This work is devoted to investigating the algebraic and analytic integrability of first order
polynomial partial differential equations via an understanding of the well-developed area
of local and global integrability of polynomial vector fields.
In the view of characteristics method, the search of first integrals of the first order
partial differential equations
P(x,y,z)∂z(x,y)
∂x
+Q(x,y,z)∂z(x,y)
∂y
= R(x,y,z),
(1)
is equivalent to the search of first integrals of the system of the ordinary differential equations
dx/dt= P(x,y,z),
dy/dt= Q(x,y,z),
dz/dt= R(x,y,z).
(2)
The trajectories of (2) will be found by representing these trajectories as the intersection
of level surfaces of first integrals of (1).
We would like to investigate the integrability of the partial differential equation (1)
around a singularity. This is a case where understanding of ordinary differential equations
will help understanding of partial differential equations. Clearly, first integrals of the
partial differential equation (1), are first integrals of the ordinary differential equations
(2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2
are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary
function of φ1and φ2.
We choose for our investigation a system with quadratic nonlinearities and such that
the axes planes are invariant for the characteristics: this gives three dimensional Lotka–
Volterra systems
x' =dx/dt= P = x(λ +ax+by+cz),
y' =dy/dt= Q = y(µ +dx+ey+ fz),
z' =dz/dt= R = z(ν +gx+hy+kz),
where λ,µ,ν 6= 0.
v
Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions
for both integrability and linearizability for three dimensional Lotka-Volterra systems for
(1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly
use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.
Also, more general three dimensional system have been investigated and necessary and
sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems.
These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder.
We also give a generalization of Singer’s theorem about the existence of Liouvillian
first integrals in codimension 1 foliations in Cnas well as to three dimensional vector
fields.
Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate
from the periodic orbits of a centre of a cubic homogeneous polynomial system using the
averaging theory of first order.
Date of Award | 2013 |
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Original language | English |
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Awarding Institution | |
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Supervisor | Colin Christopher (Other Supervisor) |
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- Lineraizability
- Three dimensional Lotka-Volterra systems
- First integrals
- Darboux
- Liouvillian
- Invariant algebraic curves
- Exponential factors
- Inverse Jacobi multiplier
- Poincare' domain
- Riccati equation
- Monodromy
- Quasi–homogeneous
- Limit cycles
- Averaging theorem
- Integrability
Analytic and Algebraic Aspects of Integrability for First Order Partial Differential Equations
Aziz, W. (Author). 2013
Student thesis: PhD