Foreword to the special issue of Journal of Difference Equations and Applications on ‘Lozi, Hénon, and other chaotic attractors, theory and applications’

Abstract

This foreword presents the 32 articles of the special issue devoted to the Lozi, Hénon and other chaotic attractors proposed by 78 authors from 20 countries (Algeria, Brazil, Canada, China, Czech Republic, France, Germany, India, Italy, Jamaica, Japan, Lebanon, New Zealand, Portugal, Russia, Saudi Arabia, Spain, Ukraine, Tunisia, USA). These original research papers consider chaotic attractors, both in their theoretical and applied aspects.

Keywords:

• Lozi map
• Lozi attractor
• Hénon map
• Belykh attractor
• Chua attractor
• Shilnikov attractor
• Markov partition
• chaotic attractor
• dendrite
• fractional map
• control
• cryptography
• evolutionary techniques
• optimization
• secure communication

1. Introduction

Since the first publication of a continuous chaotic attractor, the Lorenz attractor, in 1963 [36], and a discrete chaotic attractor, the Hénon attractor in 1976 [23], the study of chaotic dynamical systems has been developed along several directions in the second half of the twentieth century, followed since the early 90' by unexpected applications of these pure mathematical objects to many fields, including biology, cryptography, optimization, economics and finance, etc.

This special issue collects original research papers addressing such themes, in particular chaotic attractors, both in their theoretical and applied aspects. It offers new insight into the analysis of systems that appear in different applied contexts. For two years (from Spring 2021 to Spring 2023), 78 authors from 20 countries (Algeria, Brazil, Canada, China, Czech Republic, France, Germany, India, Italy, Jamaica, Japan, Lebanon, New Zealand, Portugal, Russia, Saudi Arabia, Spain, Ukraine, Tunisia, USA) worked jointly to publish new results on Lozi, Hénon, Belykh and other chaotic attractors including continuous attractors like the Chua attractor. They are presented in four parts described as follows.

2. Lozi, Hénon and Belykh attractors

Nowadays, Belykh, Hénon and Lozi attractors are the most studied and used for application purposes among discrete chaotic attractors.

In the first part of this special issue, theoretical research on the corresponding maps are considered. Not all the papers publishing new research on these attractors have been collected during the two years of the conception of this special issue due to various reasons (some are PhD dissertations, some authors have published more than one article on the topic, only one being in this issue, etc.) A partial list of these articles is briefly commented in Section 6.

2.1. Control

Different aspects of control of the Hénon and Lozi maps are first presented together.

Capeáns and Sanjuán [17] considered both maps for a choice of parameters where they show transient chaos. They applied a new approach of the partial control method that they have previously developed, to prevent the escape of orbits close to the chaotic saddle which behave chaotically for a while to eventually escape to an external attractor. 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 this escape. They highlighted the strong connection between the safety function and the classical safe set. To illustrate that, they computed the safety function for both maps, and also showed the strong dependence of this function on the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits.

Braverman and Rodkina, [16] explored the stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters was explored. The results were tested on the Hénon and the Lozi maps.

Rybalova, Schöll and Strelkova [50], studied numerically the spatio-temporal dynamics of ring networks of coupled discrete-time systems in the presence of additive noise. The robustness of chimera states with respect to noise perturbations was explored for two ensembles in which the individual elements are described by either logistic maps or Hénon maps in the chaotic regime. The influence of noise on the behaviour of solitary states has been investigated for a ring of non-locally coupled Lozi maps. They found that there is a finite and sufficiently wide region with respect to the coupling strength and the noise intensity where the probability of observing the chimeras and solitary states is high.

2.2. Statistics and Markov partition

Statistical methods including Markov partition are also involved in understanding the chaotic dynamics of these two maps.

