One-dimensional Piecewise Linear Map Research Articles

Basic Bifurcation of Artificial Spiking Neurons with Triangular Base Signal

This paper studies a spiking neuron circuit with triangular base signal. The circuit can output rich spike-trains and the dynamics can be analyzed using a one-dimensional piecewise linear map. This system exhibits period doubling bifurcation, tangent bifurcation, super-stable periodic orbit bifurcation and so on. These phenomena can be characterized based on the inter-spike intervals. Using the maps, we can analyze the phenomena precisely. By presenting a simple test circuit, typical phenomena are confirmed experimentally.

NETWORK AS A CHAOTIC DYNAMICAL SYSTEM

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius–Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure.

On detection of multi-band chaotic attractors

In this work, we present two numerical methods for the detection of the number of bands of a multi-band chaotic attractor. The first method is more efficient but can be applied only for dynamical systems with a continuous system function, whereas the second one is applicable for dynamical systems with a discontinuous system function as well. Using the developed methods, we investigate a one-dimensional piecewise-linear map and report for both cases of a continuous and a discontinuous system functions some new bifurcation scenarios involving multi-band chaotic attractors.

Multi-parametric bifurcations in a piecewise–linear discontinuous map

In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained.

On multi-parametric bifurcations in a scalar piecewise-linear map

In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multi-parametric bifurcations. It is demonstrated that an infinite number of two-parametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.

Lyapunov spectrum of a lattice of chaotic systems with local and non-local couplings

We consider a one-dimensional chaotic piecewise linear map lattice with periodic boundary conditions and two types of interactions: (i) local couplings between nearest and next-to-the-nearest neighbors; and (ii) non-local couplings randomly chosen along the lattice according to a specified probability. The chaoticity of the lattice is described by means of its Lyapunov spectrum, which furnishes also information about the system global attractor in a high-dimensional phase space. We study in particular the dependence of this spectrum with the coupling parameters, as well as make comparisons with limiting cases, for which the Lyapunov spectrum is known. 2005 Elsevier Ltd. All rights reserved.

Directed Network as a Chaotic Piece-Wise Linear One-Dimensional Map and Its Large-Deviation Properties

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius-Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large-deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure. Recurrence time statistics and their relationships to double time correlation functions and to power spectra are also considered.

Life Expectancy of Transient Microchaotic Behaviour

The application of digital control may lead to so-called transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system, which may create such vibrations. As a consequence of the digital effects, i.e., the sampling and the round-off error, the behaviour of this system can be described by a one-dimensional piecewise linear map. The lifetime of chaotic transients is usually characterized by the so-called escape rate. In the literature, the reciprocal of the escape rate is considered to be the expected duration of the transient chaotic phenomenon. We claim that this approach is not always fruitful, and present a different way of calculating the mean lifetime in the case of one-dimensional piecewise linear maps. Our method might also be used to solve diffusion problems in one-dimensional models of periodic arrays.

Systems of Coupled Piecewise-Linear Maps with Central Element: Stability of a Synchronized State

We investigate the stability of a synchronized state in systems of coupled one-dimensional piecewise-linear maps that, by construction, contain a central element, i.e., an element that interacts with every other (peripheral) element of the system. Two types of systems are considered, namely, systems with and without interaction between peripheral elements. We establish conditions for the strong and weak stability of a synchronized state in systems considered.

Stability of synchronized and clustered states in a system of coupled piecewise-linear maps

Parameter regions for different types of stability of synchronized and clustered states are obtained for two interacting ensembles of globally coupled one-dimensional piecewise-linear maps. We analyze the strong (asymptotic) and weak (Milnor) stability of the synchronized state, as well as its instability. We establish that the stability and instability regions in the phase space depend only on parameters of the individual skew tent map and do not depend on the ensemble size. In the simplest nontrivial case of four coupled chaotic maps, we obtain stability regions for coherent and two-cluster states. The regions appear to be large enough to provide an efficient control of coherent and clustered chaotic regimes. The transition from desynchronization to synchronization is identified to be qualitatively different in smooth and piecewise-linear models.

