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Elegant chaos. Algebraically simple chaotic flows. Dedicated to the memory of Edward Norton Lorenz
Abstract
This heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos. No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study. © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
... with σ = 10, ρ = 28, and β = 8/3, which results in that the dynamical system has a Lyapunov spectrum of λ 1 = 0.901, λ 2 = 0, and λ 3 = −14.6 [21], and the Halvorsen system [21] d dt ...
... with σ = 10, ρ = 28, and β = 8/3, which results in that the dynamical system has a Lyapunov spectrum of λ 1 = 0.901, λ 2 = 0, and λ 3 = −14.6 [21], and the Halvorsen system [21] d dt ...
... with a = 1.3. The Lyapunov spectrum of the Halvorsen system is λ 1 = 0.69, λ 2 = 0, and λ 3 = −4.9 when the considered parameters are used [21]. We obtain a time series by discretizing the dynamical systems (11) and (12) with a sampling rate δt = 0.1. ...
Echo-state networks are simple models of discrete dynamical systems driven by a time series. By selecting network parameters such that the dynamics of the network is contractive, characterized by a negative maximal Lyapunov exponent, the network may synchronize with the driving signal. Exploiting this synchronization, the echo-state network may be trained to autonomously reproduce the input dynamics, enabling time-series prediction. However, while synchronization is a necessary condition for prediction, it is not sufficient. Here, we study what other conditions are necessary for successful time-series prediction. We identify two key parameters for prediction performance, and conduct a parameter sweep to find regions where prediction is successful. These regions differ significantly depending on whether full or partial phase space information about the input is provided to the network during training. We explain how these regions emerge.
... This has revealed the presence of chaos in systems ranging from the fields of chemistry and biology to turbulence, e.g. Yamada and Ohkitani (1988); Gallez and Babloyantz (1991); Manneville (1995) and Sprott (2010). In the early days the investigations essentially dealt with low-order systems, but later the scope was broadened to include spatially distributed systems with a large number of degrees of freedom, in coupled maps, e.g. ...
The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean-atmosphere model on a beta-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation scheme and the resolution on the spectra of LEs. The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing. The increase of the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The convergence rate of the rate functional for the large deviation law of the finite-time Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric time-scale separation in such a model. The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated to the ocean dynamics, is not fully resolved because of its associated long time scales, even at intermediate orders. This paper highlights the need to investigate the natural variability of the atmosphere-ocean coupled dynamics by associating rate of growth and decay of perturbations to the physical modes described using the formalism of the covariant Lyapunov vectors and to consider long integrations in order to disentangle the dynamical processes occurring at all time scales.
... LE is related to the fast divergence or convergence of the exponent on adjacent orbits in phase space. Generally, a system with one positive LE is defined as a chaotic system, and a system with two or more positive LEs is defined as a hyperchaotic system [37,38]. For a conservative chaotic system, the volume of its phase space is conservative, and the sum of all LEs is zero [39]. ...
Compared to dissipative chaotic systems, conservative chaotic systems have gained attention because they can avoid reconstruction attacks due to the absence of attractors. This paper reports a general method for constructing 5D Hamiltonian conservative hyperchaotic systems, mainly by coupling three 5D sub-rigid bodies with two identical axes to obtain 5D Euler equations, and then combining Hamiltonian energy and Casimir energy analysis to obtain a 5D conservative hyperchaotic system. This method is general and convenient, and the constructed conservative hyperchaotic system has good performance. In addition, this paper investigates the impact of parameters and initial values on system performance using energy analysis and proposes a simple signal amplitude adjustment method. This method has no restrictions on the mathematical models of chaotic systems, can quickly adjust signal amplitudes, and enhances the hyperchaotic characteristics of the system based on this method. Finally, the correctness of the theoretical and simulation analysis is verified using a DSP hardware platform.
... The Lyapunov exponent spectrums of the attractors of the memristive map (2) will be calculated by using the Wolf methods. [44,45] The iteration length of the memristive map (2) is chosen as 10 5 . To show the bifurcation mechanism of dynamical transition of the memristive map (2), the two-parameter bifurcation analysis are carried out for a ∈ [−0.7, 0.7], b ∈ [−0.95, 1.75] and (c, d, e) = (1, 1, 1). Figure 2 presents several two-parameter bifurcation curves of main low-periodic solutions. ...
In this work, we present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are demonstrated and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map has been carried out to reveal the bifurcation mechanism of the dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors. They can exhibit extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, rarely shown in memristive maps. Potentially, this work can be used for secure communication, such as data and image encryption.