Paula Viveiros (de) [46] considered non-classical characterizations of strange attractors which assign occupancy times and numbers to each of the hypercubes that cover them, local information, and generate statistical distributions over such quantities, global information, at the expense of losing all the local ones. He presented statistical distributions in the Lozi map of the mentioned times and numbers, and quantities derived from them, linked to each hypercube of a set that corresponds to a small fraction of their total, which confirms the correspondence between local and global statistical properties.

Baptista [9], studied iterations of two-dimensional maps, in particular iterations of Lozi maps in the region of the parameter space where it has a strange attractor. Using symbolic dynamics techniques for two-dimensional maps, based on the kneading theory of Milnor and Thurston and also in the symbolic dynamic formalism developed by Sousa Ramos, through the kneading sequence for the Lozi maps, he characterized the region in the parameter space that contains the kneading curves and present a method to define a Markov partition for the Lozi attractors. Consequently, the topological entropy for the Lozi map was computed.

2.3. Generalization

Lozi, Hénon and Belykh maps have been generalized in many ways since their initial publication. In this special issue, four articles are linked to these generalizations.

Sushko, Avrutin and Gardini [54] investigated how the bifurcation structure of the Lozi map is incorporated into the bifurcation structure of the 2D border collision normal form using an analytical representation of the boundaries of the largest periodicity regions related to the cycles with rotation number 1/n, $n\ge 3$. At the centre bifurcation boundary of the stability domain of the fixed point both maps are conservative which leads to a quite intricate bifurcation structure near this boundary.

Bilal and Ramaswamy [12] generalized 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. They carried out a bifurcation analysis and investigated the dynamics through both numerical and analytical means. The analysis was extended to a sequence of approximations that smooth the discontinuity of the derivatives in the Lozi map.

Belykh, Barabash and Grechko [11] introduced a 2D map with a single piecewise-smooth nonlinear function that is suitable for analytic studies of its strange attractor. This 2D piecewise-smooth nonlinear map represents a broad range of chaotic maps, including a hybrid Lozi-Hńon map and the Belykh map. To prove the existence of a strange attractor in the 2D map, they

- (i)	constructed its trapping region that contains all limit sets and
- (ii)	demonstrated that the invariant set's trajectories have negative and positive Lyapunov exponents.

Moreover, they developed an invariant cone approach to establish the latter property, which involves constructing expanding and contracting cones as bounds for the eigenvectors of the variational equations along the chaotic trajectories. They applied their approach to analyse the chaotic hybrid Lozi-Hénon map, the original Lozi map, and the Belykh map.

Linero Bas and Nieves Roldàn [35] revised a transformation that links generalized Lozi maps with max-type difference equations. In this view, according to the technique of topological conjugation, they related the dynamics of a concrete Lozi map with a complete uniparametric family of max-equations, and they applied this fact to investigate the dynamics of two particular families.

Ibrahim [24], suggested three different systems: the k-symbol fractional Lozi system (FLS), the Lozi system (FLS) and the k-symbol fractional Lozi system. He examined several of these systems' crucial dynamics and explored the sufficient and necessary conditions for the k-symbol fractional dynamic systems to be stable and asymptotically stable, providing the sufficient conditions on the k-symbol FLS to attain steady and asymptotically steady zero resolutions. Additionally, he suggested control laws designed to stabilize a combination of these systems.

3. General properties of attractors

3.1. Properties of 2D-maps

There are three articles devoted to the topological properties of 2D-maps.

Gardini and Radi [20], considered a duopoly model characterized by a two-dimensional non-invertible continuous map T given by piecewise linear functions, with several partitions, defining a duopoly game. The structure of the game is such that it has a separate second iterate so that its dynamics can be studied via a one-dimensional composite function, that is piecewise linear with multiple partitions in which the definition of the map changes. The number of partitions may change from 2 to 5, depending on the parameters. The dynamics are characterized by degenerate bifurcations and border-collision bifurcations, which are typical in maps having kink points. Here, the peculiarity is the multiplicity of the partitions, which leads to bifurcations different from those occurring in maps with only one kink point. They showed several bifurcations, coexistence of cycles, attracting and superstable, as well chaotic attractors and chaotic repellors, related to the outcome of particular border collision bifurcations.

