Abstract:
Differential equation have important application and are powerful tool in the study of
many problems in the natural sciences and in technology; they are extensively employed
in mechanics, astronomy, physics, and in many problems of chemisty and biology. Direct
resolution of a differential equation is usually difficult or impossible.
However, another way out it possible. This is the qualitative study of differential equations. This study makes it possible to provide information on the behavior of the solutions
of a differential equation without the need to solve it explicit, and it consists in examining
the properties and the characteristics of the solutions of this equation, and to justify among
these solution, the existence or non existence of an isolated closed curve form called limit
cycle.
An important problem of the qualitative theory of differential equations is to determine
the limit cycles of a systems of a differential equations.
Usually, we ask for the number of such limit cycles as orbits, and an even more difficult
problem is to give an explicit expression of them.
The limit cycles introduced for the first time by Henri Poincaré in 1881 in his "Dissertation on the curves defined by a differential equation" [6]. Poincaré was interested in
the qualitative study of the solutions of the differential equations, i.e. points equilibrium,
limit cycles and their stability.
This makes it possible to have an overall idea of the other orbits of the studied systems.
The mathematician David Hilbert presented at the second international congress of mathematics ([3], 1900), 23 problems whose future awaits resolution through new methods that
will be discovered in the century that begins. The problem number 16 is to know the maximum number and relative position of the limit cycles of a planar polynomial differential
systems of degree n. We denote Hn this maximum number. Dulac [2] in 1923, offered
a proof that Hn is finite. In recent years, several papers have studied the limit cycles of
planar polynomial differential systems. The main reason for this study is Hilbert 16-th
unsolved problem. Later on Van der Pol [7] in 1926, Liénard[4] in 1928 and Andronov
[1] in 1929 shown that the periodic solution of self-sustained oscillation of a circuit in a
vacuum tube was a limit cycle.
The objective of this work is to give a quintic polynomial differential system of the
2
form:
x˙ = x − (γ(2y − ax) + α(x
2 + y
2
)(ax − 4y))Q(x, y),
y˙ = y − (−γ(2x + ay) + α(x
2 + y
2
)(4x + ay))Q(x, y).
(1)
Where Q(x, y) is homogeneous polynomial of degrees 2 where α, γ and a are real
constants. The main motivation of this dissertation is to prove that these systems are
integrable. Moreover, we determine sufficient conditions for a polynomial differential
systems to possess at most two limit cycles, one of them algebraic and the other one
is non-algebraic, counted two explicit limit cycles. Concrete examples exhibiting the
applicability of our result are introduced.
This dissertation is structured in three chapters. The first chapter is dedicated to reminders of some preliminary concepts on the planar differential system. In the second
chapter we put α = 0 and we get a system of degree 3 as follows:
x˙ = x − γ(2y − ax)Q(x, y),
y˙ = y − γ(−2x − ay)Q(x, y).
(2)
We solve this system and we use the available conditions in the theories to prove that this
system possesses at most one limit cycle. In the last chapter, we solve the system in the
case α 6= 0 and apply the theorems conditions in order to prove that the system possesses
at most two limit cycles, and we prove as well the limit cycles algebraic or not.