Scientists have been interested in designing and analyzing new dynamical systems with desirable characteristics in recent decades. Such systems can be used to study a wide range of complex phenomena. This study presents the design of a new two-dimensional mega-stable nonlinear system, which is highly sensitive to initial conditions. Depending on these initial conditions, the system can exhibit an infinite number of coexisting limit cycles. Several of these attractors are self-excited, as their basins contain a fixed point, while many others are hidden. These trajectories form a mosaic pattern in the phase space. The system also possesses numerous equilibria, with some being stable and many others unstable. In addition to a comprehensive investigation of this mega-stable system, its forced version is also explored. By applying a periodic force, the system’s dynamics become more complex. Depending on the amplitude of the force term, the system’s attractor can exhibit a toroidal, chaotic, or periodic behavior. The Hamilton energy of the mega-stable system is also investigated. It demonstrates how the energy drops as the system’s frequency increases.
Discovering simple systems with the minimal number of terms is both rare and challenging. This paper introduces a new, simple five-term dissipative 3D chaotic system derived from a third-order ordinary differential equation (ODE) in a single variable. The system is characterized by a single non-hyperbolic equilibrium point or critical value, with both real eigenvalues equal to zero. The dynamic properties of this system are thoroughly analyzed, including: equilibria points, bifurcation diagram, Lyapunov exponents and NIST test. Moreover, an electronic circuit for this system is designed using Multisim 14.3 software, and the results obtained from Multisim show excellent agreement with those from MATLAB 2021. Finally, image encryption and decryption algorithms rely on the new system are proposed. The effectiveness of these algorithms is validated through statistical analyses, including histograms and correlation analysis. Security tests and experimental results confirm that these encryption and decryption algorithms offer high security and are promising candidates for practical image applications.
Chaos is a significant concept in nonlinear dynamics, featuring numerous real-world examples and diverse applications, including cryptography. This study presents a novel chaotic system with a single fully unstable equilibrium point, where the equilibrium point is unstable in all directions — a property that is rare in chaotic systems, where the unstable equilibrium is typically a saddle. The dynamical properties of the proposed system are comprehensively analyzed through stability analysis, forward and backward bifurcation diagrams, Lyapunov exponent diagrams, and basin of attraction studies. Our analyses reveal that when the system’s equilibrium point is completely unstable, it generates monostable and multistable self-excited solutions, encompassing periodic and chaotic attractors. Conversely, in scenarios where the equilibrium point is stable, the system demonstrates periodic orbits, qualifying as hidden attractors. In the subsequent phase, we propose a sequential image encryption algorithm tailored for encrypting RGB images, utilizing the novel chaotic system as its foundation. The algorithm is rigorously evaluated using three prominent benchmark RGB images and subjected to various analyses and metric calculations, assuring its resilience against differential, statistical, and brute-force attacks.
In the topic of discrete variable-order systems governed by fractional difference equations, this study makes a significant contribution by introducing two innovative variable-order versions of the fractional Grassi–Miller system. These new formulations are aimed at deepening our understanding of the complex dynamics that such systems exhibit. The research specifically delves into the chaotic dynamical behaviors manifested by these systems: one version being the fractional Grassi–Miller map with commensurate variable order and the other being the fractional Grassi–Miller map with incommensurate variable order. To provide a comprehensive analysis, this study incorporates a variety of variable orders, encompassing both exponential and sinusoidal functions. These variable orders are crucial in exploring how different functional forms influence the behavior of the system. By varying these orders, the research seeks to uncover the patterns and chaotic dynamics that emerge under different conditions. A suite of advanced numerical methods is employed to rigorously analyze and validate the presence of chaotic attractors in these newly proposed variable fractional versions of the Grassi–Miller system. The methods used include bifurcation diagrams, phase portraits, Lyapunov exponents, approximate entropy, C0 complexity, and 0–1 test for chaos. Through the application of these numerical methods, the study thoroughly validates the existence of chaotic attractors in the proposed variable fractional versions of the Grassi–Miller system. The findings underscore the rich and complex behaviors that arise from different variable orders, offering new insights into the dynamics of fractional-order systems.
The presence of nonlinear feedback is an effective method for generating chaos in dynamical systems. Although physics provides a plethora of nonlinear relationships that can be exploited for this purpose, engineering chaotic electronic circuits with prescribed nonlinear terms remain a formidable challenge. In particular, the implementation of fractional exponentiation represents a particularly challenging task. In this paper, we present a solution to this problem in the form of a new circuit topology, which we have designated as the “444 circuit”, comprising only four transistors, four operational amplifiers, and four resistor pairs. In addition to multiplication and division, this circuit can perform complex algebraic operations, including the calculation of fractional powers. After introducing its operating principles and some conceptual aspects of its implementation, we demonstrate the performance of this unifying circuit structure by realizing a family of Lorenz-like systems.
