Weird quasiperiodic attractors in a piecewise linear Belykh map

Discontinuous maps occur in many practical systems, and yet bifurcation phenomena in such maps is quite poorly understood. In this paper, we report some important results that help in analyzing the border collision bifurcations that occur in n-dimensional discontinuous maps. For this purpose, we use the piecewise linear approximation in the neighborhood of the plane of discontinuity. Earlier, Feigin had made a similar analysis for general n-dimensional piecewise smooth continuous maps. In this paper, we extend that line of work for maps with discontinuity to obtain the general conditions of existence of period-1 and period-2 fixed points before and after a border collision bifurcation. The application of the method is then illustrated using a specific example of a two-dimensional discontinuous map.

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map.

The new approach for identification of synchronous generator using Hammerstein model and with piecewise linear map is investigated in this paper. In this method, synchronous generator model consists of a linear-dynamic block and a nonlinear-static block. The identification method simultaneously approximates these blocks without requiring prior assumptions on the form of the static non-linearity. In this study, the field voltage is considered as the input and the active output power and the terminal voltage are considered as the outputs of the synchronous generator. Experimental results show good accuracy of the identified model.

The statistical properties of a wide class of 1D piecewise linear Markov maps are compiled. The method used enables one to address analytically the inverse problem of designing a map with a prescribed correlation function. This class of piecewise linear maps is then used as a system of reference to analyze non-Markov piecewise linear maps and to design maps with given invariant measure and correlation function.

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.

We present a solvable two-dimensional piecewise linear chaotic map which arises from the duplication map of a certain tropical cubic curve. Its gener al solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the nontrivial ultradiscrete limit of the solutio n in spite of a problem known as “the minus-sign problem.”

Recently, the chaos based cryptographic algorithms have suggested some meaningful techniques to develop efficient image encryption schemes. In this paper, we propose a new digital image encryption scheme based on the piecewise linear chaotic map (PWLCM) in order to meet the requirements of the secure encryption. The proposed scheme is described in detail, along with the security analyses including key space analysis, sensitivity analysis, information entropy analysis and differential attack analysis. The results show that the suggested image encryption scheme has some properties desirable in a good security cryptosystem.

A piecewise linear chaotic map and sequential quadratic programming based robust hybrid particle swarm optimization

Trajectories associated with parabolic and hyperbolic periodic points of piecewise linear area-preserving map

We consider the identification of Hammerstein/non-linear feedback models by approximating internal non-linearities using piecewise linear static maps. The resulting method utilizes a point-slope parameterization that leads to a computationally tractable optimization problem. The computational appeal of this technique is derived from the fact that the method only requires a matrix inverse and singular value decomposition. Furthermore, the identification method simultaneously identifies the linear dynamic and static non-linear blocks without requiring prior assumptions on the form of the static non-linearity.

Analytical expressions for power spectral density issued from one-dimensional continuous piecewise linear maps with three slopes

This thesis treats the topic of chaos based broadband communication schemes, with particular emphasis on the statistical analysis and optimization of such schemes. The work is motivated by the recent advances in nonlinear system theory suggesting advantages of using chaos in digital communication applications. In particular, the inherent broadband nature of chaotic signals and their synchronization and decorrelation properties have inspired a number of chaos based communications applications. On the transmitter side, chaos based broadband communication typically involves one or more chaotic generators which are modulated in one of their system parameters or their initial state according to the information to be transmitted. Due to a number of fundamental properties of chaos, it is not difficult to conceive transmitters which benefit from the ease of generating signals with desirable statistical characteristics. However, by the same token it is in general challenging to design robust receivers that result in a scheme exhibiting good performance. A central purpose of this thesis is to optimize the choice of the transmitter part and the receiver part (before chosen in a relatively ad-hoc manner) with respect to overall efficiency criteria of the complete communication system. Considered criteria are: overall bit error rate, computational cost; with given statistical channel properties. The chaotic systems considered mainly belong to the class of discrete time systems defined by the iteration of piecewise linear maps. It is shown that, in particular on the receiver side, optimality implies exponential computational cost. Since, due to this fact, sub-optimal receivers (with linear cost) are of great interest, a significant result of this work has been the systematic establishment of efficient receivers in conjunction with an understanding about the implications of their sub-optimality. Furthermore, supported by an information theoretic argument, a number of rather general implications of the use of chaotic signals in communication applications are derived.

Spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear Markov maps and a large class of repellers. The eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal Markov partition equal the reciprocals of the zeros of the Ruelle zeta functions. The logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing. The space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit(s), so that local behavior of observables can be considered. The new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit(s). Finally, arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed.

A dynamic model of adaptive consumers with endogenous unimodal preferences

Nonlinear dynamic investigations and global analysis of a Cournot duopoly game with two different objectives