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An analysis of the passive dynamic walking of a compass-gait biped model under the OGY-based control approach using the impulsive hybrid nonlinear dynamics is presented in this paper. We describe our strategy for the development of a simplified analytical expression of a controlled hybrid Poincare map and then for the design of a state-feedback control. Our control methodology is based mainly on the linearization of the impulsive hybrid nonlinear dynamics around a desired nominal one-periodic hybrid limit cycle. Our analysis of the controlled walking dynamics is achieved by means of bifurcation diagrams. Some interesting nonlinear phenomena are displayed, such as the period-doubling bifurcation, the cyclic-fold bifurcation, the period remerging, the period bubbling and chaos. A comparison between the raised phenomena in the impulsive hybrid nonlinear dynamics and the hybrid Poincare map under control was also presented.

In this paper, a Linear Matrix Inequality- (an LMI-) based approach for designing a robust state-feedback controller for a 1-DoF, periodically forced, impact mechanical oscillator subject to double-side rigid barriers/constraints and under bounded parametric uncertainties to track another impact oscillator, as a master or reference system, is proposed. The dynamics of such impact oscillator is defined by a hybrid non-autonomous system with impulsive effects, for which the impulsive event occurs when the system state encounters the two barriers and the oscillation motion is limited between them. The main idea in the synthesis of the stability conditions lies in the use of the S-procedure Lemma and the Finsler Lemma in order to only consider the regions inside which the master-slave tracking error evolves. We show that the stability conditions of the tracking error are reformulated by a set of Bilinear Matrix Inequalities (BMIs). Via the Schur complement Lemma and the Matrix Inversion Lemma, a linearization procedure is realized to transform these BMIs into LMIs where the admissible maximum bounds of the parametric uncertainties are maximized. The effectiveness of the proposed feedback controller towards uncertainties is illustrated through simulation results.

Recently, many papers have appeared which study the power spectral density (PSD) of signals issued from some specific maps. This interest in the PSD is due to the importance of frequency in the telecommunications and transmission security. With the large number of wireless systems, the availability of frequencies for transmission and reception is increasingly uncommon for wireless communications. Also, guided media have limitations related to the bandwidth of a signal. In this paper, we investigate some properties associated to the border-collision bifurcations in a one-dimensional piecewise-linear map with three slopes and two parameters. We derive analytical expressions for the autocorrelation sequence, power spectral density (PSD) of chaotic signals generated by our piecewise-linear map. We prove the existence of strong relation between different types of the power spectral density (low-pass, high-pass or band-stop) and the parameters. We also find a relation between the type of spectrum and the order of attractive cycles which are located after the border collision bifurcation between chaos and cycles.

The compass-gait walker is a two-degree-of-freedom biped that can walk passively and steadily down an incline without any actuation. The mathematical model of the walking dynamics is represented by an impulsive hybrid nonlinear model. It is capable of displaying cyclic motions and chaos. In this paper, we propose a new approach to controlling chaos cropped up from the passive dynamic walking of the compass-gait model. The proposed technique is to linearize the nonlinear model around a desired passive hybrid limit cycle. Then, we show that the nonlinear model is transformed to an impulsive hybrid linear model with a controlled jump. Basing on the linearized model, we derive an analytical expression of a constrained controlled Poincare map. We present a method for the numerical simulation of this constrained map where bifurcation diagrams are plotted. Relying on these diagrams, we show that the linear model is fairly close to the nonlinear one. Using the linearized controlled Poincare map, we design a state feedback controller in order to stabilize the fixed point of the Poincare map. We show that this controller is very efficient for the control of chaos for the original nonlinear model.

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.

In this work, we cryptanalyse a recently chaos-based cryptosystem on DSP by proposing three different attacks to break it. We report the weakness of this cryptosystem and hence demonstrate that in its actual design, it cannot be used in the real world applications and it needs to be first enhanced by avoiding the design drawbacks reported in this work.

This paper introduces a new scheme for joint compression and encryption using the Huffman codec. A basic tree is first generated for a given message and then based on a keystream generated from a chaotic map and depending from the input message, the basic tree is mutated without changing the statistical model. Hence a symbol can be coded by more than one codeword having the same length. The security of the scheme is tested against the known plaintext attack and the brute force attack. Performance analysis including encryption/decryption speed, additional computational complexity and compression ratio are given.