scholarly journals Period-doubling bifurcation analysis and chaos control for load torque using FLC

2021 ◽  
Author(s):  
Eman Moustafa ◽  
Abdel-Azem Sobaih ◽  
Belal Abozalam ◽  
Amged Sayed A. Mahmoud
Keyword(s):  
Floquet Theory ◽  
Chaos Control ◽  
Fuzzy Logic Controller ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Parameter Variation ◽  
Period Doubling Bifurcation ◽  
Load Torque ◽  
Nonlinear Behaviors ◽  
Scientific Fields

AbstractChaotic phenomena are observed in several practical and scientific fields; however, the chaos is harmful to systems as they can lead them to be unstable. Consequently, the purpose of this study is to analyze the bifurcation of permanent magnet direct current (PMDC) motor and develop a controller that can suppress chaotic behavior resulted from parameter variation such as the loading effect. The nonlinear behaviors of PMDC motors were investigated by time-domain waveform, phase portrait, and Floquet theory. By varying the load torque, a period-doubling bifurcation appeared which in turn led to chaotic behavior in the system. So, a fuzzy logic controller and developing the Floquet theory techniques are applied to eliminate the bifurcation and the chaos effects. The controller is used to enhance the performance of the system by getting a faster response without overshoot or oscillation, moreover, tends to reduce the steady-state error while maintaining its stability. The simulation results emphasize that fuzzy control provides better performance than that obtained from the other controller.

scholarly journals Bifurcation analysis and chaos control of a second-order exponential difference equation

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 5003-5022 ◽  
Author(s):  
Q. Din ◽  
E.M. Elabbasy ◽  
A.A. Elsadany ◽  
S. Ibrahim
Keyword(s):  
Difference Equation ◽  
Chaos Control ◽  
Control Method ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Period Doubling ◽  
Second Order ◽  
Period Doubling Bifurcation ◽  
Positive Real ◽  
Second Order Difference Equation

The aim of this article is to study the local stability of equilibria, investigation related to the parametric conditions for transcritical bifurcation, period-doubling bifurcation and Neimark-Sacker bifurcation of the following second-order difference equation xn+1 = ?xn + ?xn-1 exp(-?xn-1) where the initial conditions x-1, x0 are the arbitrary positive real numbers and ?,? and ? are positive constants. Moreover, chaos control method is implemented for controlling chaotic behavior under the influence of Neimark-Sacker bifurcation and period-doubling bifurcation. Numerical simulations are provided to show effectiveness of theoretical discussion.

BIFURCATION CONTROL OF A PARAMETRIC PENDULUM

2012 ◽  
Vol 22 (05) ◽  
pp. 1250111 ◽  
Author(s):  
ALINE S. DE PAULA ◽  
MARCELO A. SAVI ◽  
MARIAN WIERCIGROCH ◽  
EKATERINA PAVLOVSKAIA
Keyword(s):  
Chaos Control ◽  
Control Method ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Bifurcation Control ◽  
Period Doubling Bifurcation ◽  
Delayed Feedback Control ◽  
Unstable Periodic Orbits ◽  
Parametric Pendulum ◽  
Feedback Control Method

In this paper, we apply chaos control methods to modify bifurcations in a parametric pendulum-shaker system. Specifically, the extended time-delayed feedback control method is employed to maintain stable rotational solutions of the system avoiding period doubling bifurcation and bifurcation to chaos. First, the classical chaos control is realized, where some unstable periodic orbits embedded in chaotic attractor are stabilized. Then period doubling bifurcation is prevented in order to extend the frequency range where a period-1 rotating orbit is observed. Finally, bifurcation to chaos is avoided and a stable rotating solution is obtained. In all cases, the continuous method is used for successive control. The bifurcation control method proposed here allows the system to maintain the desired rotational solutions over an extended range of excitation frequency and amplitude.

Bifurcation Analysis and Chaos Control in a Second-Order Rational Difference Equation

2018 ◽  
Vol 19 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Qamar Din ◽  
A. A. Elsadany ◽  
Samia Ibrahim
Keyword(s):  
Difference Equation ◽  
Bifurcation Theory ◽  
Chaos Control ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
Center Manifold Theorem ◽  
Rational Difference Equation

