The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

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CONTROL OF n-SCROLL CHUA'S CIRCUIT

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409025134 ◽

2009 ◽

Vol 19 (11) ◽

pp. 3813-3822 ◽

Cited By ~ 16

Author(s):

ABDELKRIM BOUKABOU ◽

BILEL SAYOUD ◽

HAMZA BOUMAIZA ◽

NOURA MANSOURI

Keyword(s):

Numerical Simulations ◽

Fixed Points ◽

Periodic Orbits ◽

Predictive Control ◽

Control Method ◽

Sufficient Conditions ◽

Chua’S Circuit ◽

Unstable Periodic Orbits ◽

Chua's Circuit ◽

Necessary And Sufficient

This paper addresses the control of unstable fixed points and unstable periodic orbits of the n-scroll Chua's circuit. In a first step, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed predictive control method. In addition, we show how a chaotic system with multiple unstable periodic orbits can be stabilized by taking the system dynamics from one UPO to another. Control performances of these approaches are demonstrated by numerical simulations.

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Periodic orbits analysis for the Zhou system via variational approach

Modern Physics Letters B ◽

10.1142/s0217984919502129 ◽

2019 ◽

Vol 33 (19) ◽

pp. 1950212 ◽

Cited By ~ 2

Author(s):

Chengwei Dong ◽

Lian Jia

Keyword(s):

Fixed Points ◽

Periodic Orbits ◽

Symbolic Dynamics ◽

Variational Approach ◽

Topological Classification ◽

Dissipative Systems ◽

Systematic Calculation ◽

Low Dimensional ◽

General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

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INFINITE DIMENSIONAL KRAWCZYK OPERATOR FOR FINDING PERIODIC ORBITS OF DISCRETE DYNAMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407019937 ◽

2007 ◽

Vol 17 (12) ◽

pp. 4261-4272 ◽

Cited By ~ 9

Author(s):

ZBIGNIEW GALIAS ◽

PIOTR ZGLICZYŃSKI

Keyword(s):

Dynamical Systems ◽

Fixed Points ◽

Periodic Orbits ◽

Discrete Dynamical Systems ◽

Krawczyk Operator ◽

Infinite Dimensional ◽

Dispersal Model ◽

Period 2 ◽

Existence Of Zeros

In this work, we introduce the Krawczyk operator for infinite dimensional maps. We prove two properties of this operator related to the existence of zeros of the map. We also show how the Krawczyk operator can be used to prove the existence of periodic orbits of infinite dimensional discrete dynamical systems and for finding all periodic orbits with a given period enclosed in a specified region. As an example, we consider the Kot–Schaffer growth-dispersal model, for which we find all fixed points and period-2 orbits enclosed in the region containing the attractor observed numerically.

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