general understanding of the bifurcation sequences of periodic orbits in general position from the normal modes,
parametrized by the energyE, the detuning parameter δ and the independent coefficients characterizing the nonlinear
perturbation. Changing detuning parameters, we see normal modes losing their stability and period-doubling bifurcations
arise, while as we said for the long axial orbit, losing and regaining stability two period-doubling bifurcations
arise.
We apply our results for a specific class of applications. We finalize with some final comments and other conclusions.