AbstractThis work is related to dynamics of a second-order rational difference equation. We investigate the parametric conditions for local asymptotic stability of equilibria. Center manifold theorem and bifurcation theory are implemented to discuss the parametric conditions for existence and direction of period-doubling bifurcation and pitchfork bifurcation at trivial equilibrium point. Moreover, the parametric conditions for existence and direction of Neimark–Sacker bifurcation at positive steady state are investigated with the help of bifurcation theory. The chaos control in the system is discussed through implementation of OGY feedback control method. In particular, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control. Finally, numerical simulations are provided to illustrate theoretical results. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

Dynamical behavior of nonlinear wave solutions of the generalized Newell–Whitehead–Segel equation

2020 ◽  
Vol 31 (04) ◽  
pp. 2050059
Author(s):  
Asit Saha ◽  
Amiya Das
Keyword(s):  
Nonlinear Wave ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Phase Portraits ◽  
Phase Plane Analysis ◽  
Linear Coefficient ◽  
Wave Solutions ◽  
Nonlinear Wave Solutions

Dynamical behavior of nonlinear wave solutions of the perturbed and unperturbed generalized Newell–Whitehead–Segel (GNWS) equation is studied via analytical and computational approaches for the first time in the literature. Bifurcation of phase portraits of the unperturbed GNWS equation is dispensed using phase plane analysis through symbolic computation and it shows stable oscillation of the traveling waves. Chaotic behavior of the perturbed GNWS equation is obtained by applying different computational tools, like phase plot, time series plot, Poincare section, bifurcation diagram and Lyapunov exponent. A period-doubling bifurcation behavior to chaotic behavior is shown for the perturbed GNWS equation and again it shows chaotic to periodic motion through inverse period-doubling bifurcation. The perturbed GNWS equation also shows chaotic motion through a sequence of periodic motions (period-1, period-3 and period-5) depending on the variation of the parameter of linear coefficient. Thus, the parameter of linear coefficient plays the role of a controlling parameter in the chaotic dynamics of the perturbed GNWS equation.

Nonlinear Vibrations of Buried Rectangular Plate

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Guangyang Hong ◽  
Jian Li ◽  
Zhicong Luo ◽  
Hongying Li
Keyword(s):  
Nonlinear Dynamic ◽  
Granular Layer ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Dynamic Strain ◽  
Burial Depth ◽  
Dynamic Behaviors ◽  
Jump Phenomena ◽  
Nonlinear Behaviors

We perform an investigation on the vibration response of a simply supported plate buried in glass particles, focusing on the nonlinear dynamic behaviors of the plate. Various excitation strategies, including constant-amplitude variable-frequency sweep and constant-frequency variable-amplitude sweep are used during the testing process. We employ the analysis methods of power spectroscopy, phase diagramming, and Poincare mapping, which reveal many complicated nonlinear behaviors in the dynamic strain responses of an elastic plate in granular media, such as the jump phenomena, period-doubling bifurcation, and chaos. The results indicate that the added mass, damping, and stiffness effects of the granular medium on the plate are the source of the nonlinear dynamic behaviors in the oscillating plate. These nonlinear behaviors are related to the burial depth of the plate (the thickness of the granular layer above plate), force amplitude, and particle size. Smaller particles and a suitable burial depth cause more obvious jump and period-doubling bifurcation phenomena to occur. Jump phenomena show both soft and hard properties near various resonant frequencies. With an increase in the excitation frequency, the nonlinear added stiffness effect of the granular layer makes a transition from strong negative stiffness to weak positive stiffness.

scholarly journals CONNECTING PERIOD-DOUBLING CASCADES TO CHAOS

2012 ◽  
Vol 22 (02) ◽  
pp. 1250022 ◽  
Author(s):  
EVELYN SANDER ◽  
JAMES A. YORKE
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Duffing Equation ◽  
Period Doubling Bifurcation ◽  
Bifurcation Diagrams ◽  
The Third ◽  
Infinite Collection

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value μ2 of the map at which there is chaos. We show that often virtually all (i.e. all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired — connected to exactly one other cascade, or solitary — connected to exactly one regular periodic orbit at μ2. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅). Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

scholarly journals The Effect of Convection in Nonlinear One-Zone Stellar Models

1993 ◽  
Vol 134 ◽  
pp. 355-358
Author(s):  
M. Saitou
Keyword(s):  
Limit Cycle ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Variable Stars ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Other Hand ◽  
Pulsating Variable Stars ◽  
Stellar Models

AbstractWe study the convective nonlinear one-zone models of the pulsating variable stars. In the small convective case, the solution shows chaotic behavior through the period doubling bifurcation as the temperature is lower although the temperature at which the chaos appears is lower than in the case of no convection. On the other hand, in the strong convective case, the solution only shows period-one limit cycle.

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