Predicting the dynamics of chaotic systems is crucial across various practical domains, including the control of infectious diseases and responses to extreme weather events. Such predictions provide quantitative insights into the future behaviors of these complex systems, thereby guiding the decision-making and planning within the respective fields. Recently, data-driven approaches, renowned for their capacity to learn from empirical data, have been widely used to predict chaotic system dynamics. However, these methods rely solely on historical observations while ignoring the underlying mechanisms that govern the systems' behaviors. Consequently, they may perform well in short-term predictions by effectively fitting the data, but their ability to make accurate long-term predictions is limited. A critical challenge in modeling chaotic systems lies in their sensitivity to initial conditions; even a slight variation can lead to significant divergence in actual and predicted trajectories over a finite number of time steps. In this paper, we propose a novel Physics-Guided Learning (PGL) method, aiming at extending the scope of accurate forecasting as much as possible. The proposed method aims to synergize observational data with the governing physical laws of chaotic systems to predict the systems' future dynamics. Specifically, our method consists of three key elements: a data-driven component (DDC) that captures dynamic patterns and mapping functions from historical data; a physics-guided component (PGC) that leverages the governing principles of the system to inform and constrain the learning process; and a nonlinear learning component (NLC) that effectively synthesizes the outputs of both the data-driven and physics-guided components. Empirical validation on six dynamical systems, each exhibiting unique chaotic behaviors, demonstrates that PGL achieves lower prediction errors than existing benchmark predictive models. The results highlight the efficacy of our design of data-physics integration in improving the precision of chaotic system dynamics forecasts.
The location and basins of attraction of attractors involved in nonlinear smooth dynamical systems play a crucial role in understanding the dynamics of these systems across a wide range of parameter values. Nonlinear dynamical systems, described by autonomous nonlinear coupled ordinary differential equations with three or more variables, often exhibit chaotic dynamics within specific parameter ranges. In the literature, three distinct routes to chaos are well known: period doubling, crisis, and intermittency. Recent research in the field of nonlinear dynamical systems focuses on the characterization and identification of hidden attractors. The primary objective of this study is to investigate the dynamics and the successive local and global bifurcations that give rise to chaotic behavior in a Modified Lorenz System (MLS). Our findings reveal that chaos emerges via the type II intermittency route. Furthermore, we identify the chaotic attractor as a hidden attractor, with its existence corroborated by the dissipative nature of the model beyond the subcritical Hopf bifurcation. Key contributions of this research include the derivation of analytical criteria for a degenerate pitchfork bifurcation, the numerical calculation of the first Lyapunov exponent, and a semi-analytic proof of the homoclinic bifurcation. These results enhance our understanding of the complex dynamics within the MLS and contribute to the broader field of nonlinear dynamics and chaos theory.
In dynamical systems, events that deviate significantly from usual or expected behavior are referred to as extreme events. This paper investigates the mechanism of extreme event generation in a 3D jerk system based on a generalized memristive device. In addition, regions of coexisting parallel bifurcation branches are explored as a way of investigating the multistability of the memristive system. The system is examined using bifurcation diagrams, Lyapunov exponents, time series, probability density functions of events, and inter-event intervals. It is found that extreme events occur via a period-doubling route and are due to an interior crisis that manifests itself as a sudden shift from low-amplitude to high-amplitude oscillations. Multistability is also identified when both control parameters and initial values are modified. Finally, an analog circuit based on the memristive jerk system is designed and experimentally realized. To our knowledge, this is the first time that extreme events have been reported in a memristive jerk system in particular and in jerk systems in general.
Vehicle-to-Everything (V2X) communication, essential for enhancing road safety, driving efficiency, and traffic management, must be robust against cybersecurity threats for successful deployment and acceptance. This survey comprehensively explores V2X security challenges, focusing on prevalent cybersecurity threats such as jamming, spoofing, Distributed Denial of Service (DDoS), and eavesdropping attacks. These threats were selected due to their prevalence and ability to compromise the integrity and reliability of V2X systems. Jamming can disrupt communications, spoofing can lead to data and identity manipulation, DDoS attacks can saturate system resources, and eavesdropping can compromise user privacy and information confidentiality. Addressing these major threats ensures that V2X systems are robust and secure for successful deployment and widespread acceptance. An extensive review uncovered a global landscape of V2X cybersecurity research. We highlight contributions from China in scientific publications and the United States in patent innovations, with notable advancements from leading corporations such as Qualcomm, LG, Huawei, Intel, Apple, and Samsung. This work educates and informs on the current state of V2X cybersecurity and identifies emerging trends and future research directions based on a year-by-year analysis of the literature and patents. The findings underscore the evolving cybersecurity landscape in V2X systems and the importance of continued innovation and research in this critical field. The survey navigates the complexities of securing V2X communications, emphasizing the necessity for advanced security protocols and technologies, and highlights innovative approaches within the global scientific and patent research context. By providing a panoramic view of the field, this survey sets the stage for future advancements in V2X cybersecurity.
In this work, we employ simulations to investigate deterministic and stochastic dynamics of a system governed by four coupled nonlinear ordinary differential equations (ODEs) derived from the canonical three-dimensional Chua model, replacing the piecewise nonlinearity with a cubic function. We employed two simulation approaches: (i) using software - where we performed numerical simulations (digital computation) in the ODEs model, and (ii) using hardware - where we performed experimental studies (analog computation) in the ODEs model. In the initial software-based approach for simulations, we performed stability and bifurcation analyses of the trivial equilibrium point, which were analytically conducted, along with the numerical continuation method.
