Attractor Research Articles

A Novel Memristor-Based Multi-Vortex Hyper-Chaotic Circuit Design and its Application in Image Encryption

This paper proposes a new four-dimensional hyper-chaotic system capable of generating multi-wing chaotic attractors by introducing active magnetron memristors, multi-segmented square functions and trigonometric functions. The dynamical properties of this new hyper-chaotic system, such as equilibrium point, dissipation, Lyapunov exponential spectrum, bifurcation diagram and Poincaré cross-section and attraction basin, are analyzed theoretically and simulated numerically, and the complexity of this system with different parameters is analyzed. It is observed that this hyper-chaotic system has periodic, chaotic and hyper-chaotic variations with an infinite number of equilibria and coexisting attractors under different parameter conditions. The circuit simulation was performed using Multisim and the results obtained were consistent with the numerical analysis of the dynamics, and the chaotic circuit system is designed by FPGA to verify the realizability of the system. Finally, an image encryption algorithm is designed in conjunction with the DNA algorithm to enable a new system chaotic sequence for image encryption. The results show that the hyper-chaotic system has rich dynamical behavior and has high-security performance when applied to image encryption with strong chaotic key and plaintext sensitivity and large key space in image encryption.

A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor

Recently, the application of memristors in chaotic systems has been extensively studied. Unfortunately, there is limited literature on the introduction of discrete memristors into chaotic maps, especially into non-classical multidimensional maps. For this reason, this paper establishes a new three-dimensional parabolic chaotic map model; in order to improve the complexity and randomness of the map, it is coupled with a square-charge-controlled discrete memristor to design a new four-dimensional memristive hyperchaotic map. Firstly, the stability of the two maps is discussed. And their dynamical properties are compared using Lyapunov exponential spectra and bifurcation diagrams. Then, the phase diagram and iteration sequence of the 4D memristive hyperchaotic map are obtained. Meanwhile, we investigate the hyperchaotic states, the transient chaos, state transfer and attractor coexistence phenomena of the four-dimensional memristive map. In particular, the special state transfer phenomenon of switching from a periodic attractor to a quasi-periodic attractor and the special coexistence phenomenon of a quasi-periodic attractor coexisting with a quasi-periodic attractor around fixed points are found, which have not been observed in other systems. Finally, the phase-track diagrams and iterative sequence diagrams of the four-dimensional memristive map are verified on a digital experimental platform, revealing its potential for practical applications.

Dynamical analysis and circuit implementation of a memristive chaotic system with infinite coexisting attractors

In order to obtain a chaotic system with more complex dynamic characteristics and more suitable for engineering applications, this paper combines a general memristor containing a hyperbolic tangent function with a simple three-dimensional chaotic system to construct a four-dimensional memristive chaotic system with infinite coexisting attractors. The memristive chaotic system is thoroughly studied through numerical simulations of various nonlinear systems, including the Lyapunov exponent spectra, bifurcation diagram, C0 complexity, two-parameter bifurcation diagram and basins of attraction. The analysis reveals that this system has complex dynamical behavior. It includes not only periodic limit loops and chaotic attractors that depend on the variation of system parameters, but also the extreme multi-stability phenomenon of infinite coexisting attractors that depend on the variation of the initial conditions of the system. In addition, the chaos degradation and offset boosting control of the system are also studied and analyzed. Finally, the correctness and realizability of the memristive chaotic system are verified by circuit simulation and hardware circuit fabrication.The experimental results show that this memristive chaotic system can lay the foundation for practical engineering fields such as secure communication and image encryption.

Bifurcation, Hidden Chaos, Entropy and Control in Hénon-Based Fractional Memristor Map with Commensurate and Incommensurate Orders

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system’s sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation.

A new chaotic jerk system with hidden heart-shaped attractor: dynamical analysis, multistability, connecting curves and its application in image encryption

Application of chaos in modeling natural phenomena and encryption encourages researchers to design new chaotic systems with exciting features. Here a chaotic jerk oscillator with different properties is proposed. Previous studies mainly used non-polynomial and piecewise linear terms to design the attractors’ shape. In the paper, the heart-shaped attractor is designed using just polynomial terms. This system is studied by considering its bifurcation diagram, Lyapunov exponent, and basin of attraction. These tools show that the proposed system has chaotic and periodic attractors that coexist in some parameter intervals. The oscillator does not have an equilibrium and has a heart-shaped attractor. Moreover, the connecting curves of the oscillator are considered to explore other structural properties. Numerical results confirm the analytical solutions for the system’s connecting curves. The interesting dynamics of the oscillator are used in an encryption and compression application.

Reliability and robustness of oscillations in some slow-fast chaotic systems.

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are, thus, consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.

Grazing, Homoclinic Orbits and Chaos in a Single-Loop Feedback System with a Discontinuous Function

Bifurcations and chaos of a three-dimensional single-loop feedback system with a discontinuous piecewise linear feedback function are examined. Chaotic attractors are generated at the same time of the destabilization of foci accompanied with grazing. Multiple periodic solutions are connected with homoclinic orbits based at a pseudo saddle-focus, which satisfies the condition of Shil’nikov chaos formally. The generation of chaotic oscillations is shown in a circuit experiment on a linear ring oscillator with a comparator. The homoclinic bifurcations and chaos are also shown in a ring neural network with a nonmonotonic neuron.

