Based on the results discussed on the invariant planes in the known literature Universe 2022, 8, 365 for the flat FLRW space-time universe model with ideal fluid under the gravity of f(R,T)=ξRα+ζ−T, ...this paper continues to describe the global dynamics of this model in the three-dimensional space containing infinity through dynamic system analysis. Moreover, the cosmological solutions of all the physical significance regions restricted by three invariant planes are also fully discussed.
We study the behavior of the normalized Ricci flow of invariant Riemannian metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti ...number equal to one, and the Stiefel manifolds
V
2
R
n
and
V
1
+
k
2
R
n
, with
n
=
1
+
k
2
+
k
3
. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
In this work we study the dynamics at infinity of a piecewise smooth vector fields in R3. We provide conditions for the 24-parameter family to undergo a local bifurcations of codimension zero and one ...at infinity. These results can be useful to understand the global dynamics of piecewise smooth vector fields. In addition, we apply our results to two families of systems from control theory, in one of them studying the existence of limit cycles at infinity.
In this paper by using the Poincaré compactification in
we make a global analysis of the Maxwell-Bloch system
with
, b, c and
. We give the complete description of its dynamics on the sphere at ...infinity. For some values of the parameters, this system has first integrals and invariant algebraic surfaces. For these sets, we provide the global phase portraits of the Maxwell-Bloch system in the Poincaré ball (i.e. in the compactification of
with the sphere
at infinity).
In this paper, the behavior of dynamics ‘at infinity’ of a four-dimensional autonomous food web system has been investigated. For this, a topological method has been developed to understand the ...geometry of the Poincaré compactification which investigate the behavior of the vector field at infinity. The global phase portrait has been shown on the Poincaré disc.
In Artés et al. (Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case. Birkhäuser, Basel, 2019) the authors proved that ...there are 1765 different global geometrical configurations of singularities of quadratic differential systems in the plane. There are other 8 configurations conjectured impossible, all of them related with a single configuration of finite singularities. This classification is completely algebraic and done in terms of invariant polynomials and it is finer than the classification of quadratic systems according to the topological classification of the global configurations of singularities, the goal of this article. The long term project is the classification of phase portraits of all quadratic systems under topological equivalence. A first step in this direction is to obtain the classification of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1765 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological configurations of singularities for the whole quadratic class. The 8 global geometrical configurations conjectured impossible do not affect this number of 208. From here the next goal would be to obtain a bound for the number of possible different phase portraits, modulo limit cycles.
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