Noninvertible Two-dimensional Maps Research Articles

Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus

We research global dynamic behaviors, multi-stability and synchronization in a dynamic duopoly Cournotian model with consumer surplus on the basis of nonlinear demand function and bounded rationality. This game generalizes the traditional dynamic duopoly Cournot game. The aim of enterprises is to optimize their own profits and the social welfare. This game is characterized by discrete difference equations incorporated in the competition model's optimization problem. The stability of equilibrium points and complicated dynamical phenomena are studied, such as flip bifurcation for unique Nash equilibrium point. The global bifurcation of non-invertible two-dimensional map is analyzed through critical curves, which is an important method to further study global properties. Multi-stability is observed due to the presence of multiple complex attractors. An inspection of the basins of attraction is provided for the situation of several attractors coexisting. Since this game has symmetry, it can be stated that a one-dimensional invariant sub-manifold is the diagonal of the system by analyzing the nonlinear map x(t+1)=μx(t)(1−x(t)). The properties of this nonlinear map is completely different from the logistic map although both of them have similar form. Along the invariant diagonal, the synchronization phenomenon of the game is discussed in the end.

Chaos-based cryptosystem on DSP

We present a numeric chaos-based cryptosystem, implemented on a Digital Signal Processor (DSP), which resists all the attacks we have thought of. The encryption scheme is a synchronous stream cipher. Its security arises from the properties of the trajectories in a chaotic attractor, reinforced by the use of a nonlinear non-invertible two-dimensional map, the introduction of jumps between successive points of the orbits and the retaining of only one bit of the representation of real values. We describe the results obtained through a cryptanalytic study, we detail how to adjust the different parameters of the cryptosystem in order to ensure security, and we apply the NIST (National Institute of Standards and Technology) standard tests for pseudo-randomness to our construction. The originality of this work lies in the end in the way we were able to improve the security of our system, so that it is from now on possible to envisage the use, in more general cryptographic purposes, of other recurrences than those classically employed.

An investigation of the global properties of a two-dimensional competing species model

In this paper we demonstrate how the global dynamics of a biological model can be analysed. In particular, as an example, we consider a competing species population model based on the discretisation of the original Lotka-Volterra equations. We analyse the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in the cases when the interspecific competition is considered to be “weak”, “strong” and “mixed”. The main results of this paper are derived from the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.

Dynamical properties of discontinuous and noninvertible two-dimensional map

This paper reports five different features in piecewise continuous and noninvertible system described by two conservative maps or one conservative map and another dissipative map. The features are as follows: the stochastic web bounded by images of discontinuous borderline is the only chaotic trajectory; phase collapse is caused by the irreversibility which makes some points to have two pre-images; fat fractal forbidden web induced by irreversibility; riddled-like attraction basin in stochastic web and forbidden web, when there are different attractors coexisting; and the impossibility to tell to which attractor an initial condition will approach. In an integrate-and-fire circuit, we find that the features still exist in the circuit system described by two dissipative maps, so they are common features in discontinuous and noninvertible maps.

Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps

We review critical situations, linked with period-doubling transition to chaos, which require using at least two-dimensional maps as models representing the universality classes. Each of them corresponds to a saddle solution of the two-dimensional generalization of Feigenbaum-Cvitanovic equation and is characterized by a set of distinct universal constants analogous to Feigenbaum’s α and δ. One type of criticality designated H was discovered by several authors in 80-th in the context of period doubling in conservative dynamics, but occurs as well in dissipative dynamics, as a phenomenon of codimension 2. Second is bicritical behavior, which takes place in systems allowing decomposition onto two dissipative period-doubling subsystems, each of which is brought by parameter tuning onto a threshold of chaos. Types of criticality designated as FQ and C occur in non-invertible two-dimensional maps. We present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems. In particular, we indicate a possibility for realization of the H type criticality without vanishing dissipation, but with its compensation in a self-oscillatory system. Next, we present a number of examples (coupled Henon-like maps, coupled driven oscillators, coupled chaotic self-oscillators), which manifest bicritical behavior. For FQ-type we indicate possibility to arrange it in non-symmetric systems of coupled period-doubling subsystems, e.g. in Henon-like maps and in Chua’s circuits. For C-type we present examples of its appearance in a driven Rossler oscillator at the period-doubling accumulation on the edge of syncronization tongue and in a model map with the Neimark–Sacker bifurcation

Dynamical Behavior in a DPCM System with an Order 2 Predictor. Case of a Constant Input

This paper deals with a differential pulse code modulation (DPCM) transmission system. It includes an encoder which uses a coupling between a transversal predictor and a nonlinear quantizer. This paper studies, from a dynamical point of view, an order 2 DPCM system modeled via a noninvertible two-dimensional map. Chaotic phenomena can be observed; depending on the kind of application, it can be interesting to avoid chaos or to use it. In both cases, it is necessary to understand the mechanisms which give rise to it. For that, the usual tools of chaotic dynamics are used (bifurcation structures, critical and invariant manifolds).

Modifying fractal basin boundaries by reshaping periodic terms

A generic route is described for the modification of fractal basin boundaries in nonlinear systems by changing only the shape of a periodic (autonomous or non-autonomous) term in the dynamics equations. Two examples are used to illustrate the route: a non-invertible two-dimensional map, and a driven dissipative oscillator with a cubic potential that typically models a metastable system close to a fold.

Equilibrium selection in a nonlinear duopoly game with adaptive expectations

We analyze a nonlinear discrete time Cournot duopoly game, where players have adaptive expectations. The evolution of expected outputs over time is generated by the iteration of a noninvertible two-dimensional map. The long-run behavior is characterized by multistability, that is, the presence of coexisting stable consistent beliefs, which correspond to Nash equilibria in the quantity space. Hence, a problem of equilibrium selection arises and the long run outcome strongly depends on the choice of the players’ initial beliefs. We analyze the basins of attraction and their qualitative changes as the model parameters vary. We illustrate that the basins might be nonconnected sets and reveal the mechanism which is responsible for this often-neglected kind of complexity. The analysis of the global bifurcations which cause qualitative changes in the topological structure of the basins is carried out by the method of critical curves.

Analysis of global bifurcations in a market share attraction model

In this paper we demonstrate how the global dynamics of an economic model can be analyzed. In particular, as an application, we consider a market share attraction model widely used in the analysis of interbrand competition in marketing theory. We analyze the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in discrete time. The main result of this paper is given by the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.

Universality and scaling in non-invertible two-dimensional maps

Two new types of period-doubling universality are discussed for non-invertible two-dimensional maps. The first type may appear when the map possesses the simplest fold singularity. The second type is associated with a singularity arising from projecting Whitney's umbrella.