Three-dimensional Quadratic System Research Articles

Lower bounds for the cyclicity of centers of quadratic three-dimensional systems

We consider quadratic three-dimensional differential systems having a Hopf singular point. We study their cyclicity when the singular point is a center on the center manifold using higher-order developments of the Lyapunov constants. As a result, we make a chart of the cyclicity by establishing the lower bounds for several known systems in the literature, including the Rössler, Lorenz, and Moon-Rand systems. Moreover, we construct an example of a jerk system to obtain 12 limit cycles bifurcating from the center, which is a new lower bound for three-dimensional quadratic systems.

About the Jacobi Stability of a Generalized Hopf–Langford System through the Kosambi–Cartan–Chern Geometric Theory

In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point.

Cyclicity of rigid centres on centre manifolds of three-dimensional systems

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.

Integrability and linearizability of symmetric three-dimensional quadratic systems

<p style='text-indent:20px;'>We study local integrability and linearizability of polynomial and analytic systems of ODEs. It is proven that in the case of non-degenerate singularity if an analytic system is completely integrable and one equation is linearizable, then the system is linearizable in a neighborhood of the singularity. Some integrable and linearizable systems in a family of three-dimensional quadratic autonomous systems of ODEs depending on ten parameters are found.</p>

Weak Bi-Center and Bifurcation of Critical Periods in a Three-Dimensional Symmetric Quadratic System

This paper focuses on the weak bi-center and bifurcation of critical periods in a class of three-dimensional quadratic systems with a special type of symmetry. Center manifold theory is applied to prove that the system admits either a rough bi-center or a weak bi-center of infinite order (i.e. isochronous bi-center) rather than a weak bi-center of finite order. As a consequence, no bifurcation of critical periods can be generated from the bi-center.

Zero-Hopf bifurcation of periodic orbits in the generalized Rössler system

We apply a technique of Llibre based on the averaging method to a Rössler-type three-dimensional quadratic system and we prove the existence of a periodic orbit.

Integrability and linearizability of a family of three-dimensional quadratic systems

We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of – resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpose is to understand the mechanisms of local integrability. By computing some obstructions, knowing as resonant focus quantities, first we present necessary conditions that guarantee the existence of two independent local first integrals at the origin. For this reason Gröbner basis and some other algorithms are employed. Then we examine the cases where the origin is linearizable. Some techniques like existence of invariant surfaces and Jacobi multipliers, Darboux method, properties of linearizable nodes of two dimensional systems and power series arguments are used to prove the sufficiency of the obtained conditions. For a particular three-parametric subfamily, we provide conditions on the parameters to guarantee the non-existence of a polynomial first integral.

INTEGRABILITY AND LIMIT CYCLES OF A SYMMETRIC 3-DIM QUADRATIC SYSTEM

<abstract> We study periodic solutions and first integrals in a three-dimensional quadratic system of ODEs. Coefficient conditions for existence of centers on center manifolds are obtained. Some bounds of the number of limit cycles bifurcating from the centers under small perturbations are given. </abstract>

Bifurcation analysis on a class of three-dimensional quadratic systems with twelve limit cycles

This paper concerns bifurcation of limit cycles in a class of 3-dimensional quadratic systems with a special type of symmetry. Normal form theory is applied to prove that at least 12 limit cycles exist with 6–6 distribution in the vicinity of two singular points, yielding a new lower bound on the number of limit cycles in 3-dimensional quadratic systems. A set of center conditions and isochronous center conditions are obtained for such systems. Moreover, some simulations are performed to support the theoretical results.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

A Class of Three-Dimensional Quadratic Systems with Ten Limit Cycles

Our work is concerned with a class of three-dimensional quadratic systems with two symmetric singular points which can yield ten small limit cycles. The method used is singular value method, we obtain the expressions of the first five focal values of the two singular points that the system has. Both singular symmetric points can be fine foci of fifth order at the same time. Moreover, we obtain that each one bifurcates five small limit cycles under a certain coefficient perturbed condition, consequently, at least ten limit cycles can appear by simultaneous Hopf bifurcation.