In [52], Simpson, detected invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps. He showed how to formally identify chaotic attractors in continuous, piecewise-linear maps on ${\mathbb{R}}^N$. For such a map f, this is achieved by constructing three objects. First, ${\mathrm{\Omega }}_{trap}\subset {\mathbb{R}}^N$ is trapping region for f. Second, $\mathbf{W}$ is a finite set of words that encodes the forward orbits of all points in ${\mathrm{\Omega }}_{trap}$. Finally, $C\subset T{\mathbb{R}}^N$is an invariant expanding cone for derivatives of compositions of f formed by the words in $\mathbf{W}$. The existence of ${\mathrm{\Omega }}_{trap}$, $\mathbf{W}$, and C implies f has a topological attractor with a positive Lyapunov exponent. He developed an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of ${\mathrm{\Omega }}_{trap}$ and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. He also observed how the failure of C to be expanding can coincide with a bifurcation of f.

Ladjeroud and Zeraoulia [32] proved the quasi-hyperbolicity of a certain family of 2-D piecewise linear maps by applying the theorem of Shilnikov and Gavrilov concerning the hyperbolicity of some types of smooth maps.

3.2. Fractional maps

The new paradigm of fractional maps is explored in one article of this issue.

Wu, Wei and Luo [59] introduced in their study a right fractional calculus method for inverse problem modelling and proposed a concept of inverse-time fractional chaotic maps. First, a simple right fractional linear differential equation's terminal value problem and the solutions were investigated. Then, some basics of the right discrete fractional calculus were introduced and the idea was extended to the discrete case. Right fractional sum equations were derived and numerical schemes were provided for dynamical analysis. Discrete chaos does exist in an inverse-time fractional logistic map and Hénon map, respectively. Their local stability conditions were given. They concluded that this right fractional calculus method is simpler but more efficient than the left one.

3.3. Predictive control

In parallel to the control of Hénon and Lozi maps (see Section 2.1), the control of the generalized tent map is studied by Ayers et al. [6], who described a predictive control method to search for its unstable periodic orbits. The invariant set containing periodic orbits is a repelling set with a complicated Cantor-like structure. They showed that for certain values of the control parameter, both the local behaviour and the global behaviour of solutions change in the controlled system; in particular, the invariant set enlarges to become an interval or the entire real axis. The computational particularities of using the control system were considered, and necessary conditions for the orbit to be periodic were given. Moreover, they fully investigated the local asymptotic stability of subcycles of the controlled system's stable cycles, and some statistical properties of the subset of the classical Cantor middle thirds set that was determined by the periodic points of the generalized tent map were described.

3.4. 3D-maps and Shilnikov attractors

Bakhanova et al. [7] described scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of three-dimensional flows and maps. The presented results were illustrated with various specific examples.

3.5. Evolutionary techniques

Chaos, an inherent phenomenon manifested by both continuous and discrete dynamic nonlinear systems, has become an essential component of engineering design. The field of evolutionary computing has been no exception. These heuristic optimization techniques (also known as metaheuristics) have seen researchers endeavouring to incorporate elements of chaos to enhance optimization, and, simultaneously, seeking to employ metaheuristics to control and harness chaotic systems.

The article of Zelinka and Senkerik [60] delved into the intersection between discrete dynamical systems and bio-inspired metaheuristic algorithms. They examined the relationship between chaos and bio-inspired metaheuristic algorithms from two distinct angles: the presence of chaos within the realm of algorithms, and the application of bio-inspired algorithms for the identification, control, or synthesis of complex systems. Furthermore, their paper offered an in-depth exploration of the interplay between metaheuristics and complex nonlinear dynamics, including potential avenues for future research. Their presented simulation studies, along with the design of objective functions for optimization and the implementation of metaheuristics, serve as a valuable foundation for subsequent experiments involving discrete nonlinear systems and complex systems with time delays. Such systems may benefit from multiparametric or multicriteria optimization approaches, paving the way for novel advancements in the field.

