A simple chaotic system with only one nonlinearity and five terms was introduced by Sprott. We consider the generalized Sprott differential system. We study the local stability of equilibrium points ...and local bifurcation, in particular, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcation occur in the system, and presented a formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solution by applying normal form theory. Moreover, we study the dynamics near and at infinity by using the Poincar´e compactification to describe the global dynamics of the trajectories of the system. Our results show that the real parameters do not affect the global dynamics at infinity of the system.
We provide the normal forms of all $\mathbb{Z}_2$-symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin. Furthermore, we complete the classification of ...the global phase portraits in the Poincare disk of the above systems initiated by Dias, Llibre and Valls 9.
In this paper, we study the global phase portraits of the discontinuous planar piecewise linear differential system of the focus-center type and the center-center type with a straight line of ...separation. We obtain sufficient conditions for the existence and number of the crossing limit cycles and sliding limit cycles of the system. We prove the system of focus-center type and center-center type have 45 and 15 topologically different global phase portraits, respectively.
A conjugate Lorenz-like system which includes only two quadratic non-linearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their ...stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.
An invariant algebraic surface is calculated for a 3D autonomous quadratic system. Also, the dynamics near finite singularities and near infinite singularities on the invariant algebraic surface is ...analyzed. Furthermore, pitchfork bifurcation is analyzed using center manifold theorem and a first integral of this quadratic system for some special parameters is provided. Finally, the dynamics of this system at infinity using the Poincare compactification in
R
3
is investigated and the singularly degenerate heteroclinic cycles are presented by a first integral and verified by numerical simulations.
In this work we consider the problem of classifying all configurations of singularities, both finite and infinite of quadratic differential systems, with respect to the geometric equivalence relation ...defined in 3. This relation is deeper than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci of different orders. Such distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows to incorporates all these important geometric features which can be expressed in purely algebraic terms. This equivalence relation is also deeper than the qualitative equivalence relation introduced in 17. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in 4 where the classification was done for systems with total multiplicity $m_f$ of finite singularities less than or equal to one. In this article we continue the work initiated in 4 and obtain the geometric classification of singularities, finite and infinite, for the subclass of quadratic differential systems possessing finite singularities of total multiplicity $m_f=2$. We obtain 197 geometrically distinct configurations of singularities for this family. We also give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants. The results can therefore be applied for any family of quadratic systems in this class, given in any normal form. Determining the geometric configurations of singularities for any such family, becomes thus a simple task using computer algebra calculations.
Hopf bifurcation, dynamics at infinity and robust modified function projective synchronization (RMFPS) problem for Sprott E system with quadratic perturbation were studied in this paper. By using the ...method of projection for center manifold computation, the subcritical and the supercritical Hopf bifurcation were analyzed and obtained. Then, in accordance with the Poincare compactification of polynomial vector field in R3, the dynamical behaviors at infinity were described completely. Moreover, a RMFPS scheme of this special system was proposed and proved based on Lyapunov direct method. The simulation results demonstrate the correctness of the dynamics analysis and the effectiveness of the proposed synchronization strategy.
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