A class of quadratic system with six limit cycles

In this paper, the Hopf bifurcation for a class of three-dimensional quadratic system is investigated by making use of singular values methods. Studied system lies in symmetrical vector field with regard to planar y = x and it has two symmetrical singular points (1,2,1) and (2,1,1). We give the expressions of the first three focal values of the singular point (1,2,1) and show that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can become a fine focus of third order at the same time. Moreover, we obtain that each one of the two singular points (1,2,1) and (2,1,1) of the investigated system can bifurcate three small limit cycles under a certain coefficient's disturbed condition. In sum, six limit cycles can bifurcate from system (1). In terms of Hopf bifurcation of a three-dimensional quadratic system, our results are new and interesting.

Local integrability of a family of three-dimensional quadratic systems

We investigate the local integrability of a family of three dimensional quadratic systems in a neighborhood of (0:−1:1) resonant singular point. We find the necessary and sufficient conditions for the existence of two functionally independent first integrals of the system. Mechanisms for integrability for the systems are either time-reversibility or Darboux integrability.

Determining nonchaotic parameter regions in some simple chaotic jerk functions

In this paper we apply Theorem 2.1 in [Heidel J, Zhang F. Nonchaotic and chaotic behaviour in the three-dimensional quadratic systems: five-one conservative cases, in press] to some simple chaotic jerk functions listed in [Sprott JC. Simple chaotic systems and circuits. Am J Phys 2000;68(8):758–63; Sprott JC. Algebraically simple chaotic flows. Int J Chaos Theory Appl 2000;5(2):1–20] to locate the parameter regions at which they are nonchaotic. We show that for each of the twenty chaotic systems studied here there are some nonchaotic parameter regions. This indicates that our theorem will help reduce the amount of work searching for parameters causing chaos. We also generalize Theorem 2.1 to include systems with exponential functions.

Si’lnikov homoclinic orbits in a new chaotic system

Abstract In the present paper, a new chaotic system is considered, which is a three-dimensional quadratic system and exhibits two 1-scroll chaotic attractors simultaneously with only three equilibria for some parameters. The existence of Si’lnikov homoclinic orbits in this system has been proven by using the undetermined coefficient method. As a result, the Si’lnikov criterion guarantees that the system has Smale horseshoes. Moreover, the geometric structures of attractors are determined by these homoclinic orbits.

Boundedness, invariant algebraic surfaces and global dynamics for a spectral model of large-scale atmospheric circulation

We consider a three-dimensional quadratic system S in R3 with six parameters which appears in geophysical fluid dynamics (atmospheric blocking). In this paper we start its systematic study from the point of view of dynamical systems. First, we reduce the number of its parameters from six to three. Thus, we must study a three-dimensional quadratic system with three parameters, which recalls us the famous Lorenz-63 system. Traditionally, system S has been studied by considering two subcases, called the conservative and the dissipative case, as the parameter responsible for dissipation is zero or not. In the conservative case, we reduce system S to systems without parameters. Among these there are two interesting systems: one is homeomorphic to the simple pendulum, and the other is a perturbation of it. In the latter system the saddle point corresponding to topographic instability is connected to two homoclinic orbits to it. In the dissipative case we prove that all trajectories of system S enter in an ellipsoid for any values of the parameters. We characterize their invariant algebraic surfaces of degree 2, and for those systems having such invariant algebraic surfaces we describe their global phase portraits.

Non-chaotic behaviour in non-dissipative quadratic systems

It is shown that some three-dimensional non-dissipative quadratic systems of ordinary differential equations with four terms, posed by Zhang Fu and Jack Heidel, do not exhibit Chaos. This partially supports the conjecture that three-dimensional quadratic systems of ordinary differential equations with four terms are not chaotic.

Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case

It is shown that almost all three-dimensional conservative quadratic systems of ordinary differential equations with a total of four terms on the right-hand side of the equations do not exhibit chaos. A previous paper showed the same thing for dissipative systems. PACS Number: 0545

Non-chaotic behaviour in three-dimensional quadratic systems

It is shown that three-dimensional dissipative quadratic systems of ordinary differential equations with a total of four terms on the right-hand side of the equations do not exhibit chaos. This complements recent work of Sprott who has given many examples of chaotic quadratic systems with as few as five terms on the right-hand side of the equations. PACS Number: 0545