Pluhacek et al. [47] presented an overview of the history and recent efforts in combining chaos theory and evolutionary computation techniques. Various algorithms from the evolutionary computation domain, also known as metaheuristic algorithms, have been successfully enhanced with chaotic components in the past. Numerous ways to incorporate chaos have been examined, and many impressive results have been reported. Implementations of discrete chaotic maps such as Lozi, Hénon, and logistic map as generators of chaotic pseudo-random sequences for controlling evolution operators in metaheuristics have achieved significant popularity. In their survey, they focussed on the research field itself and performed a bibliographical analysis to show how broad and active is nowadays the research field of chaos-enhanced metaheuristics and what are some of the most recent works published.

4. Influence of the topology of attractors and phase spaces on dynamics of maps

In this second part of this issue, dynamics of maps are considered from a more general point of view.

4.1. Geometric properties

Efremova [19] considered $C^1$-smooth self-maps of a cylinder close to $C^1$-smooth skew products (and satisfying some additional conditions) and studied a geometric property of these maps, as the existence of $C^1$-smooth invariant local lamination, and applied this geometric property to the proof of their geometric integrability. The obtained results allowed her to construct an example of such a family of maps, so that each map of this family admits a global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. It consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

Barinova, Grines et Pochinka, [10] considered a class of A-diffeomorphisms given on a 3-manifold, assuming that all the basic sets of the diffeomorphisms are two dimensional. It is known that such basic sets are either attractors or repellers and they are two types only: surface or locally direct product of 2-disc by Cantor set. They proved that different types of two-dimensional basic sets do not coexist in the non-wandering set of the same 3-diffeomorphism. They also constructively proved the existence of an energy function for systems of this class. This result was illustrated by examples that the two-dimensionality of the basic sets is essential in this matter and a decrease in the dimension can lead to the absence of the energy function for a diffeomorphism.

In [62] Zhukova and Korotkov highlighted some remarks on the chaotic behaviour of countable products of homeomorphism groups. They investigated relationships between chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products. They provided a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as non-compact surfaces, like triangulable closed manifolds of an arbitrary dimension.

4.2. Dendrites

Makhrova [38], considering X a dendrite and $f:X⟶X$ a continuous map that has a minimal set M in X, obtained conditions on the structure of a dendrite X and the set M, under which f has an arc horseshoe, and, hence a positive topological entropy. She showed that this result is not correct both for continuous maps with an empty set of periodic points, and for continuous maps with a non-empty set of periodic points on finite graphs.

Abdelli, Marzougui and Mchaalia [1] Considered recurrence and non-wandering sets of local dendrite maps. Given a local dendrite X with set $End(X)$ of endpoints of X countable and $f:X⟶X$ be a continuous map, let $P(f)$, $AP(f)$, $R(f)$, $\mathrm{\Omega }(f)$, $EP(f)$, $EAP(f)$ the sets of periodic points, almost periodic points, recurrent points, nonwandering points, eventually periodic points and eventually almost periodic points of f, respectively. They showed that $R(f)\subset \overline{EAP(f)}$ and that $\mathrm{\Omega }(f)\subset \overline{EAP(f)}$, whenever $P(f)=\mathrm{\varnothing }$. As a consequence, they proved that if the set $B(X)$ of branch points is finite, then $\overline{R(f)}=\overline{AP(f)}$. They gave a counter-example showing that these results do not hold whenever $B(X)$ is infinite or $End(X)$ is uncountable.