We solved the model using digital computation and explored its intricate dynamical behavior through nonlinear dynamics techniques applied to the parameter planes. In the second approach, employing hardware for simulations through analog computation, we designed an electronic circuit using analog electronics to solve the model in a continuous-time integration. The experimental setup mirrored the numerical simulations, yielding comparable results in terms of dynamical behaviors. This study underscores significant agreement between numerical and experimental findings, highlighting the robustness of the model across digital and analog contexts, and offering practical insights into the various nonlinear dynamics presented in this system.
To improve the complexity of the chaotic system and achieve the effective transmission of image information, in this paper, a five-dimensional memristive chaotic system with cubic nonlinear terms is constructed, which has four pairs of symmetric coordinates. First, the cubic nonlinear memristive chaotic system is analyzed using the Lyapunov exponential map, bifurcation map, and attractor phase diagram. The experimental results show that under four pairs of symmetric coordinates, the system exists not only parameter-dependent symmetric rotational coexisting attractor and transient chaotic phenomena but also exists super-multistationary with alternating chaotic cycles dependent on the initial value of the memristor. Then, it is proposed to add a constant term to the linear state variable to explore the effect of the offset increment of the linear state variable on the system in four pairs of symmetric coordinates, while circuit simulation of the five-dimensional chaotic system is carried out using Simulink to verify its existence and realisability. Finally, the synchronization of the dimensionality reduction system and the confidential transmission of the image are achieved, using the control voltage of the system to replace the internal state variables of the memristor to achieve the one-dimensional reduction process, and an adaptive synchronization controller is designed to synchronize the system before and after the dimensionality reduction. Based on the above, an image to be transmitted is modulated into a one-dimensional array and then subjected to the fractional and cyclic operations and combined with the linear encryption and decryption functions and the chaotic masking technique, the simple encryption and decryption of the image processes are realized.
Benefiting from trigonometric and hyperbolic functions, a nonlinear megastable chaotic system is reported in this paper. Its nonlinear equations without linear terms make the system dynamics much more complex. Its coexisting attractors’ shape is diamond-like; thus, this system is said to have diamond-shaped oscillators. State space and time series plots show the existence of coexisting chaotic attractors. The autonomous version of this system was studied previously. Inspired by the former work and applying a forcing term to this system, its dynamics are studied. All forcing term parameters’ impacts are investigated alongside the initial condition-dependent behaviors to confirm the system’s megastability. The dynamical analysis utilizes one-dimensional and two-dimensional bifurcation diagrams, Lyapunov exponents, Kaplan–Yorke dimension, and attraction basin. Because of this system’s megastability, the one-dimensional bifurcation diagrams and Kaplan–Yorke dimension are plotted with three distinct initial conditions. Its analog circuit is simulated in the OrCAD environment to confirm the numerical simulations’ correctness.
Developing chaotic systems-on-a-chip is gaining much attention due to its great potential in securing communication, encrypting data, generating random numbers, and more. The digital implementation of chaotic systems strives to achieve high performance in terms of time, speed, complexity, and precision. In this paper, the focus is on developing high-speed Field Programmable Gate Array (FPGA) cores for chaotic systems, exemplified by the Lorenz system. The developed cores correspond to numerical integration techniques that can extend to the equations of the sixth order and at high precision. The investigation comprises a thorough analysis and evaluation of the developed cores according to the algorithm complexity and the achieved precision, hardware area, throughput, power consumption, and maximum operational frequency. Validations are done through simulations and careful comparisons with outstanding closely related work from the recent literature. The results affirm the successful creation of highly efficient sixth-order Lorenz discretizations, achieving a high throughput of 3.39 Gbps with a precision of 16 bits. Additionally, an outstanding throughput of 21.17 Gbps was achieved for the first-order implementation coupled with a high precision of 64 bits. These outcomes set our work as a benchmark for high-performance characteristics, surpassing similar investigations reported in the literature.
Countable systems with infinite attractors are called megastable systems and found significance for its wide application. In this research work we are dealing with a Quadratic megastable oscillator (QMO) and exposing various dynamical behaviors which will enhance the complexity of the system. Usually the megastable oscillators were designed with external excitation whereas we proposed a Quadratic megastable oscillator without external excitation. For different initial conditions, nested types of multiple attractors are generated and presented in the form of phase portrait. The sustainability of the chaotic oscillation is studied using influential parameter bifurcation plots. The proposed system has intricate basins of attraction for an infinite number of coexisting attractors. Certain initial conditions may settle in far-flung destinations after eluding the swirling of local attractors. The system is implemented to perform biometric fingerprint image encryption and its effectiveness are discussed. Based on the findings of the statistical and attack analyses of the encryption design, the QMO system has been successful for chaotic system-based encryption applications.
