This paper provides a review of the principles of chaotic digital communications using correlator receivers. Modulation schemes using one and two chaotic basis functions, as well as coherent and noncoherent correlation receivers, are discussed. The performance of differential chaos shift keying (DCSK) in multipath channels is characterized. Results are presented for DCSK with multiuser capability and multiple bits per symbol.

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Chaotic communications with correlator receivers: theory and performance limits

Assessment of EEG synchronization based on state-space analysis.

Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist. The analysis is conducted by combining numerical continuation techniques with theoretical arguments.

Belyakov Homoclinic Bifurcations in a Tritrophic Food Chain Model

We present a classification of border-collision bifurcations in one-dimensional discontinuous maps depending on the parameters of the piecewise linear approximation in the neighborhood of the point of discontinuity. For each range of parameter values we derive the condition of existence and stability of various periodic orbits and of chaos. This knowledge will help in understanding the bifurcation phenomena in a large number of practical systems which can be modeled by discontinuous maps in discrete domain.

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Border-collision bifurcations in one-dimensional discontinuous maps

A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feedback structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay attention to obtain equivalent state equations containing both a minimum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canonical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of different electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology.

On simplifying and classifying piecewise-linear systems

This paper deals with the implications of considering a first-order approximation of the circuit nonlinearities in circuit simulation and design. The Colpitts oscillator is taken as a case study and the occurrence of discontinuous bifurcations, namely, border-collision bifurcations, in a piecewise-linear model of the oscillator is discussed. In particular, we explain the mechanism responsible for the dramatic changes of dynamical behavior exhibited by this model when one or more of the circuit parameters are varied. Moreover, it is shown how an approximate one-dimensional (1-D) map for the Colpitts oscillator can be exploited for predicting border-collision bifurcations. It turns out that at a border collision bifurcation, a 1-D return map of the Colpitts oscillator exhibits a square-root-like singularity. Finally, through the 1-D map, a two-parameter bifurcation analysis is carried out and the relationships are pointed out between border-collision bifurcations and the conventional bifurcations occurring in smooth systems.

Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator

In this work we consider the Colpitts oscillator as a paradigm for sinusoidal oscillation and we investigate its nonlinear dynamics In particular, we carry out a two-parameter bifurcation analysis of a model of the oscillator This analysis is conducted by combining numerical continuation techniques and normal form theory First, we show that the birth of the harmonic cycle is associated with a Hopf bifurcation and we discuss the effects of idealization in the model Various families of limit cycles are identified and their bifurcations are analyzed in detail In particular, we demonstrate that the bifurcation diagram in the parameter space is organized by an infinite family of homoclinic bifurcations Finally, local and global coexistence phenomena are described

The colpitts oscillator: families of periodic solutions and their bifurcations

Nonsmooth bifurcations in a piecewise linear model of the Colpitts Oscillator

Keywords: Non-Linear Signal Processing ; Communication a Chaos Reference LANOS-CONF-2000-010 Record created on 2004-12-03, modified on 2017-05-12

T-CSK: A robust approach to chaos-based communications

In this paper we consider an approximate one-dimensional model for describing the non-linear dynamics of the Colpitts oscillator. It has already been shown that this model preserves the qualitative dynamical behavior of the Colpitts oscillator. Using the theory of snap-back repellors we prove that the one-dimensional model is chaotic for certain values of its parameters.

Gian Mario Maggio

Papers

Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator

The colpitts oscillator: families of periodic solutions and their bifurcations

Nonsmooth bifurcations in a piecewise linear model of the Colpitts Oscillator

T-CSK: A robust approach to chaos-based communications

Devaney Chaos in an Approximate One-Dimensional Model of the Colpitts Oscillator