The Poincaré compactification is an extension of a polynomial vector field to a compact manifold. We generalize this construction to weight-homogeneous vector fields with weight exponent
(
1
,
ℓ
)
...and different weight-degrees. Then we apply this generalization of the Poincaré compactification to obtain new developments in the real Jacobian conjecture.
Dynamics of a Generalized Rayleigh System Baldissera, Maíra Duran; Llibre, Jaume; Oliveira, Regilene
Differential equations and dynamical systems,
07/2024, Letnik:
32, Številka:
3
Journal Article
Recenzirano
Odprti dostop
Consider the first order differential system given by
x
˙
=
y
,
y
˙
=
-
x
+
a
(
1
-
y
2
n
)
y
,
where
a
is a real parameter and the dots denote derivatives with respect to the time
t
. Such system is ...known as the generalized Rayleigh system and it appears, for instance, in the modeling of diabetic chemical processes through a constant area duct, where the effect of adding or rejecting heat is considered. In this paper we characterize the global dynamics of this generalized Rayleigh system. In particular we prove the existence of a unique limit cycle when the parameter
a
≠
0
.
In Leonov and Kuznetsov (2013), the authors shown numerically the existence of a limit cycle surrounding the unstable node that system (1) has in the positive quadrant for specific values of the ...parameters. System (1) is one of the Belousov–Zhabotinsky dynamical models. The objective of this paper is to prove that system (1), when in the positive quadrant Q has an unstable node or focus, has at least one limit cycle, and when
f=2/3,
q=ϵ2/2, and ϵ > 0 sufficiently small this limit cycle is unique.
We classify the global dynamics of a one-parameter family of planar quadratic polynomial differential systems which for some interval of values of its parameter describes the evolution of a static ...star. The characterization of their distinct topological phase portraits is done in the Poincaré disc. In this way we can describe the dynamics of these systems near infinity and to provide their global phase portrait.
•It is a well-known model that describes the time variation of the interest rate, the investment demand and the price index.•We study its global dynamics in R3.•We show that there is a global ...attractor.
Recently several works have studied the following model of financex˙=z+(y−a)x,y˙=1−by−x2,z˙=−x−cz,where a, b and c are positive real parameters. We study the global dynamics of this polynomial differential system, and in particular for a one–dimensional parametric subfamily we show that there is an equilibrium point which is a global attractor.
We present a general mechanism of generation of limit cycles in planar piecewise polynomial differential systems with two zones by means of a transcritical bifurcation at infinity and from a global ...centre. This study justifies the existence of limit cycles that arise through the intersection of the separation boundary with the one that characterizes the global centre.
In this paper we are interested in a structurally orthotropic stringer shell system, which can be transformed into a planar reversible polynomial Hamiltonian system via the Galerkin method. Numerous ...articles have employed numerical methods to study its dynamics, but only partial dynamical behaviors are obtained due to the limitation of numerical methods. To capture all possible dynamical behaviors of this shell system, we resort to the Poincaré compactification method and the homogeneous directional blow up technique in the qualitative theory of dynamical systems. After reducing the original system to a normal form with less parameters, we prove that this shell system exhibits exactly 14 non-equivalent global dynamics. Moreover, the parameter range corresponding to each kind of global dynamics is accurately determined, and bifurcation diagrams are drawn.
•We obtain all global dynamics of the considered shell system.•The parameter range corresponding to each kind of global dynamics is accurately determined.•Some qualitative methods of dynamical systems are adopted to overcome the limitation of numerical results.
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