Offset boosting is a crucial passage for chaotic signal modification in chaos-based engineering. Searching an offset controller for a 3D chaotic system is usually complex, let alone two independent nonbifurcating offset constants. This paper constructs a class of chaotic systems, providing a single constant posing direct offset boosting for two dimensions. This offset boostable chaotic system regime has multiple typical control modes, including a system variable single control, synchronous common control, reverse control, and differential control. This new type of chaotic systems also finds two-dimensional offset boosting combined with amplitude control.
In the theory of gene networks, the mathematical apparatus that uses dynamical systems is fruitfully used. The same is true for the theory of neural networks. In both cases, the purpose of the simulation is to study the properties of phase space, as well as the types and the properties of attractors. The paper compares both models, notes their similarities and considers a number of illustrative examples. A local analysis is carried out in the vicinity of critical points and the necessary formulas are derived.
This paper presents the synchronization of a class of hyperchaotic systems using a robust underactuated approach. The proposed scheme guarantees the convergence in finite time of the slave system trajectories to the master system based on Lyapunov theory. The main novelty of the method is its simplicity resulting from the underactuated strategy and its robustness due to the presence of disturbances in the stability analysis. Simulations are presented to show the performance of the proposed method and its advantages compared with another recent study in the literature. In addition, a secure communication example is considered to illustrate the simple application of the synchronizer.
This work introduces a three-dimensional system with quasi-periodic solutions for special values of parameters. The equations model the interactions between genes and their products. In gene regulatory networks, quasi-periodic solutions refer to a specific type of temporal behavior observed in the system. We show the dynamics of Lyapunov exponents. Visualizations are provided. It is important to note that the study of gene regulatory networks is a complex interdisciplinary field that combines biology, mathematics, and computer science.
This paper presents the fractional-order projection of a chaotic system, which delivers a collection of self-excited and hidden chaotic attractors as a function of a single system parameter. Based on an integer-order chaotic system and the proposed transformation, the fractional-order chaotic system obtains when the divergence of integer and fractional vector fields flows in the same direction. Phase portraits, bifurcation diagrams, and Lyapunov exponents validate the chaos generation. Apart from these results, two passivity-based fractional control laws are designed effectively for the integer and fractional-order chaotic systems. In both cases, the synchronization schemes depend on suitable storage functions given by the fractional Lyapunov theory. Several numerical experiments confirm the proposed approach and agree well with the mathematical deductions.
In this paper, the stability of a fractional-order chaotic Jerk system is investigated, using the Routh–Hurwitz criteria. Hopf bifurcation analysis versus the fractional order and a parameter is done. Under the bifurcation parameter, some conditions ensuring the Hopf bifurcation, in the fractional order and its counterpart integer order, are proposed. Furthermore, numerical investigations are performed by taking commensurate and incommensurate fractional chaotic systems, it has been observed that under the influence of fractional order, the location of the critical Hopf bifurcation value changes, causing the system to exhibit reduced chaotic behaviour. Tables and some figures are given to illustrate and verify the results.
In the literature, hyperjerk systems raised up meaningful interest due to their simple and elegant structure as well as their complex dynamical features. In this work, we propose a novel 4D autonomous hyperjerk system which the particularity resides on the type of its nonlinearity namely the Van der Pol nonlinearity. The dynamics of this hyperjerk system is assessed thanks to the well-known nonlinear dynamic tools such as time series, bifurcation diagrams, Lyapunov exponent spectrum, two-parameter phase diagram, and phase portraits. As important result, it is established that the system presents a particular phenomenon of hysteretic dynamics that leads to the coexistence of attractors. Next, through the calculation of the Hamiltonian energy, we show that this latter depends on all the variables of the novel hyperjerk system. Furthermore, basing on an adaptive backstepping method whose target is a function of the states of the error system, a new controller is designed to carry out from t = 30, complete chaotic synchronization of the identical novel hyperjerk chaotic systems. Likewise, PSpice (9.2 full version) based simulations are presented in detail to confirm the feasibility of the theoretical model. One of the key points of this work is the designing in PSpice environment of this new adaptive backstepping controller to validate both theoretical and numerical synchronization results. Finally, our experimental measurements in the laboratory are in good agreement with the numerical and analog results.