5. Applications

The first paradigm shift in the study of chaotic maps came in 1987, when instead of focussing on the theoretical study of their mathematical properties or the search for generalizations, some electrical engineers used them to build new devices, like Rodriguez-Vasquez et al. [48] who presented a special-purpose analog computer made of switched-capacitor circuit for analysing chaos and bifurcation phenomena in nonlinear discrete dynamical systems modelled by discrete maps. They published results for four maps: the logistic map, a piece-wise linear map, the Hénon and Lozi maps. Since then, and until today, these maps are continuously involved in the field of electronics or electrical devices like memristors, in secure communications, biological models and even economy. Some of them are presented in this special issue.

5.1. Chua attractor and memristor

In 1971, Leon Chua, then a professor at the University of California, Berkeley, predicted the existence of a missing fourth passive circuit element, in addition to the three classical ones: resistor, inductor and capacitor [18]. He called this new element ‘memristor’ meaning it is a resistor with memory. It is characterized by a nonlinear constitutive relationship between the charge q and the flux φ. Such a physical device would not be reported until 2008, when a physical model of a two-terminal hp device behaving like a memristor was announced [53] sparkling intense research with thousands of papers published to date. A general Ohm's law for theorizing this device was published ten years ago [2].

In 1983, Chua invented a very simple electric circuit producing chaos [41]. The advantage of this circuit is that the variables of the mathematical equations correspond to voltage and current and can be viewed on the screen of an oscilloscope.

These two important improvements in the theory of electrical circuits are studied in three articles on this issue.

Letellier, Minati and Barbot [33], considered the control of continuous-time and discretized Chua circuit, governed by differential and difference equations. They showed that selecting the measured variable and the actuator can be guided by considering the observability and controllability symbolic coefficients and applying the notion of flatness. The feedback linearization was compared to a homogeneous and a passivity-based control laws, the flat control laws being more efficient than the others. It was thus shown that the proposed flat control law by feedback linearization is very efficient. The authors suggested that most likely, these results could be extendable to a large group of natural and experimental systems.

Boudjerida, Abdelouahab and Lozi [15], reported a modified hyperchaotic memristor-based Chua's circuit and its generalized discrete model and explored their dynamics using different classical tools, showing that the proposed circuit model possesses a line of equilibrium and exhibits rich dynamics, including the coexistence of regular and chaotic attractors, and mixed-mode oscillations. Using the Jury criterion, they constructed a useful stability test to localize some stable regions in three parameter planes of the discrete model. The presence of an exact periodic solution was numerically highlighted, and the occurrence of chaotic behaviour was confirmed using the largest Lyapunov exponent and the 0-1 test.

Bao et al. [8] presented a unified memristor-based Gauss mapping model by coupling discrete memristors with a Gauss map and then four two-dimensional (2-D) memristive Gauss maps using four different memristance functions were derived. These maps possess two specific cases of infinitely many line fixed points and no fixed points, resulting in the appearance of hidden dynamics or self-exited dynamics. Their results demonstrated that the discrete memristor can make the four 2-D memristive Gauss maps produce complex dynamical behaviours and greatly improve the chaos complexity of the one-dimensional (1-D) Gauss map.

5.2. Secure communication, cryptography and technology

Nowadays, chaos-based cryptography systems are widely used to protect sensitive data. A variety of chaos-based cryptosystems with software implementation have been published in the literature. However, due to the rapid growth of connected devices, the development of hardware secure and low-resource cryptographic systems is necessary.

In [37] Madani et al. enhanced the design of one of their already published software cryptosystems and then addressed its FPGA-based implementation. The improvement in the design consists essentially to introduce an Initial Vector (IV) entry to the Chaotic Pseudo Random Number Generator (CPRNG), thus allowing to generate new dynamic keys for the confusion layer (using a modified 2-D cat map) and diffusion layer (using a logistic map) with each new execution of the system. The proposed cryptosystem, which operates in Cipher Block Chaining (CBC) mode, is controlled by a Finite State Machine (FSM) to ensure that both the encryption and decryption are performed by the same reconfigurable architecture. The system was implemented on a Xilinx XC7Z020 ZYNQ FPGA platform.

