Periodic Orbits, Invariant Tori and Chaotic Behavior in Certain Nonequilibrium Quadratic Three-Dimensional Differential Systems

2018 ◽  
pp. 299-326
Author(s):  
Alisson C. Reinol ◽  
Marcelo Messias
Keyword(s):  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Differential Systems

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750205 ◽  
Author(s):  
Tonghua Zhang ◽  
Jibin Li
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Torus Knot ◽  
Cubic System ◽  
Theoretical Results

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

scholarly journals Periodic Orbits Bifurcating from a Nonisolated Zero–Hopf Equilibrium of Three-Dimensional Differential Systems Revisited

2018 ◽  
Vol 28 (05) ◽  
pp. 1850058 ◽  
Author(s):  
Murilo R. Cândido ◽  
Jaume Llibre
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Three Dimensional ◽  
Averaging Theory ◽  
Chaotic Attractors ◽  
Differential Systems ◽  
Polynomial Differential System

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in [Formula: see text]. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in [Formula: see text] having [Formula: see text]-scroll chaotic attractors.

INVARIANT TORI FULFILLED BY PERIODIC ORBITS FOR FOUR-DIMENSIONAL $\mathcal{C}^2$ DIFFERENTIAL SYSTEMS IN THE PRESENCE OF RESONANCE

2010 ◽  
Vol 20 (10) ◽  
pp. 3341-3344 ◽  
Author(s):  
JAUME LLIBRE ◽  
ANA CRISTINA MEREU ◽  
MARCO A. TEIXEIRA
Keyword(s):  
Linear System ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Differential Systems ◽  
Perturbed System

We provide an algorithm for studying invariant tori fulfilled by periodic orbits of a perturbed system which emerge from the set of periodic orbits of an unperturbed linear system in p : q resonance. We illustrate the algorithm with an application.

scholarly journals Zero–Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

2020 ◽  
Vol 30 (13) ◽  
pp. 2050189
Author(s):  
Jaume Llibre ◽  
Marcelo Messias ◽  
Alisson de Carvalho Reinol
Keyword(s):  
Equilibrium Point ◽  
Periodic Orbits ◽  
Quadratic Differential ◽  
Stable Equilibrium ◽  
Three Dimensional ◽  
Period Doubling ◽  
Stable Equilibrium Point ◽  
Differential Systems ◽  
Saddle Type ◽  
Route To Chaos

In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in [Formula: see text] with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in [Formula: see text] depending on a real parameter [Formula: see text], which, for [Formula: see text], coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value [Formula: see text]. We prove that, for [Formula: see text], all the 23 considered systems have a nonisolated zero–Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero–Hopf bifurcation takes place at this point for [Formula: see text], which leads to the creation of three periodic orbits bifurcating from it for [Formula: see text] small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when [Formula: see text] are obtained by period-doubling route to chaos.

KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS

2011 ◽  
Vol 21 (09) ◽  
pp. 2505-2523 ◽  
Author(s):  
JIBIN LI ◽  
FENGJUAN CHEN
Keyword(s):  
Periodic Orbits ◽  
Invariant Torus ◽  
Vector Fields ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Symmetric Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed Systems

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

BIFURCATION OF ONE-PARAMETER PERIODIC ORBITS OF THREE-DIMENSIONAL DIFFERENTIAL SYSTEMS

2013 ◽  
Vol 23 (07) ◽  
pp. 1350121
Author(s):  
YULIAN AN ◽  
CHIGUO WANG
Keyword(s):  
Periodic Orbits ◽  
Poincaré Map ◽  
Averaging Method ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Poincare Map ◽  
Differential Systems ◽  
Invariant Surfaces

In this paper, we study two classes of three-dimensional differential systems. By using the Poincaré map and the averaging method, we give sufficient conditions for the existence of invariant surfaces consisting of periodic orbits for the systems. Illustrative examples are given for our main results.

scholarly journals Periodic Orbits for a Three-Dimensional Biological Differential Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Renato Colucci ◽  
Daniel Nuñez
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Differential System ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Differential Systems

We study the existence of periodic orbit for a differential system describing the effects of indirect predation over two preys. Besides discussing a generalized version of the model, we present some remarks and numerical experiments for the nonautonomous version of the two models.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

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