This work concerns the description of all possible global dynamics on the Poincaré disc associated to a family of planar semi-homogeneous polynomial vector fields of the form
X
=
(
(
x
+
y
)
y
,
x
3
...+
B
x
2
y
+
C
x
y
2
+
D
y
3
)
with three real parameters
B
,
C
,
D
where
B
-
C
+
D
≠
1
.
This study focuses on the integrability and qualitative behaviors of a quadratic differential system
x
˙
=
a
+
y
z
,
y
˙
=
-
y
+
x
2
,
z
˙
=
b
-
4
x
.
We provide some new perspectives on the system ...and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.
We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the x-axis having a nilpotent ...center at the origin.
In this paper by using the Poincaré compactification of R
3
we make a global analysis of the model
x′ = − ax
+
y + yz, y′ = x − ay + bxz, z′ = cz − bxy.
In particular we give the complete description ...of its dynamics on the infinity sphere. For
a + c
= 0 or
b
= 1 this system has invariants. For these values of the parameters we provide the global phase portrait of the system in the Poincaré ball. We also describe the
α
and
ω
-limit sets of its orbits in the Poincaré ball.
In this paper by using the Poincaré compactification in ℝ
3
we make a global analysis of the model
x
′ =
z, y
′ =
b
(
x
-
dy
),
z
′ =
x
(
x
2
- 1 ) +
y + cz
with
b
∈ ℝ and
c, d
∈ ℝ
+
, here known as ...the three-dimensional Newell-Whitehead system. We give the complete description of its dynamics on the sphere at infinity. For some values of the parameters this system has invariant algebraic surfaces and for these values we provide the dynamics of the system restricted to these surfaces and its global phase portrait in the Poincaré ball. We also include the description of the
α
-limit and
ω
-limit set of its orbits in the Poincaré ball including its boundary, that is, in the compactification of ℝ
3
with the sphere at the infinity. We recall that the restricted systems are not analytic and so in this paper we overcome this difficulty by using the blow-up technique.
In this paper, we study the dynamics of the FitzHugh–Nagumo system
x
˙
=
z
,
y
˙
=
b
x
-
d
y
,
z
˙
=
x
x
-
1
x
-
a
+
y
+
c
z
having invariant algebraic surfaces. This system has four different types ...of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in Valls (J Nonlinear Math Phys 26:569–578, 2019). Using the quasi-homogeneous directional blow-up and the Poincaré compactification, we describe the dynamics of the FitzHugh–Nagumo system having the two remaining classes of invariant algebraic surfaces. Moreover, for these FitzHugh–Nagumo systems we prove that they do not have limit cycles.
We consider the radial symmetric stationary solutions of
u
t
=
Δ
u
-
|
x
|
q
u
-
p
. We first give a result on the existence of the negative value functions that satisfy the radial symmetric ...stationary problem on a finite interval for
p
∈
2
N
,
q
∈
R
. Moreover, we give the asymptotic behavior of
u
(
r
) and
u
′
(
r
)
at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at
r
=
0
for
p
∈
N
and
q
≥
-
2
. In fact, the behavior of solutions for
q
>
-
2
and
q
=
-
2
are different. In each case of
q
=
-
2
and
q
>
-
2
, we derive the asymptotic behavior for
r
→
0
and
r
→
∞
. These facts are studied by applying the Poincaré compactification and basic theory of dynamical systems.
Using the qualitative theory of the differential equations we describe the global dynamics of the cosmological model based on Hořava–Lifshitz gravity in a Friedmann–Lemaître–Robertson–Walker space ...time with zero curvature and without the cosmological constant term.
PDF