Khalifa and Benrejeb [27], proposed a secure communication system based on mixed discrete-time chaotic systems. It is composed of two chaotic systems on the transmitter and receiver sides, to increase signal transmission security over the public channel. The case of electrocardiogram encryption and decryption is considered as an application to illustrate the effectiveness of the proposed chaotic encryption approach.

Rosalie and Chaumette [49] focussed on chaotic dynamics and reviewed their applications in the design of mobility models. They also provided a review of the nonlinear tools used to characterize mobility models, as it can be found in the literature. Finally, they proposed a method to generate traces for a given scenario involving moving people, using tools from the nonlinear analysis domain usually dedicated to topological analysis of chaotic attractors.

5.3. Models in life science and economy

Chaotic attractors have also applications in life science and economy. Two articles are related to the Covid-19 disease and the dynamics of cereal crops, and one in duopoly game.

Othman Almatroud et al. [44] made a contribution to the studies on epidemic mathematical models based on fractional-order operators by introducing a novel Covid-19 model described by noninteger-order difference equations. Their system has two fixed points at most, i.e. a disease-free fixed point which is proved globally stable, indicating that the pandemic will disappear when a simple condition on the system parameters is satisfied. Numerical simulations confirmed these results.

Mangiarotti and Le Jean [39] recalled that chaotic attractors were captured from remote sensing time series of the dynamics of cereal crops in semi-arid region in the early 2010s. They questioned the generality of this result by applying the same modelling approach – the global modelling technique – to four provinces in coastal and inland Morocco: Safi, El Jadida, Khourigba and Khenifra. Several experimentations were considered, applying the analyses at the provinces scale, or to several provinces in association and in aggregation. The obtained models confirmed the results obtained in the early 2010s: the dynamics of cereal crops is chaotic and can be approximated by a weakly dissipative three-dimensional dynamical systems.

Meskine, Abdelouahab and Lozi [43] investigated the dynamics of a remanufacturing duopoly game model with heterogeneous players and different competition strategies, assuming that the original equipment manufacturer has bounded rationality and aims to maximize profit, and the third-party remanufacturer has an adaptive nature and aims to maximize share in order to get a certain profit. The model behaviours were discussed mainly from the point of view of stability, bifurcations and chaos. By utilizing Jury's stability criteria, the Cournot-Nash equilibrium's asymptotic stability was examined. Under certain conditions, when consumer willingness to pay was taken as a bifurcation parameter, the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation. Lyapunov exponents were used to demonstrate that the system becomes chaotic through each of the previous bifurcations.

6. Additional recent literature on Lozi and Hénon maps

Research on chaotic attractors is a hot topic today. Between April 2021 and June 2023, significant literature has been published. We give a brief overview of the contents of those articles that were not published in this special issue, although related to the same topic.

Kucharski [30] studied the case b<0 of the Lozi map and presented a rigorous proof that for a set of parameters quasi-symmetric to the set $\mathcal{M}$ defined by Misurewicz in 1978 in the case b>0, this map has a strange attractor. With Boroński, and Ou [13] he rigorously determined an open region $\mathcal{B}$ in the parameter space for which this map exhibits periodic points of least period n, for all n>13.

Ou [45] proved that all periodic orbits of this map can be classified by using symbolic dynamics and that the bifurcation parameters of some types of periodic orbits form analytic curves in the parameter space. Kilassa Kvaternik [29] in his doctoral dissertation studied the tangential homoclinic points locus of the Lozi map and applications.

Wormell [57] defined a new property of chaotic attractor: the conditional mixing, and conjectured that for generic parameters $(a,b)$ and Lebesgue-almost all $x_0\in R$, the Lozi map has conditional mixing with respect to level curves $x=x_0$ (that is, the measures $\rho (\cdot |x_0)$). She has shown strong and direct numerical evidence in favour of this conjecture.