Hidden chaotic attractors is a fascinating subject of study in the field of nonlinear dynamics. Jerk systems with a stable equilibrium may produce hidden chaotic attractors. This paper seeks to enhance our understanding of hidden chaotic dynamics in jerk systems of three variables (x,y,z) with nonlinear terms from a predefined set: {−xy + dz2,−xy + dztanh(z)}, where d is a real parameter. The behavior of the systems is analyzed using rigorous Hopf bifurcation analysis and numerical simulations, including phase portraits, bifurcation diagrams, Lyapunov spectra, and basins of attraction. For certain jerk systems with a subcritical Hopf bifurcation, adjusting the coefficient of a linear term can lead to hidden chaotic behavior. The adjustment modifies the subcritical Hopf equilibrium, transforming it from an unstable state to a stable one. One such jerk system, while maintaining its equilibrium stability, experiences a sudden transition from a point attractor to a stable limit cycle. The latter undergoes a period-doubling route to chaos, which may be followed by a reverse route. Therefore, by perturbing certain jerk systems with a subcritical Hopf equilibrium, we can gain insights into the formation of hidden chaotic attractors. Furthermore, adjusting the coefficient of the nonlinear term ztanh(z) in certain systems with a stable equilibrium can also lead to period-doubling routes or reverse period-doubling routes to hidden chaotic dynamics. Both findings are significant for our understanding of the hidden chaotic dynamics that can emerge from nonlinear systems with a stable equilibrium.
In-depth analysis of a novel multiple scroll memristive-based hyperchaotic system with no equilibrium is provided in this work. We identify a family of more complicated nth-order multiple scroll hidden attractors for a unique, enhanced 4-dimensional Sprott-A system. The system is particularly sensitive to initial conditions with coexistence and multistability of attractors when changing the associated parameters and the finite transient simulation time. The complexity (CO), spectral entropy (SE) algorithms, and 0-1 complexity characteristics was thoroughly discussed. On the other hand, the outcomes of the electronic simulation are validated by theoretical calculations and numerical simulations.
Three-dimensional steady-state Arnold–Beltrami–Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier–Stokes (NS) equations with an external force per unit mass. It is well known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ‘butterfly effect’, their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when Re=50 with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.
We introduce the notion of a strong pair of periodic segments over [0, T] and we show its applications in detecting chaotic dynamics. We prove a various number of fixed point index formulas concerning periodic points of the Poincaré map.
Undoubtedly, multistability represents one of the most followed venues for researchers working in the field of nonlinear science. Multistability refers to the situation where a combination of two or more attractors occurs for the same rank of parameters. However, to the best of our knowledge, the situation encountered in the relevant literature is never one where periodicity, chaos and hyperchaos coexist. In this article, we study a fourth-order autonomous dynamical system composing of a Duffing oscillator coupled to a van der Pol oscillator. Coupling consists in disturbing the amplitude of one oscillator with a signal proportional to the amplitude of the other. We exploit analytical and numerical methods (bifurcation diagrams, phase portraits, basins of attraction) to shed light on the plethora of bifurcation modes exhibited by the coupled system. Several ranks of parameters are revealed where the coupled system exhibits two or more competing behaviors. In addition to the transient dynamics, the most gratifying behavior reported in this article concerns the coexistence of four attractors consisting of a limit cycle of period-n, a pair of chaotic attractors and a hyperchaotic attractor. The impact of a fractional-order derivative is also examined. A physical implementation of the coupled oscillator system is performed and the PSpice simulations confirm the predictions of the theoretical study conducted in this work.
We consider a four-prototype Rossler system introduced by Otto Rössler among others as prototypes of the simplest autonomous differential equations (in the sense of minimal dimension, minimal number of parameters, minimal number of nonlinear terms) having chaotic behavior. We contribute towards the understanding of its chaotic behavior by studying its integrability from different points of view. We show that it is neither Darboux integrable, nor C1-integrable.
A new 4D memristive chaotic system with an infinite number of equilibria is proposed via exhaustive computer search. Interestingly, such a new memristive system has a plane of equilibria and two other lines of equilibria. Lyapunov exponent and bifurcation analysis show that this system has chaotic solutions with coexisting attractors. The basins of attraction of the coexisting attractors show chaos, stable fixed-points, and unbounded solutions. Furthermore, the 2D parameter space of the system is explored to find the optimum values of the parameters using the ALO (Ant Lion Optimizer) optimization algorithm.
In a previous publication, we established a new paradigmatic model of laser with feedback including the minimal nonlinearity leading to chaos. In this paper, the jerk dynamics of this minimal universal model of laser is presented. It is proved that two equivalent forms of the model in jerk dynamics can be derived. The electronic circuit of the simpler dynamics is designed and implemented. The link between the minimal universal model of laser and the search for simple jerk circuits is established.
The dynamics of a model of neural networks is studied. It is shown that the dynamical model of a three-dimensional neural network can have several attractors. These attractors can be in the form of stable equilibria and stable limit cycles. In particular, the model in question can have two three-dimensional limit cycles.
In this paper, a new image encryption algorithm is proposed by using the chaotic maps. This cryptosystem is used to encrypt the color images of any size using a dynamic permutation and a large substitution table. The permutation is used to perform the first phase of the encryption process. This permutation is generated by using three parameters calculated from the pixel’s values of three color channels (Red, Green and Blue) of the plaintext image. The second phase consists to make a substitution of the pixels of each channel using a large substitution table generated by three chaotic permutations, chaotic control vector and chaotic rotation vector. In order to increase the avalanche effect of our system, an enhanced cipher block chaining ECBC is used. The results obtained after several simulations carried out by our algorithm on images of different sizes, prove the security of our algorithm against statistical, brutal and differential attacks.
