Lozi Map Research Articles

Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution

The efficiency in the controller performance of a BLDC motor in an uncertain environment highly depends on the adaptability of the controller gains. In this paper, the chaotic adaptive tuning strategy for controller gains (CATSCG) is proposed for the speed regulation of BLDC motors. The CATSCG includes two sequential dynamic optimization stages based on identification and predictive processes, and also the use of a novel chaotic online differential evolution (CODE) for providing controller gains at each predefined time interval. Statistical comparative results with other tuning approaches evidence that the use of the chaotic initialization based on the Lozi map included in CODE for the CATSCG can efficiently handle the disturbances in the closed-loop system of the dynamic environment.

Controlling chaotic transients in the Hénon and the Lozi map with the safety function

In this work, we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work, we will apply a new approach of the partial control method that has been recently developed. This new approach is based on finding a special function called the safety function which allows to automatically find the minimum control necessary to avoid the escape of the orbits. Furthermore, we will show the strong connection between the safety function and the classical safe set. To illustrate that, we will compute for the first time, safety functions for the two-dimensional Hénon and Lozi maps, where we also show the strong dependence of this function with the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits.

A higher-dimensional generalization of the Lozi map: bifurcations and dynamics

We generalize the two-dimensional Lozi map in order to systematically obtain piecewise continuous maps in three and higher dimensions. Similar to higher dimensional generalizations of the related Hénon map, these higher dimensional Lozi maps support hyperchaotic dynamics. We carry out a bifurcation analysis and investigate the dynamics through both numerical and analytical means. The analysis is extended to a sequence of approximations that smooth the discontinuity of the derivatives in the Lozi map.

Response of solitary states to noise-modulated parameters in nonlocally coupled networks of Lozi maps.

We study numerically the impact of heterogeneity in parameters on the dynamics of nonlocally coupled discrete-time systems, which exhibit solitary states along the transition from coherence to incoherence. These partial synchronization patterns are described as states when single or several elements demonstrate different dynamics compared with the behavior of other elements in a network. Using as an example a ring network of nonlocally coupled Lozi maps, we explore the robustness of solitary states to heterogeneity in parameters of local dynamics or coupling strength. It is found that if these network parameters are continuously modulated by noise, solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence are constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient.

Chaos Driven Evolutionary Algorithm: a Novel Approach for Optimization

This research deals with the initial investigations on the concept of a chaos-driven evolutionary algorithm Differential evolution. This paper is aimed at the embedding of simple two-dimensional chaotic system, which is Lozi map, in the form of chaos pseudo random number generator for Differential Evolution. The chaotic system of interest is the discrete dissipative system. Repeated simulations were performed on standard benchmark Schwefel’s test function in higher dimensions. Finally, the obtained results are compared with canonical Differential Evolution.

Center Bifurcation in the Lozi Map

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number [Formula: see text]. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form.

Analytic Programming – a Novel Tool for Synthesis of Controller for Chaotic Lozi Map

In this paper, it is presented a utilization of a novel tool for symbolic regression, which is analytic programming, for the purpose of the synthesis of a new feedback control law. This new synthesized chaotic controller secures the fully stabilization of selected discrete chaotic systems, which is the two-dimensional Lozi map. The paper consists of the descriptions of analytic programming as well as selected chaotic system, used heuristic and cost function design. For experimentation, Self-Organizing Migrating Algorithm (SOMA) and Differential evolution (DE) were used. Two selected experiments are detailed described.