Random templex encodes topological tipping points in noise-driven chaotic dynamics.

Random attractors are the time-evolving pullback attractors of deterministically chaotic and stochastically perturbed dynamical systems. These attractors have a structure that changes in time and that has been characterized recently using Branched Manifold Analysis through Homologies cell complexes and their homology groups. This description has been further improved for their deterministic counterparts by endowing the cell complex with a directed graph (digraph), which encodes the order in which the cells in the complex are visited by the flow in phase space. A templex is a mathematical object formed by a cell complex and a digraph; it provides a finer description of deterministically chaotic attractors and permits their accurate classification. In a deterministic framework, the digraph of the templex connects cells within a single complex for all time. Here, we introduce the stochastic version of a templex. In such a random templex, there is one complex per snapshot of the random attractor and the digraph connects the generators or "holes" of successive cell complexes. Tipping points appear in a random templex as drastic changes of its holes in time, through their birth, splitting, merging, or death. This paper introduces random templexes and computes them for the noise-driven Lorenz system's random attractor.

Grazing Bifurcations, Homoclinic Orbits and Chaos in a Ring Network of Linear Neuron-Like Elements with a Single Nonmonotonic Piecewise Constant Output Function

The bifurcations and chaos of a ring of three unidirectionally coupled neuron-like elements are examined as a minimal chaotic neural network. The output function of one neuron is nonmonotonic and piecewise constant while those of the other two neurons are linear. Two kinds of nonmonotonic output functions are considered and it is shown that periodic solutions undergo grazing bifurcations owing to discontinuity in the nonmonotonic functions. Chaotic attractors are created directly through a grazing bifurcation and homoclinic orbits based at pseudo steady states are generated. It is shown that homoclinic/heteroclinic orbits satisfying the condition of Shil’nikov chaos are caused by overshoot in the nonmonotonic functions.

Arnold Tongue-Like Structures and Coexisting Attractors in the Memristive Muthuswamy–Chua–Ginoux Circuit Model

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors’ dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs [Formula: see text]. These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors.

CCDH: Complexity based Causal Discovery of Hidden common cause in time series

Causal discovery based on temporal observations poses a significant challenge, especially when dealing with causal relationships among dynamical systems exhibiting chaotic attractors. Existing solutions are data-intensive and unable to detect hidden common drivers (also known as common causes or confounders). To address these limitations, we propose a novel method that overcomes both issues. Our method relies on Takens’ embedding theorem and assesses the complexity of rank order patterns in the embedded series by measuring compressibility through non-sequential recursive pair substitution. Remarkably, this approach is effective even with short data samples and has the capability to detect both unidirectional and bidirectional causation, as well as hidden common causes. To validate its performance, we apply the method to synthetic datasets and human electrophysiological data obtained from an epileptic patient. The method successfully provides insights into the involvement of the left and right hippocampus before, during, and after an epileptic seizure. Consequently, our method may offer valuable additional information for decision-making medical panels in determining the optimal intervention locations. Due to its advantages and simplicity, this method holds promise for application in various scientific and practical domains with successful outcomes.

Fractional-order projection of a chaotic system with hidden attractors and its passivity-based synchronization

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.

Construction and implementation of discrete memristive hyperchaotic map with hidden attractors and self-excited attractors

The investigation of discrete memristor and its application in the discrete map is valuable and meaningful. This paper proposes a new one-dimensional map and couples it with a discrete memristor to construct a memristive hyperchaotic map. The most distinctive feature of this map is that it can generate self-excited and hidden chaotic attractor (HCA), the coexistence dynamics of these two types of attractors are different under the conditions of their generations. The relationship between the fixed points and the parameters is analyzed, revealing that it has line fixed points or no fixed points. Moreover, due to the introduction of the memristor, it has a more extensive chaotic range than the original map and can generate hyperchaotic attractors (HA) with better performance metrics. Finally, the map is implemented in a digital circuit to verify the experimental results.

The Third Type of Chaos in a System of Adaptively Coupled Phase Oscillators with Higher-Order Interactions

Adaptive network models arise when describing processes in a wide range of fields and are characterized by some specific effects. One of them is mixed dynamics, which is the third type of chaos in addition to the conservative and dissipative types. In this work, we consider a more complex type of connections between network elements—simplex, or higher-order adaptive interactions. Using numerical simulation methods, we analyze various characteristics of mixed dynamics and compare them with the case of pairwise couplings. We found that mixed dynamics in the case of simplex interactions is characterized by a very high similarity of a chaotic attractor to a chaotic repeller, as well as a stronger closeness of the sum of the Lyapunov exponents of the attractor and repeller to zero. This means that in the case of three elements, the conservative properties of the system are more pronounced than in the case of two.