When high-dimensional non-uniformly hyperbolic chaotic systems undergo dynamical perturbations, their long-time statistics are generally observed to respond differentiably with respect to the perturbation. In [58] considering this map as a family of the mathematically tractable class of piecewise smooth hyperbolic maps she proved that the existence of a formal derivative of the response reduces to an exponential mixing property of the SRB measure when conditioned on the map's singularity set.

Li and Gu [34], in the conservative case studied the hierarchical structure of a chaotic sea and the existence condition and delimitation method of island chain for this map.

Many new generalizations of the Hénon and the Lozi maps have been proposed.

Boronski and Stimac [14] showed that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions of maps on metric trees with dense set of branch points. In consequence, these trees approximate very well the topology of the attractors, and the maps on them give good models of the dynamics. Kucharski [31] generalized the Lozi-like family introduced by Misiurewicz and Stimac which encompasses in particular certain Lozi-like maps, orientation preserving or reversing Lozi maps or large parameter regions of 2-dimensional border collision normal forms.

Ibrahim and Baleanu [25] using a linear interpolation of quadratic and absolute value functions defined a combined Hénon-Lozi fractional map and studied it. Al-Saidi et al. [5] extended the definition of fractional map to the case where the order of derivation is variable using Lozi map as an example. Joshi et al. [26] considered the definition of fractional map to the case where the order of derivation ν is a complex number with $0<Re(\nu )<1$.

A canonical example of 2D border-collision normal form family of difference equations was presented by Glendinning and Simpson [22], which reduces to the Lozi map for certain values of parameters. Ghosh, Mc Lachlan, and D. Simpson [21] extended their results to the non-orientation-preserving cases. In [3] Aliwi and Ajeena proposed an unconventional generalization of this map based on the Sine Chaotic Map concept, and on the Cosine Chaotic map in [4].

Menasri [42] replaced the piecewise linear term in the first equation of the Lozi map by the max of two arbitrary functions.

Applications of chaotic attractors have been published also during this period.

Recently, hidden attractors have also been discovered in some discrete memristors based maps by Zhang et al. [61]. Wang et al. [56] included a discrete-time memristor to create a memristive Lozi map. This new 3-D memristor-based Lozi map was established by coupling a discrete memristor to the original 2-D Lozi map. Khennaoui et al. [28] using the Caputo-like delta difference obtained another memristor-based fractional Lozi map.

In [51] Sahnoune and Berkani presented a new method to design a deterministic interleaver with random-like behaviour based on Lozi map. Urrea et al. [55] compared the performances of such a chaotic interleaver with short block lengths, with a new chaotic interleaver based on Duffing map which showed slightly better performances.

Martinez-Vargas et al. [40] proposed a leader-follower optimization algorithm based on the combination of five different Population-based Algorithms (PAs) to tackle the high-dimensional optimization problem known as the base station (BS) deactivation problem in telephony. Among these five algorithms, chaotic binary particle swarm optimization (BPSO) has a Lozi map, embedded in.

7. Conclusion

Since the first example of a chaotic attractor, sixty years ago, research on such objects is still relevant. This special issue is only one step in this process which will continue for a long time.

To conclude this extraordinary adventure, seven of the most experienced authors agreed to tell how they became interested in mathematics and carried out research on chaotic dynamical systems. Two of them received the Bern Aulbach Prize awarded every two years by the International Society of Difference Equations. Two other winners of this prize joined them to write a general article which will be published in link to this special issue.

Acknowledgments

We wish to thank Saber Elaydi editor in chief of the Journal of Difference Equations and Applications and President of the International Society of Difference Equations during the two years of this effort, for giving us not only the opportunity but also a strong motivation, for editing this special issue. Our warmful thanks also go to Justin Robinson and Micaela Gail Villaseñor for their efficient assistance in this endeavour. We would also like to thank all the reviewers of these 32 articles who made it possible to produce this special issue.

Disclosure statement

No potential conflict of interest was reported by the author(s).