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.

Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter

1997 ◽  
Vol 07 (12) ◽  
pp. 2755-2771 ◽  
Author(s):  
M. Di Bernardo ◽  
E. Fossas ◽  
G. Olivar ◽  
F. Vasca
Keyword(s):  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Main Sequence ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Bifurcation Diagrams ◽  
Synchronous Switching ◽  
Route To Chaos ◽  
Secondary Bifurcations

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.

CYCLIC-FOLD BIFURCATION AND BOUNDARY CRISIS IN DYNAMIC WALKING OF BIPED ROBOTS

2012 ◽  
Vol 22 (10) ◽  
pp. 1250257 ◽  
Author(s):  
HASSENE GRITLI ◽  
SAFYA BELGHITH ◽  
NAHLA KHRAIEF
Keyword(s):  
Period Doubling ◽  
Biped Robot ◽  
Dynamic Walking ◽  
Small Perturbations ◽  
Biped Robots ◽  
Transition Mechanism ◽  
Inertial Parameters ◽  
Fold Bifurcation ◽  
Boundary Crisis ◽  
Inclined Surface

The aim of this paper is to show that the onset/destruction of bipedal chaos in the dynamic walking of a passive compass-gait biped robot and a semi-passive torso-driven biped robot walking down a slope can occur via a transition mechanism known as boundary crisis. It is known that such biped robots exhibit a scenario of period-doubling bifurcations route to chaos as one of their geometrical or inertial parameters changes. In this paper, we show that a cyclic-fold bifurcation is the key of the occurrence of a double boundary crisis. We demonstrate through bifurcation diagrams how the same period-three unstable periodic orbit generated from the cyclic-fold bifurcation causes the sudden birth/death of the bipedal chaos in the dynamic walking of the two biped robots. We stress that a double boundary crisis is responsible for the fall of each biped robot while walking down an inclined surface and as some bifurcation parameter varies. Stability of the cyclic-fold bifurcation under small perturbations is also discussed.

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

Further Investigation of the Period-Three Route to Chaos in the Passive Compass-Gait Biped Model

2015 ◽  
pp. 279-300 ◽  
Author(s):  
Hassène Gritli ◽  
Nahla Khraief ◽  
Safya Belghith
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Limit Cycle ◽  
Basin Of Attraction ◽  
Biped Robot ◽  
Computation Method ◽  
Dynamic Walking ◽  
Route To Chaos ◽  
Inclined Surface ◽  
The Basin Of Attraction

This chapter presents further investigations into the period-three route to chaos exhibited in the passive dynamic walking of the compass-gait biped robot as it goes down an inclined surface. This discovered kind of route in the passive bipedal locomotion was found to coexist with the conventional period-one passive hybrid limit cycle. The further analysis on the period-three route chaos is realized by means of the Lyapunov exponents and the fractal Lyapunov dimension. Numerical computation method of these two tools is presented. The first return Poincaré map of the chaotic attractor and its basin of attraction are presented. Furthermore, the further study of the period-three passive gait is realized. The analysis of the period-three hybrid limit cycle is given. The balance between the potential energy and the kinetic energy of the biped robot is illustrated. In addition, the basin of attraction of the period-three passive gait is also presented.

Experimental Stabilization of Unstable Periodic Orbit in Magneto-Elastic Chaos by Delayed Feedback Control

1997 ◽  
Vol 07 (12) ◽  
pp. 2837-2846 ◽  
Author(s):  
Takashi Hikihara ◽  
Masato Touno ◽  
Toshiaki Kawagoshi
Keyword(s):  
Feedback Control ◽  
Periodic Orbit ◽  
Chaotic Attractor ◽  
Control Method ◽  
Elastic Beam ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Unstable Periodic Orbit ◽  
Beam System ◽  
Onset Timing

In our previous paper, it was confirmed that the unstable periodic orbit embedded in the chaotic attractor in magneto-elastic beam system can be stabilized by delayed feedback control experimentally. It seems an advantage that the control method does not require any exact model of the system. However, the application of the control raises the problem that we cannot predict the stabilized unstable periodic orbit until it converges. In this paper, an "onset window" is introduced to determine the onset timing for targeting the desired orbit embedded in the chaotic attractor experimentally. Moreover, the dependence of the stabilization on the delay and the gain parameters is also discussed based on the experimental results.