Destabilization of a seasonal synchronization in a population model with a seasonally varying Allee effect

Climate change causing large seasonal fluctuations is likely to lead to an increase in the average threshold of the Allee effect as well as an increase in its seasonal variability. In this paper, we show that a seasonally synchronized predator-prey system can be strongly destabilized by these changes in the threshold of the Allee effect. The typical result first leads to two-year and multiyear cycles by the principle of period doubling on a folded Möbius strip, up to the emergence of a chaotic and hyper-chaotic attractor living close to the trivial equilibrium corresponding to the extinction of populations. Moreover, this instability of the ecosystem can be hidden for a long time and the transition to its basin of attraction can occur by external perturbation in a random and irreversible manner. We also reveal a double folding of the 1:1 synchronized cycle manifold inside the corresponding Arnold tongue and hysteresis similar to well-known Duffing oscillator.

Investigating a new conservative 4-dimensional chaotic system

This study analyzed a modified conservative 4D chaotic system is investigated using both integer and non-integer order derivatives. Several dynamical aspects of the said model are explored, such as stable equilibrium points, Lyapunov spectra (LS), attractor projection, Poincare, bifurcations and phase portrait. The system is also analyzed using a singular fractional operator, and the theory of the existence of solutions is established through functional analysis. To obtain numerical results of the fractional order system, a numerical method based on Newton polynomial is applied. The study reveals the presence of hidden and fixed point chaotic attractors for certain fractional order values.

Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator

Purpose:This study aims to explore the intricate and concealed chaotic structures of meminductor systems and their applications in applied sciences by utilizing fractal fractional operators (FFOs). Methods:The dynamical analysis of a three-dimensional meminductor system with FFO in the Caputo sense is presented, and a unique solution for the system is obtained via a novel contraction in an orbitally complete metric space. Numerical results are derived using a newly developed method based on Newton polynomial. The analysis includes variations in fractional order and fractal dimension, with a presentation of properties such as Lyapunov spectra, bifurcations, Poincar’e sections, attractor projection, inversion property, and time series analysis. Results:The numerical simulations uncovered intricate hidden attractors for certain values of fractional and fractal orders. This study provides insight into the effects of FFOs on analyzing hidden chaotic attractors and presents a novel approach to understanding and analyzing the dynamics of meminductor systems. The study’s findings contribute to the understanding of the complex and hidden structures of meminductor systems using FFOs. The novel method developed in this study could be applied to other dynamical systems, leading to further advancements in the analysis of complex systems.

A novel grid multi-structure chaotic attractor and its application in medical image encryption

Grid multi-scroll/wing chaotic systems are complex non-linear dynamic systems, which are widely used in secure communication. The grid multi-scroll/wing chaotic systems are usually realized by using the function control method, which has a complex realization method, many control parameters, and a simple unit attractor structure. In this paper, based on the Hopfield neural network, a memristive Hopfield neural network model is proposed by using the memristor synapse control method. The model can generate novel grid multi-structure chaotic attractors, which have the characteristics of a simple implementation method, few control parameters, and complex unit attractor structure. Firstly, the generation mechanism of the grid multi-structure chaotic attractors is analyzed by the equilibrium points and stability. Secondly, its basic dynamical characteristics including the Lyapunov exponent spectrum, fractal dimension, time series, power spectrum, bifurcation diagram, and Poincaré section are analyzed. Thirdly, an analog circuit of the neural network model is designed and realized by Multisim. Finally, combined with the chaos encryption principle, an image encryption scheme is designed based on the generated grid multi-structure attractors. Experimental results show that compared with the existing schemes, the proposed scheme has larger information entropy, higher key sensitivity, and a good application prospect.

Dynamical analysis of a new 5D hyperchaotic system

This paper reports a new five-dimensional (5D) autonomous hyperchaotic system that is obtained by introducing two linear controllers to the Rabinovich system. The dynamical behaviors, including the boundedness, dissipativity and invariance, existence and stability of nonzero equilibrium points are studied and analyzed. The existences of the hyperchaotic and chaotic attractors are numerically verified through analyzing phase trajectories, Lyapunov exponent spectrum, bifurcations and Poincaré maps. The results indicate that the new 5D Rabinovich system can exhibit rich and complex dynamical behaviors. Finally, the existence of Hopf bifurcation, the stability and expression of the Hopf bifurcation are investigated by using the normal form theory and symbolic computations. Some cases are employed to test and verify the theoretical results.

Can a border collision bifurcation of a chaotic attractor lead to its expansion?

Recently, we reported that a chaotic attractor in a discontinuous one-dimensional map may undergo a so-called exterior border collision bifurcation, which causes additional bands of the attractor to appear. In the present paper, we suppose that the chaotic attractor’s basin boundary contains a chaotic repeller, and discuss a bifurcation pattern consisting of an exterior border collision bifurcation and an expansion bifurcation (interior crisis). In the generic case, where neither the border point the chaotic attractor collides with, nor any of its images belong to the chaotic repeller, the exterior border collision bifurcation is followed by the expansion bifurcation, and the distance between both bifurcations may be arbitrarily small but positive. In the non-generic (codimension-2) case, these bifurcations occur simultaneously, so that a border collision bifurcation of a chaotic attractor leads directly to its expansion.