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 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.

BIFURCATIONS OF HOMOCLINIC ORBIT CONNECTING TWO NONLEADING EIGENDIRECTIONS

2007 ◽  
Vol 17 (03) ◽  
pp. 823-836 ◽  
Author(s):  
TIANSI ZHANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Period Doubling ◽  
Homoclinic Bifurcation ◽  
Period Doubling Bifurcation ◽  
Dimensional System ◽  
Bifurcation Surface

Bifurcations of homoclinic orbit connecting the strong stable and strong unstable directions are investigated for four-dimensional system. The existence, numbers, co-existence and incoexistence of 1-homoclinic orbit, 2n-homoclinic orbit, 1-periodic orbit and 2n-periodic orbit are obtained, and the bifurcation surfaces (including codimension-1 homoclinic bifurcation surfaces, double periodic orbit bifurcation surfaces, homoclinic-doubling bifurcation surfaces, period-doubling bifurcation surfaces and codimension-2 triple periodic orbit bifurcation surface, and homoclinic and double periodic orbit bifurcation surface) and the existence regions are also located.

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3689-3701 ◽  
Author(s):  
YANCONG XU ◽  
DEMING ZHU ◽  
FENGJIE GENG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Bifurcation ◽  
Reversible Systems ◽  
Reversible System ◽  
Bifurcation Diagrams ◽  
Symmetric Periodic Orbits ◽  
The Continuum

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

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.

INTERMITTENCY AND INTERIOR CRISIS AS ROUTE TO CHAOS IN DYNAMIC WALKING OF TWO BIPED ROBOTS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HASSENE GRITLI ◽  
SAFYA BELGHITH ◽  
NAHLA KHRAEIF
Keyword(s):  
Periodic Orbit ◽  
Chaotic Attractor ◽  
Period Doubling ◽  
Biped Robot ◽  
Type I ◽  
Dynamic Walking ◽  
Unstable Periodic Orbit ◽  
Bifurcation Diagrams ◽  
Biped Robots ◽  
Route To Chaos

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position. In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.

Quantum signature of a period-doubling bifurcation and scars of periodic orbits

1993 ◽  
Vol 47 (3) ◽  
pp. 1625-1632 ◽  
Author(s):  
C. P. Malta ◽  
M. A. M. de Aguiar ◽  
A. M. Ozorio de Almeida
Keyword(s):  
Periodic Orbits ◽  
Period Doubling ◽  
Quantum Signature ◽  
Period Doubling Bifurcation

Effects of noise on periodic orbits of the logistic map

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Feng-guo Li
Keyword(s):  
Periodic Orbit ◽  
White Noise ◽  
Probability Density ◽  
Periodic Orbits ◽  
Gaussian Noise ◽  
Dynamical Behavior ◽  
Gaussian White Noise ◽  
Logistic Map ◽  
Period Doubling ◽  
Colored Gaussian Noise

AbstractNoise can induce an inverse period-doubling transition and chaos. The effects of noise on each periodic orbit of three different period sequences are investigated for the logistic map. It is found that the dynamical behavior of each orbit, induced by an uncorrelated Gaussian white noise, is different in the mergence transition. For an orbit of the period-six sequence, the maximum of the probability density in the presence of noise is greater than that in the absence of noise. It is also found that, under the same intensity of noise, the effects of uncorrelated Gaussian white noise and exponentially correlated colored (Gaussian) noise on the period-four sequence are different.

Bifurcations of Periodic Orbits of a One-Dimensional Pre-Compressed Granular Array

2018 ◽  
Author(s):  
Gizem Dilber Acar ◽  
Balakumar Balachandran
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Bifurcation Point ◽  
Granular Media ◽  
Period Doubling ◽  
Initial Step ◽  
Hertzian Contact ◽  
One Dimensional ◽  
The Stability ◽  
External Forces

Bifurcations of periodic orbits of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is considered without any damping or external forces. By using the Hertzian contact model, and confining the system’s total energy to a certain level, changes in in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution, and to detect the bifurcation location. Although the trajectory of period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced as one moves away from the bifurcation point. This work is meant to serve as an initial step towards understanding how pre-compression may introduce qualitative changes in system dynamics of granular media.

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.

The Dynamics of Symmetric Bimodal Maps

1997 ◽  
Vol 07 (12) ◽  
pp. 2735-2744 ◽  
Author(s):  
Thomas Lofaro
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Period Doubling ◽  
General Setting ◽  
The Family ◽  
Pitchfork Bifurcations ◽  
Parameter Values ◽  
Kneading Theory ◽  
Period Doubling Bifurcations

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.

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