A new idea of using one-dimensional PWL map in digital secure communications-dual-resolution approach

In this brief, a new approach of using 1-D map in digital chaotic encryption systems is presented. This scheme utilizes a class of 1-D piecewise linear (PWL) map with different discretization on its input and output terminals to realize finite-precision chaotic encryption and decryption systems through the inverse system approach. This kind of map has the property of different discrete-value sets on its input and output terminals, and is called a dual-resolution map. The general structure of encryption systems that utilize a dual-resolution map is proposed, in which a dynamical system /spl Sigma/(/spl middot/) is used to connect the n-order delay line with the input terminal of the dual-resolution map. An example encoder with a four-order delay line is given to promote an understanding of the performance of this new type nonlinear digital encryption system. Simulation results shows that the zero-response of the encoder has a d-like auto-correlation function and the response sequences converge to cyclic sequences with long period. The decoder (inverse system of the encoder) is sensitive to all parameter mismatches, so it can provide high security of the encrypted signals and can be used in real-time digital secure communications when high security is needed.

Critical sets and two-dimensional topologically mixing attractors of a pair of coupled “Tent maps”

Endomorphisms of the plane generated by coupling two identical one-dimensional unimodal piece-wise linear maps are proved to have two-dimensional topologically mixing attractors.

The existence of two-dimensional topologically mixed attractors for some piecewise linear maps of the plane

We prove the existence of two-dimensional topologically mixed attractors for plane endomorphisms formed by coupling two identical one-dimensional unimodal piecewise linear maps.

Information processing in 1-D systems with chaos

Mathematical models are proposed, based on one-dimensional piecewise-linear maps, in which complex dynamics, bifurcation phenomena and chaos are used for information processing.

Hole-induced crisis in a piece-wise linear map

An interior crisis with striking new characteristics has been observed in a one-dimensional piece-wise linear map. Induced by the collision between a chaotic attractor and a “hole”, the crisis exhibits a scaling law of the characteristic time as 〈 τ〉 ∼ ϵ −1. The analytical calculation shows that this scaling behavior sensitively depends on the function form of the map and suggests that a different and more complicated scaling behavior may be observed in some non-linear maps.

ANALYTICAL DESCRIPTION OF A FRACTAL SET GENERATED BY THE TIME-DELAYED CHUA’S CIRCUIT

Using the thermodynamic formalism of Feigenbaum et al. [1986] we calculate the properties of a fractal set generated by a piecewise-linear one-dimensional map, introduced by Sharkovsky et al. [1993] in studying the time-delayed Chua’s circuit.

Decay of metastable states with discrete dynamics.

We consider the escape from invariant sets of one-dimensional piecewise linear maps which are additively disturbed by weak Gaussian white noise. The escape rates from point attractors and from strange invariant sets in the vicinity of the crisis at fully developed chaos are analytically determined and compared with results from numerical simulations. Both situations are combined resulting in a model with a point attractor which has a strange invariant set as basin boundary. Numerically a nonexponential decay of the attractor is found that can be described by a Markovian three-state model with transition rates known from the previous analysis.

Ergodic properties of piecewise linear maps on fractal repellers

We construct a class of one-dimensional piecewise linear maps admitting fractal invariant sets and uncountably many invariant measures and show that they are ergodic (exact). A physical measure is selected as the invariant measure with maximum information dimension. We also briefly discuss the spectral decomposition of the Frobenius-Perron operator.

Diffusion in uniformly hyperbolic one-dimensional maps and Appell polynomials

We construct the eigenpolynomials and the eigendistributions associated with Ruelle resonances in a piecewise-linear one-dimensional map model of deterministic diffusion which is uniformly hyperbolic. We show that the eigenpolynomials belong to the class of Appell polynomials and that the eigendistributions are given by series of derivatives of the Dirac distribution. The expansion on the eigenpolynomials is shown to converge for initial densities which are entire functions of exponential type.

Variational formulation and ergodic invariants

A family of ergodic invariants for discrete time dynamical systems is constructed from a variational formulation of the dynamics. Some examples are presented for the case of one-dimensional piecewise linear maps.