A system of ordinary differential equations is considered, which arises in the modeling of genetic networks and artificial neural networks. Any point in phase space corresponds to a state of a network. Trajectories, which start at some initial point, represent future states. Any trajectory tends to an attractor, which can be a stable equilibrium, limit cycle or something else. It is of practical importance to answer the question of whether a trajectory exists which connects two points, or two regions of phase space. Some classical results in the theory of boundary value problems can provide an answer. Some problems cannot be answered and require the elaboration of new approaches. We consider both the classical approach and specific tasks which are related to the features of the system and the modeling object.
The subject system of this paper is of no particular interest or importance. Rather the intent is to illustrate how artificial intelligence (AI) might someday find new dynamical systems with special properties, analyze their behavior, and prepare a publication reporting the results. In fact, this entire paper was written automatically without human intervention using a template and computer program developed by the author.
We investigate the large-scale influence of numerical noises as tiny artificial stochastic disturbances on a sustained turbulence. Using two-dimensional (2-D) turbulent Rayleigh–Bénard convection (RBC) as an example, we solve numerically the Navier–Stokes equations, separately, by means of a traditional algorithm with double precision (denoted RKwD) and the so-called clean numerical simulation (CNS). The numerical simulation given by RKwD is a mixture of the ‘true’ physical solution and the ‘false’ numerical noises that are random and can be regarded as a kind of artificial stochastic disturbances; unfortunately, the ‘true’ physical solution is mostly at the same level as the ‘false’ numerical noises. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the ‘false’ numerical noises are negligible compared with the ‘true’ physical solution, thus the CNS solution can be used as a ‘clean’ benchmark solution for comparison. It is found that the numerical noises as tiny artificial stochastic disturbances could indeed lead to large-scale deviations of simulations not only in spatio-temporal trajectories but also even in statistics. In particular, these numerical noises (as artificial stochastic disturbances) even lead to different types of flows. The shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter; however, the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Thus we provide rigorous evidence that numerical noises as a kind of small-scale artificial stochastic disturbances have quantitatively and qualitatively large-scale influences on a sustained turbulence, i.e. the 2-D turbulent RBC considered in this paper.
Memristive circuits and systems have been widely studied in the last years due to their potential applications in several technological areas. They are capable of producing nonlinear periodic and chaotic oscillations, due to their locally-active characteristics. In this paper, we consider a cubic four-parameter differential system which models a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and a locally-active current-controlled generic memristor. This system has a saddle-focus equilibrium point at the origin, whose global stable and unstable manifolds are, respectively, the x-axis and the plane x = 0, which are invariant sets where the dynamic is linear. We show that this structure can generate two twin Rössler-type chaotic attractors symmetrical with respect to the plane x = 0. We describe the mechanism of creation of these chaotic attractors, showing that, although being similar to the Rössler attractor, the twin attractors presented here have simpler structural mechanism of formation, since the system has no homoclinic or heteroclinic orbits to the saddle-focus, as presented by the Rössler system. The studied memristive system has the rare property of having chaotic dynamics and an invariant plane with linear dynamic, which is quite different from other chaotic systems presented in the literature that have invariant surfaces filled by equilibrium points. We also present and discuss the electronic circuit implementation of the considered system and study its dynamics at infinity, via the Poincaré compactification, showing that all the solutions, except the ones contained in the plane x = 0, are bounded and cannot escape to infinity.
The paper studies a 3D Chaotic Jerk oscillator with fractional derivatives. An approach is proposed to implement it on a PIC16F877A microcontroller in order to reduce the requirements for multiple analogue electronic components such as resistors, capacitors, coils, multipliers, operational amplifiers, which are very bulky and consume a lot of power. The behaviours of the underlying system are analysed analytically, numerically and experimentally. It comes from this analysis that the fractional model exhibits chaotic dynamics when for parameters for which the equivalent integer derivative system exhibits limit-cycles. The synchronization under two closed initial conditions is also studied, highlighting one of the most common applications of the chaos concept.
This paper studies the dynamics and integrability of two generalisations of a 3D Swinging Atwood’s Machine with additional Coulomb’s interactions and Hooke’s law of elasticity. The complexity of these systems is presented with the help of Poincaré cross sections, phase-parametric diagrams and Lyapunov exponents spectrums. Amazingly, such systems possess both chaotic and integrable dynamics. For the integrable cases we find additional first integrals and we construct general solutions written in terms of elliptic functions. Moreover, we present bifurcation diagrams for the integrable cases and we find resonance curves, which give families of periodic orbits of the systems. In the absence of the gravity, both models are super-integrable.
