Narrow Results

We investigate the cosmic dynamics of Rastall gravity in nonflat Friedmann–Robertson–Walker (FRW) space–time with barotropic fluid. In this context, we are concerned about the class of model satisfying the affine equation of state. We derive the autonomous system for the Rastall model with barotropic fluid. We apply the derived system to investigate the critical points for expanding and bouncing cosmologies and their stable nature and cosmological properties. The expanding cosmology yields de Sitter universe at late times with decelerating past having matter and radiation-dominated phases. Investigation of autonomous system for bouncing cosmology yields oscillating solutions in positive spatial curvature region. We also investigate the consequences of setting-up a model of nonsingular universe by using energy conditions. The distinct features of the Rastall model compared to the standard model have been discussed in detail.

In this paper by using the Poincaré compactification of ℝ³ we describe the global dynamics of the Lorenz system

We use the Poincaré compactification for a polynomial vector field in ℝ³ to study the dynamics near and at infinity of the classical Chua's system with a cubic nonlinearity. We give a complete description of the phase portrait of this system at infinity, which is identified with the sphere 𝕊² in ℝ³ after compactification, and perform a numerical study on how the solutions reach infinity, depending on the parameter values. With this global study we intend to give a contribution in the understanding of this well known and extensively studied complex three-dimensional dynamical system.

In this paper, we perform a global analysis of the dynamics of the Chen system

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

A one-parameter family of differential systems that bridges the gap between the Lorenz and the Chen systems was proposed by Lu, Chen, Cheng and Celikovsy. The goal of this paper is to analyze what we can say using analytic tools about the dynamics of this one-parameter family of differential systems. We shall describe its global dynamics at infinity, and for two special values of the parameter a we can also describe the global dynamics in the whole ℝ³ using the invariant algebraic surfaces of the family. Additionally we characterize the Hopf bifurcations of this family.

In this article, we consider the class of all real quadratic differential systems , with gcd(p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in ${ℝ}^3$ having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree three with a nilpotent saddle at the origin and ${\mathbb{Z}}_2$-symmetric with $(x,y)\mapsto (-x,y)$.

We provide normal forms and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-equivariant symmetry. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

We provide the normal forms, the bifurcation diagrams and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-symmetries.

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system

Memristive circuits and systems have been widely studied in the last years due to their potential applications in several technological areas. They are capable of producing nonlinear periodic and chaotic oscillations, due to their locally-active characteristics. In this paper, we consider a cubic four-parameter differential system which models a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and a locally-active current-controlled generic memristor. This system has a saddle-focus equilibrium point at the origin, whose global stable and unstable manifolds are, respectively, the $x$-axis and the plane $x=0$, which are invariant sets where the dynamic is linear. We show that this structure can generate two twin Rössler-type chaotic attractors symmetrical with respect to the plane $x=0$. We describe the mechanism of creation of these chaotic attractors, showing that, although being similar to the Rössler attractor, the twin attractors presented here have simpler structural mechanism of formation, since the system has no homoclinic or heteroclinic orbits to the saddle-focus, as presented by the Rössler system. The studied memristive system has the rare property of having chaotic dynamics and an invariant plane with linear dynamic, which is quite different from other chaotic systems presented in the literature that have invariant surfaces filled by equilibrium points. We also present and discuss the electronic circuit implementation of the considered system and study its dynamics at infinity, via the Poincaré compactification, showing that all the solutions, except the ones contained in the plane $x=0$, are bounded and cannot escape to infinity.

In this paper, integrability and the global dynamics of two chaotic systems, Coullet and Malasoma systems, are studied. We mainly use the contradiction technique to show that both systems have no polynomial, Darboux and rational first integrals. Moreover, it is proved that the Cullet system has no analytic first integrals if some conditions on the parameters are satisfied. We also give a complete description of the dynamics at infinity by Poincaré compactification technique for both aforementioned systems.

One of the most important tasks in the qualitative theory of differential equations in the plane is the study of global asymptotic stability: an equilibrium point that is globally attractive. It is known that if an equilibrium point of a planar vector field is globally asymptotically stable, then the vector field is bounded. A planar vector field is said to be bounded if the forward orbit of every point enters and remains in a compact set. In this paper, we study cubic Kukles systems that are bounded which is the first step toward the characterization of cubic Kukles systems that are globally asymptotically stable. We emphasize that these systems form a seven-parameter family of polynomial differential systems. We obtain a total of 25 cubic Kukles sub-systems that are bounded and we provide 11 phase portraits of such sub-systems in a neighborhood of infinity.

In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.

This paper deals with the global dynamics of planar piecewise smooth differential systems constituted by two different vector fields separated by one straight line that passes through the origin. From a quasi-canonical family of piecewise quadratic differential systems with a pseudo-focus point at the origin, which has six parameters, we investigate the subfamilies where the origin is indeed a pseudo-center. For such subfamilies, we classify its global phase portraits in the Poincaré disk and the associated bifurcation sets.

We characterize the global phase portraits in the Poincaré disc of all the planar Lotka–Volterra quadratic polynomial differential systems having a Darboux invariant.

We apply the central extension technique of Poincaré to dynamics involving an interacting mixture of pressureless matter and vacuum near a finite-time singularity. We show that the only attractor solution on the circle of infinity is the one describing a vanishing matter-vacuum model at early times.