Topological and Smooth Dynamics on Surfaces

Because of the pandemia, the workshop on “Topological and Smooth Dynamics on Surfaces” could not be realized in the usual format or in the new hybrid format. Instead, a subgroup of 3 participants used the week mostly for informal discussions, collaborations and research in the topic of the workshop. Namely, a study of a parametric family of planar homeomorphisms was carried out by Boroński and Štimac. Recently a novel approach to the study of parametric families of disiaptive diffeomorphisms on surfaces was introduced by Crovisier and Pujals in their seminal paper [1], in which they initiate the study of the class of strongly dissipative diffeomorphisms . These maps are shown in [1] to be very close in a certain sense to 1-dimensional maps. In particular, Crovisier and Pujals showed that for any of them there exists a reduction to a 1-dimensional model, that consists of a metric tree and a continuous map on it that is semi-conjugate to the original diffeomorphism. The authors also showed that the class of strongly dissipative diffeomorphisms contains maps in the Hénon family, for an open set of parameters. The study carried out by Boroński and Štimac at MFO attempted on generalizing the aforementioned results to a new setting. Namely, a Crovisier- Pujals-like 1-dimensional model was constructed for a Lozi family of maps [3], within the Misiurewicz parameter set [4]. This model conjugates each Lozi map in the studied parameter set to the natural extension of a continuous map on a metric tree, thus extending the earlier result to the C^0 case, but also improving semi-conjugation to a conjugacy. Further results in this direction are expected to come in a timely fashion. The Ph.D. student Kilassa Kvaternik studied and presented the paper [2].

Load frequency control of a microgrid employing a 2D Sine Logistic map based chaotic sine cosine algorithm

This paper proposes a maiden application of a two dimensional Sine Logistic map based chaotic sine cosine algorithm (2D-SLCSCA) optimized classical PID controller for load frequency control (LFC) of an islanded microgrid (MG). In comparison to random variables and 1D chaotic sequences, the 2D chaotic sequences are more ergodic and possess a wider chaotic range, thereby enhancing the global convergence speed and search capability of an algorithm. Initially, the proposed 2D-SLCSCA is tested on eight classical benchmark test functions and its performance is compared with 1D Logistic map based chaotic SCA (1D-LCSCA), 1D Sine map based chaotic SCA (1D-SCSCA), and the SCA incorporating random variables. Test results reveal that the proposed algorithm exhibits better convergence characteristics, statistics, and execution time. Finally, the proposed 2D-SLCSCA is implemented for the LFC analysis of the islanded MG. To establish the competence of the proposed algorithm in this regard, its performance is compared with 2D Hénon map based chaotic SCA, 2D Lozi map based chaotic SCA, improved salp swarm algorithm (ISSA), SCA, grey wolf optimizer (GWO), and particle swarm optimization (PSO) algorithm considering diverse load disturbance patterns in the MG. Simulation results affirm that the proposed control scheme augments the frequency response of the MG exhibiting a maximum percentage improvement of 78.89%, 78.86%, and 96.51% in peak overshoot (OSpeak), peak undershoot (USpeak) and objective function (OFITSE) value, respectively as compared to the other algorithms. Furthermore, sensitivity of the proposed 2D-SLCSCA is validated considering ± 30% variation in the MG parameters.

A novel fractional piecewise linear map: regular and chaotic dynamics

In this paper, a new piecewise linear map of the plan and its fractional version deduced from the Lozi map is introduced and analysed. The main attention is paid to the study of fixed points and their stability, cycles of period two and their stability regions and the type of bifurcation that occur in the dynamical behaviours of this map. The routes to chaos and some chaotic attractors that exist in the behaviour of the integer map are discussed. Finally, the chaotic behaviour of the associated proposed fractional map is analysed by means of bifurcations diagrams.

Estimating the Interval Length for Vertical Monotonicity of Topological Entropy of the Lozi Mappings

In this short note, we prove some lower and upper bounds for the interval length function for vertical monotonicity of topological entropy of the Lozi mappings.

Chaotic optimization algorithm based on the modified probability density function of Lozi map

Chaos optimization algorithms (COAs) usually utilize different chaotic maps(logistic, tent, Hénon, Lozi,...) to generate the pseudo-random numbers mapped as the design variables for global optimization. In this paper we are going to propose new technique to improve the chaotic optimization algorithm by using some transformations to modify the density of the map instead of changing it.

On the performance of chaotic interleaver for turbo codes

It is well known that the interleaver plays a critical role in the performance of turbo codes and its design using random and deterministic permutations. In this paper, we present a new method to design a deterministic interleaver with random-like behavior based on two-dimensional chaotic map, so called lozi map. The designed interleaver is called chaotic interleaver. The statistical properties and the performance of such interleaver have been investigated and compared with random interleaver and dithered golden interleaver. Chaotic interleaver results in lowering the latency and the complexity of implementation and enhance the reliability and the security of the communication system.