What makes artificial intelligence techniques so remarkable in the field of computer science is undoubtedly their success in producing effective solutions to difficult computational problems. In particular, metaheuristic optimization algorithms are a unique example of using artificial intelligence techniques to generate approximate solutions to problems that cannot be solved in polynomial time, called NP. Obtaining a substitution box (s-box) structure that will satisfy the desired requirements in cryptography is an example of these NP problems. In the literature, it is a hot topic to optimize the s-box structures obtained from chaotic entropy sources with heuristic algorithms to improve their cryptographic properties. The study with the highest nonlinearity value (110.25) based on optimization algorithms to date has been published in 2020. In this study, a method with a higher nonlinearity value than the algorithms previously proposed in the literature is developed. It has been shown that the nonlinearity value can be increased to 111.75. These results will be a basis for new research on the chaos-based s-box literature and will motivate new studies to develop alternative optimization algorithms in the future to obtain s-box structures based on the random selection equivalent to the AES s-box.
Due to its fundamental and technological relevance, the system formed by a van der Pol oscillator coupled to a Duffing oscillator is the subject of increasing research interest. In this contribution, we study the dynamics of a system composed of a van der Pol oscillator coupled to a Duffing oscillator with two asymmetric potential wells. In the considered coupling scheme, the van der Pol oscillator is disturbed by a signal proportional to the difference of position while the Duffing oscillator is driven by a signal proportional to the speed difference. We use analytical and numerical methods to shed light on the complex behaviors exhibited by the coupled system. We highlight striking phenomena such as hysteresis, parallel bifurcation branches, the coexistence of multiple (i.e., three, four or five) self-excited and hidden attractors, and the coexistence of symmetric and asymmetric bifurcation bubbles depending on the numerical values of initial conditions and parameters. An in-depth study of the coexistence of solutions is carried out by constructing basins of attraction. Sample experimental results captured from STM32F407ZE microcontroller-based digital implementation of the coupled system verify the striking dynamics features observed during the theoretical analysis. It should be noted that the coexistence of a hidden attractor with self-excited others is unique for the coupled oscillators system considered in the context of this work and thus deserves dissemination.
A unique mega-stable oscillator with a memristor device is constructed, with the special properties of parity invariant, approximate symmetry, and attractor growing. The introduced memristor device with sinusoidal functions holds the polarity when each variable switches from positive to negative, and correspondingly approximate symmetric coexisting quasi-periodic attractors are coined. More importantly, with the two periodic functions, the new memristive mega-stable oscillator originates the remarkable phenomenon of attractor growing with an infinite number of coexisting attractors. Such behaviors are investigated by using phase portraits, Poincaré section, and time series. The Hamilton energy balance based on Helmholtz’s theorem explains the dependence of the initial conditions on the mega-stability mechanism. Moreover, we show how the isosurface for which this Hamilton energy is a constant strongly depends on state variables.
This paper is reporting on electronic implementation of a three-dimensional autonomous system with infinite equilibrium point belonging to a parabola. Performance analysis of an adaptive synchronization via relay coupling and a hybrid steganography chaos encryption application are provided. Besides striking parabolic equilibrium, the proposed three-dimensional autonomous system also exhibits hidden chaotic oscillations as well as hidden chaotic bursting oscillations. Electronic implementation of the hidden chaotic behaviors is done to confirm their physical existence. A good qualitative agreement is shown between numerical simulations and OrCAD-PSpice results. Moreover, adaptive synchronization via relay coupling of three three-dimensional autonomous systems with a parabolic equilibrium is analysed by using time histories. Numerical results demonstrate that global synchronization is achieved between the three units. Finally, chaotic behavior found is exploited to provide a suitable text encryption scheme by hidden secret message inside an image using steganography and chaos encryption.
A discrete map can be obtained from Chua's circuit with a discontinuous nonlinearity. Chaotic behavior is observed in the simulation of Chua's circuit with a discontinuous piecewise-linear element. The Poincaré map can be used to study the stability behavior and opens the possibility for analyzing the chaotic behavior by determining the existence of a snap-back repeller.
It is shown that the solution of the Rikitake two-discdynamo system may be described by an orbiting point which, for sufficiently large time, lies arbitrarily close to a limit surface of bounded area. Reversal of the field currents of the dynamo coils are shown to occur in the juxtaposed regions of the two sheets of the limit surface. The two singular points of the system are shown, by Liapounov's direct method, to be unstable foci. An asymptotic theory is developed for the case of small difference in dynamo velocities; the theory is applied to cases of small dissipation in which orbits tend to become nearly periodic with a single oscillation between reversals.
This paper investigates the role of the function x|x| as a chaos
generator in nonautonomous systems. A Duffing-like nonautonomous
oscillator is used for illustration. It is rigorously proven via the
Melnikov function method that this particular quadratic function induces
Smale horseshoes to the Duffing-like system. Moreover, its physical
meaning as an energy function is demonstrated, which provides a critical
value for the emergence of chaos. Simulations with bifurcation analysis
are given for better understanding of the underlying dynamics