Raghib Abu-Saris

Fathi Allan

Mustafa Kulenovic

Alfredo Peris

Deterministic Discrete Dynamical Systems: Advances in Regular and Chaotic Behavior with Applications

First, we make the Jain's derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain's method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.

Farahnaz Soleimani

Fazlollah Soleymani

Stanford Shateyi

Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

The study of recurrent neural networks with piecewise constant transition or control functions has attracted much attention recently because they can be used to simulate many physical phenomena. A recurrent and discontinuous two-state dynamical system involving a nonnegative bifurcation parameter is studied. By elementary but novel arguments, we are able to give a complete analysis on its asymptotic behavior when the parameter varies from 0 to <svg     style="vertical-align:-0.1638pt;width:17.4625px;" id="M1" height="7.9499998" version="1.1" viewBox="0 0 17.4625 7.9499998" width="17.4625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>. It is hoped that our analysis will provide motivation for further results on large-scale recurrent McCulloch-Pitts-type neural networks and piecewise continuous discrete-time dynamical systems.

Chenmin Hou

Sui Sun Cheng

Bifurcation in a Discrete-Time Piecewise Constant Dynamical System

We investigate the global dynamics of several anticompetitive systems of rational difference equations which are special cases of general linear fractional system of the forms <svg     style="vertical-align:-4.15506pt;width:709.20001px;" id="M1" height="17.4" version="1.1" viewBox="0 0 709.20001 17.4" width="709.20001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>., where all parameters and the initial conditions <svg     style="vertical-align:-4.15506pt;width:38.849998px;" id="M2" height="12.9125" version="1.1" viewBox="0 0 38.849998 12.9125" width="38.849998"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg> are arbitrary nonnegative numbers, such that both denominators are positive. We find the basins of attraction of all attractors of these systems.

M. DiPippo

M. R. S. Kulenovi&#x107;

Global Dynamics of Three Anticompetitive Systems of Difference Equations in the Plane

We consider the second order rational difference equation <svg     style="vertical-align:-3.56265pt;width:306.76251px;" id="M1" height="17.7875" version="1.1" viewBox="0 0 306.76251 17.7875" width="306.76251"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg>&#x2009;&#x2009;<i  >n</i> = 0,1,2,&#x2026;, where the parameters <svg     style="vertical-align:-3.56265pt;width:97.775002px;" id="M2" height="17.7875" version="1.1" viewBox="0 0 97.775002 17.7875" width="97.775002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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</svg> are positive real numbers, and the initial conditions <svg     style="vertical-align:-3.3907pt;width:46.612499px;" id="M3" height="13.0875" version="1.1" viewBox="0 0 46.612499 13.0875" width="46.612499"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
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			<g transform="matrix(.012,-0,0,-.012,39.65,12.313)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g>

		
	
</svg> are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenovi&#x107; and Ladas monograph (2002).

S. Atawna

R. Abu-Saris

I. Hashim

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

A discrete-time periodic model with bang-bang feedback control is investigated. It is shown that each solution tends to one of four different types of limit 2-cycles. Furthermore, the accompanying initial regions for each type of solutions can be determined. When a threshold parameter is introduced in the bang-bang function, our results form a complete bifurcation analysis of our control model. Hence, our model can be used in the design of a control system where the state variable fluctuates between two state values with decaying perturbation.

Chengmin Hou

Limit 2-Cycles for a Discrete-Time Bang-Bang Control Model

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Discrete Dynamics in Nature and Society

Deterministic Discrete Dynamical Systems: Advances in Regular and Chaotic Behavior with Applications

Deterministic Discrete Dynamical Systems: Advances in Regular and Chaotic Behavior with Applications

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