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What is Lyapunov Exponents

Chapter 10
Handbook of Research on Advanced Intelligent Control Engineering and Automation
The Lyapunov exponents are an important tool for the characterization of an attractor of a finite-dimensional nonlinear dynamic system and their excessive sensitivity to initial conditions.
Published in Chapter:
Further Investigation of the Period-Three Route to Chaos in the Passive Compass-Gait Biped Model
Hassène Gritli (Institut Supérieur des Etudes Technologiques de Kélibia, Tunisia), Nahla Khraief (Ecole Supérieure de Technologie et d'Informatique, Tunisia), and Safya Belghith (Ecole Nationale d'Ingénieurs de Tunis, Tunisia)
Copyright: © 2015 |Pages: 22
DOI: 10.4018/978-1-4666-7248-2.ch010
Abstract
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.
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Chapter 185
Nonlinear Techniques for Signals Characterization
Quantity that characterizes the rate of separation of infinitesimally close trajectories in a dynamical system. The maximal Lyapunov exponent (MLE) determines the predictability of a dynamical system. A positive MLE means a chaotic system.
Published in Chapter: Nonlinear Techniques for Signals Characterization; From: Encyclopedia of Artificial Intelligence
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Chapter 1
Robust Control and Synchronization of Chaotic Systems with Actuator Constraints
Numbers that measure the rate of exponential attraction or separation over the time of two adjacent trajectories started with different initial conditions in the phase space. For a dynamical system with bounded trajectories, a positive Lyapunov exponent indicates the existence of a chaotic motion and if two or more exponents are positive the system is said to be hyperchaotic.
Published in Chapter: Robust Control and Synchronization of Chaotic Systems with Actuator Constraints; From: Handbook of Research on Artificial Intelligence Techniques and Algorithms
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Chapter 651
Dynamical Systems Approach for Predator-Prey Robot Behavior Control via Symbolic Dynamics Based Communication
Gives the measure of the rate of divergence of trajectories of chaotic system. It tells us the sensitivity of the system to initial conditions. Chaotic systems with positive lyapunov exponents have bounded orbits. The number of exponents equals to the dimension of the system. When the largest lyapunov exponent becomes negative, then the system is synchronized.
Published in Chapter: Dynamical Systems Approach for Predator-Prey Robot Behavior Control via Symbolic Dynamics Based Communication; From: Encyclopedia of Information Science and Technology, Third Edition
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