The Control of Chaos: Theoretical Schemes and Experimental Realizations

1998 ◽  
Vol 08 (08) ◽  
pp. 1643-1655 ◽  
Author(s):  
F. T. Arecchi ◽  
S. Boccaletti ◽  
M. Ciofini ◽  
R. Meucci ◽  
C. Grebogi
Keyword(s):  
Periodic Orbit ◽  
Chaotic Attractor ◽  
The State ◽  
Control Parameters ◽  
State Variables ◽  
Unstable Periodic Orbit ◽  
Control Of Chaos ◽  
Controlling Chaos ◽  
Feedback Techniques

Controlling chaos is a process wherein an unstable periodic orbit embedded in a chaotic attractor is stabilized by means of tiny perturbations of the system. These perturbations imply goal oriented feedback techniques which act either on the state variables of the system or on the control parameters. We review some theoretical schemes and experimental implementations for the control of chaos.

Nonlinear Dynamics of Duffing System With Fractional Order Damping

2009 ◽  
Author(s):  
Junyi Cao ◽  
Chengbin Ma ◽  
Hang Xie ◽  
Zhuangde Jiang
Keyword(s):  
Nonlinear Dynamics ◽  
Fractional Order ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Route To Chaos ◽  
Euler Methods ◽  
Period Motion

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The four order Runge-Kutta method and ten order CFE-Euler methods are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on the system dynamics is investigated using phase diagrams, bifurcation diagrams and Poincare map. The bifurcation diagram is also used to exam the effects of excitation amplitude and frequency on Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits period motion, chaos, period motion, chaos, period motion in turn when the fractional order changes from 0.1 to 2.0. A period doubling route to chaos is clearly observed.

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.

SIZE RATIO UNIVERSALITY IN CHAOTIC SYSTEMS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250005
Author(s):  
R. D. GALA ◽  
P. PARMANANDA
Keyword(s):  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Period Doubling ◽  
Size Ratio ◽  
Route To Chaos

In this paper, we attempt to predict the size of the chaotic attractor at the parameter value where the period tends to infinity for nonlinear systems that follow the period doubling route to chaos, and for which the Feigenbaum universalities hold.

Intermittency Route to Chaos of a Cantilevered Pipe Conveying Fluid With a Mass Defect at the Free End

1995 ◽  
Vol 62 (4) ◽  
pp. 903-907 ◽  
Author(s):  
C. Semler ◽  
M. P. Pai¨doussis
Keyword(s):  
Period Doubling ◽  
Type I ◽  
Pipe Conveying Fluid ◽  
Mass Defect ◽  
Lumped Mass ◽  
Route To Chaos ◽  
Cantilevered Pipe ◽  
Planar Motions ◽  
Period Doubling Bifurcations ◽  
Conveying Fluid

The nonlinear equations for planar motions of a vertical cantilevered pipe conveying fluid are modified to take into account a small lumped mass added at the free end. The resultant equations contain nonlinear inertial terms; by discretizing the system first and inverting the inertia matrix, these terms are transferred into other matrices. In this paper, the dynamics of the system is examined when the added mass is negative (a mass defect), by means of numerical computations and by the software package AUTO. The system loses stability by a Hopf bifurcation, and the resultant limit cycle undergoes pitchfork and period-doubling bifurcations. Subsequently, as shown by the computation of Floquet multipliers, a type I intermittency route to chaos is followed—as illustrated further by a Lorenz return map, revealing the well-known normal form for this type of bifurcation. The period between “turbulent bursts” of nonperiodic oscillations is computed numerically, as well as Lyapunov exponents. Remarkable qualitative agreement, in both cases, is obtained with analytical results.

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