Coexisting Infinite Orbits in an Area-Preserving Lozi Map.

Extreme multistability with coexisting infinite orbits has been reported in many continuous memristor-based dynamical circuits and systems, but rarely in discrete dynamical systems. This paper reports the finding of initial values-related coexisting infinite orbits in an area-preserving Lozi map under specific parameter settings. We use the bifurcation diagram and phase orbit diagram to disclose the coexisting infinite orbits that include period, quasi-period and chaos with different types and topologies, and we employ the spectral entropy and sample entropy to depict the initial values-related complexity. Finally, a microprocessor-based hardware platform is developed to acquire four sets of four-channel voltage sequences by switching the initial values. The results show that the area-preserving Lozi map displays coexisting infinite orbits with complicated complexity distributions, which heavily rely on its initial values.

Microcontroller-based Random Number Generator Implementation by Using Discrete Chaotic Maps

In recent decades, chaos theory has been used in different engineering applications of different disciplines. Discrete chaotic maps can be used in encryption applications for digital applications. In this study, firstly, Lozi, Tinkerbell and Barnsley Fern discrete chaotic maps are implemented based on microcontroller. Then, microcontroller based random number generator is implemented by using the three different two-dimensional discrete chaotic maps. The designed random number generator outputs are applied to NIST (National Institute of Standards and Technology) 800-22 and FIPS (Federal Information Processing Standard) tests for randomness validity. The random numbers are successful in all tests.

Relay and complete synchronization in heterogeneous multiplex networks of chaotic maps.

We study relay and complete synchronization in a heterogeneous triplex network of discrete-time chaotic oscillators. A relay layer and two outer layers, which are not directly coupled but interact via the relay layer, represent rings of nonlocally coupled two-dimensional maps. We consider for the first time the case when the spatiotemporal dynamics of the relay layer is completely different from that of the outer layers. Two different configurations of the triplex network are explored: when the relay layer consists of Lozi maps while the outer layers are given by Henon maps and vice versa. Phase and amplitude chimera states are observed in the uncoupled Henon map ring, while solitary state regimes are typical for the isolated Lozi map ring. We show for the first time relay synchronization of amplitude and phase chimeras, a solitary state chimera, and solitary state regimes in the outer layers. We reveal regimes of complete synchronization for the chimera structures and solitary state modes in all the three layers. We also analyze how the synchronization effects depend on the spatiotemporal dynamics of the relay layer and construct phase diagrams in the parameter plane of inter-layer vs intra-layer coupling strength of the relay layer.

Spatio-temporal structures in ensembles of coupled chaotic systems

We review numerical results of studies of the complex dynamics of one- and double-dimensional networks (ensembles) of nonlocally coupled identical chaotic oscillators in the form of discrete- and continuous-time systems, as well as lattices of coupled ensembles. We show that these complex networks can demonstrate specific types of spatio-temporal patterns in the form of chimera states, known as the coexistence of spatially localized domains of coherent (synchronized) and incoherent (asynchronous) dynamics in a network of nonlocally coupled identical oscillators. We describe phase, amplitude, and double-well chimeras and solitary states; their basic characteristics are analyzed and compared. We focus on two basic discrete-time models, Hénon and Lozi maps, which can be used to describe typical chimera structures and solitary states in networks of a wide range of chaotic oscillators. We discuss the bifurcation mechanisms of their appearance and evolution. In conclusion, we describe effects of synchronization of chimera states in coupled ensembles of chaotic maps.

Robust chaos and the continuity of attractors

Abstract As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for many families of piecewise-smooth maps it provides a way to think about changing structures within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map and the border-collision normal form.

Data-Rate Constrained Observers of Nonlinear Systems.

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.

A NEW FAMILY OF LOZI MAPS WITH MAX FUNCTION AND ITS APPLICATIONS

A NEW FAMILY OF LOZI MAPS WITH MAX FUNCTION AND ITS APPLICATIONS