Lyapunov based estimation of the basin of attraction of Poincare maps with applications to limit cycle walking

2019 ◽  
Vol 33 ◽  
pp. 179-194 ◽  
Author(s):  
Eric Sidorov ◽  
Miriam Zacksenhouse
Keyword(s):  
Limit Cycle ◽  
Basin Of Attraction ◽  
Poincaré Maps ◽  
Limit Cycle Walking ◽  
Poincare Maps ◽  
The Basin Of Attraction

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

Stable running of a planar underactuated biped robot

Robotica ◽  
2010 ◽  
Vol 29 (5) ◽  
pp. 657-665 ◽  
Author(s):  
Yong Hu ◽  
Gangfeng Yan ◽  
Zhiyun Lin
Keyword(s):  
Limit Cycle ◽  
Stance Phase ◽  
Control Strategies ◽  
Basin Of Attraction ◽  
Biped Robot ◽  
State Spaces ◽  
Continuous State ◽  
Event Based ◽  
Telescopic Legs ◽  
The Basin Of Attraction

SUMMARYThis paper investigates the stable-running problem of a planar underactuated biped robot, which has two springy telescopic legs and one actuated joint in the hip. After modeling the robot as a hybrid system with multiple continuous state spaces, a natural passive limit cycle, which preserves the system energy at touchdown, is found using the method of Poincaré shooting. It is then checked that the passive limit cycle is not stable. To stabilize the passive limit cycle, an event-based feedback control law is proposed, and also to enlarge the basin of attraction, an additive passivity-based control term is introduced only in the stance phase. The validity of our control strategies is illustrated by a series of numerical simulations.

Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincaré maps

2003 ◽  
Vol 76 (17) ◽  
pp. 1685-1698 ◽  
Author(s):  
Alexander V. Roup ◽  
Dennis S. Bernstein ◽  
Sergey G. Nersesov ◽  
Wassim M. Haddad ◽  
VijaySekhar Chellaboina
Keyword(s):  
Differential Equations ◽  
Limit Cycle ◽  
Impulsive Differential Equations ◽  
Poincaré Maps ◽  
Poincare Maps

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.

CHAOS AND CHAOTIC TRANSIENTS IN A FORCED MODEL OF THE ECONOMIC LONG WAVE: THE ROLE OF HOMOCLINIC BIFURCATION TO INFINITY

1995 ◽  
Vol 05 (03) ◽  
pp. 741-749 ◽  
Author(s):  
JEPPE STURIS ◽  
MORTEN BRØNS
Keyword(s):  
Limit Cycle ◽  
Basin Of Attraction ◽  
Homoclinic Bifurcation ◽  
Periodic Forcing ◽  
Stable And Unstable Manifolds ◽  
Long Wave ◽  
Unstable Manifolds ◽  
Chaotic Transients ◽  
The Basin Of Attraction

When an autonomous system of ordinary differential equations exhibits limit cycle behavior but is close in parameter space to a homoclinic bifurcation to infinity in which the limit cycle blows up to infinite amplitude and disappears, periodic forcing of the system may result in the appearance of both chaos and chaotic transients. In this paper, we use numerical techniques to map out Arnol’d tongues of a forced model of the economic long wave and illustrate how the system becomes chaotic and also exhibits chaotic transients for certain parameter combinations. Based on linearizations at infinity, we argue that infinity acts like a saddle with stable and unstable manifolds. By numerical computation, we show that chaotic transients occur when the manifolds intersect. Depending on parameters, two types of bifurcations have been identified: A chaotic attractor blows up to infinite size and disappears or the boundary of the basin of attraction of a periodic solution becomes fractal.

Poincaré Maps of “ <”-Shape Planar Piecewise Linear Dynamical Systems with a Saddle

2019 ◽  
Vol 29 (12) ◽  
pp. 1950165
Author(s):  
Qianqian Zhao ◽  
Jiang Yu
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Normal Form ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Inflection Point ◽  
Piecewise Linear ◽  
Poincaré Maps ◽  
Linear Dynamical Systems ◽  
Poincare Maps

It is important in the study of limit cycles to investigate the properties of Poincaré maps of discontinuous dynamical systems. In this paper, we focus on a class of planar piecewise linear dynamical systems with “[Formula: see text]”-shape regions and prove that the Poincaré map of a subsystem with a saddle has at most one inflection point which can be reached. Furthermore, we show that one class of such systems with a saddle-center has at least three limit cycles; a class of such systems with saddle and center in the normal form has at most one limit cycle which can be reached; and a class of such systems with saddle and center at the origin has at most three limit cycles with a lower bound of two. We try to reveal the reasons for the increase of the number of limit cycles when the discontinuity happens to a system.

Analysis of limit-cycle walking for a compass-like biped robot *

2010 ◽  
Vol 43 (14) ◽  
pp. 1181-1186
Author(s):  
Leonid B. Freidovich ◽  
Anton S. Shiriaev
Keyword(s):  
Limit Cycle ◽  
Biped Robot ◽  
Limit Cycle Walking
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