Types Of Chaotic Attractors Research Articles

Dynamic analysis of a novel hyperchaotic system based on STM32 and application in image encryption

This paper presents a novel 4D hyperchaotic system derived from a modified 3D Lorenz chaotic system. A key aspect of this system is the presence of a single equilibrium point, and its stability is carefully analyzed. The dynamic properties, including the Lyapunov exponent spectrum, bifurcation diagram, and chaotic attractors, are investigated using MATLAB simulations. The results reveal that the system displays hyperchaotic behavior across a wide range of the parameter d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d$$\\end{document}, with its chaotic attractor transitioning through four states: hyperchaotic, chaotic, periodic, and quasi-periodic, showcasing the system's complex dynamics. Experimental validation using STM32 embedded hardware successfully reproduces these four types of chaotic attractors, confirming the theoretical predictions. The proposed hyperchaotic system is then applied to image encryption, introducing a novel encryption method. The hyperchaotic key sequence generated by this system meets 15 tests of the NIST SP800-22 standard, and further experimental validation with STM32 hardware demonstrates the algorithm's effectiveness, simplicity, non-linearity, and high security. The encrypted images and sequences are rigorously tested key space analysis, histogram similarity analysis, information entropy analysis, statistical attack analysis, differential attack analysis, key sensitivity analysis, and correlation analysis, highlighting the robustness and reliability of the proposed system. This method is versatile and can be extended to various fields, including audio and video encryption, text encryption, IoT security, financial transaction security, and medical data protection.

A New Hyperchaotic Map Based on Discrete Memristor and Meminductor: Dynamics Analysis, Encryption Application, and DSP Implementation

At present, discrete memristor is a focus of chaotic mapping. However, the study of discrete memcapacitor and discrete meminductor has not begun. In order to study the dynamics of discrete meminductor and the dynamics of chaotic map containing discrete meminductor. In this paper, a five dimensional Hyperchaotic mapping is constructed by combining discrete memristor and discrete meminductor. In addition, the dynamical behavior of chaotic mapping is analyzed by means of the phase diagram of chaotic attractor, LEs, Bifurcation diagram and spectral entropy complexity. Theoretical analysis and numerical simulation show that the five -dimensional chaotic map not only has hyperchaotic phenomenon, but also can generate different types of chaotic attractors with the change of parameters. Especially, the special phenomenon of coexistence of attractors is discovered. This makes this five-dimensional hyperchaotic map more suitable for many chaotic based engineering applications. So, an application of discrete chaotic system in chaotic image encryption is proposed. The security and reliability of the encryption scheme are analyzed by NIST test, entropy, correlation and histogram. Finally, the results of the numerical simulation are verified by digital circuit. Due to its complex dynamical characteristics, the 5-D hyperchaotic map has great value in engineering applications.

Hidden chaotic mechanisms for a family of chameleon systems

<abstract><p>Chameleon chaotic systems are nonlinear dynamical systems whose chaotic attractors can transform between hidden and self-excited types by tuning system parameters to modify equilibrium points. This paper proposes a novel family of chameleon chaotic systems, which can exhibit three types of chaotic attractors: self-excited attractors with a nonhyperbolic equilibrium, hidden attractors with a stable equilibrium, and hidden attractors with no equilibrium points. Bifurcation analysis uncovers the mechanisms by which self-excited and hidden chaotic attractors arise in this family of chameleon systems. It is demonstrated that various forms of chaos emerge through period-doubling routes associated with changes in the coefficient of a linear term. An electronic circuit is designed and simulated in Multisim to realize a hidden chaotic system with no equilibrium points. It is demonstrated that the electronic circuit simulation is consistent with the theoretical model. This research has the potential to enhance our comprehension of chaotic attractors, especially the hidden chaotic attractors.</p></abstract>

A novel type of chaotic attractor with a multiunit structure: from multiscroll attractors to multi-bond orbital attractors

A novel type of chaotic attractor with a multiunit structure: from multiscroll attractors to multi-bond orbital attractors

Multistability in a 3D Autonomous System with Different Types of Chaotic Attractors

This paper investigates multistability in a 3D autonomous system with different types of chaotic attractors, which are not in the sense of Shil’nikov criteria. First, under some conditions, the system has infinitely many isolated equilibria. Moreover, all equilibria are nonhyperbolic and give the first Lyapunov coefficient. Furthermore, when all equilibria are weak saddle-foci, the system also has infinitely many chaotic attractors. Besides, the Lyapunov exponents spectrum and bifurcation diagram are given. Second, under another condition, all the equilibria constitute a curve and there exist infinitely many singular degenerated heteroclinic orbits. At the same time, the system can show infinitely many chaotic attractors.

Multi-frequency sinusoidal chaotic neural network and its complex dynamics

A large number of animal experiments show that there is irregular chaos in the biological nervous systems. An artificial chaotic neural network is a highly nonlinear dynamic system, which can realize a series of complex dynamic behaviors, optimize global search and neural computation, and generate pseudo-random sequences for information encryption. According to the superposition theory of sinusoidal signals with different frequencies of brain waves, a non-monotone activation function based on the multifrequency-frequency conversion sinusoidal function and a piecewise function is proposed to make a neural network more consistent with the biological characteristics. The analysis shows that by adjusting the parameters, the activation function can exhibit the EEG signals in its different states, which can simulate the rich and varying brain activities when the brain waves of different frequencies and types work at the same time. According to the activation function we design a new chaotic cellular neural network. The complexity of the chaotic neural network is analyzed by the structural complexity based SE algorithm and C0 algorithm. By means of Lyapunov exponential spectrum, bifurcation diagram and basin of attraction, the effects of the activation function’s parameters on its dynamic characteristics are analyzed in detail, and it is found that a series of complex phenomena appears in the chaotic neural network, such as many different types of chaotic attractors, coexistent chaotic attractors and coexistence limit cycles, which improves the performance of the chaotic neural network, and proves that the multi-frequency sinusoidal chaotic neural network has rich dynamic characteristics, so it has a good prospect in information processing, information encryption and other aspects.

A physically extended Lorenz system.

The Lorenz system is a simplified model of Rayleigh-Bénard convection, a thermally driven fluid convection between two parallel plates. Two additional physical ingredients are considered in the governing equations, namely, rotation of the model frame and the presence of a density-affecting scalar in the fluid, in order to derive a six-dimensional nonlinear ordinary differential equation system. Since the new system is an extension of the original three-dimensional Lorenz system, the behavior of the new system is compared with that of the old system. Clear shifts of notable bifurcation points in the thermal Rayleigh parameter space are seen in association with the extension of the Lorenz system, and the range of thermal Rayleigh parameters within which chaotic, periodic, and intermittent solutions appear gets elongated under a greater influence of the newly introduced parameters. When considered separately, the effects of scalar and rotation manifest differently in the numerical solutions; while an increase in the rotational parameter sharply neutralizes chaos and instability, an increase in a scalar-related parameter leads to the rise of a new type of chaotic attractor. The new six-dimensional system is found to self-synchronize, and surprisingly, the transfer of solutions to only one of the variables is needed for self-synchronization to occur.

A Novel Type of Chaotic Attractor for Quadratic Systems Without Equilibriums

In this paper, a new chaotic dynamic system without equilibriums is presented. A conducted research of the qualitative properties of the discovered system reveals a noncompliance between the bifurcation behavior of the system and the Feigenbaum–Sharkovskii–Magnitsky theory. Additional research of known systems confirms the discrepancy.

On the Synchronization of Coupled Forced Negative Conductance Circuits: A Numerical Study

In this paper, a numerical study on the complete synchronization phenomenon exhibited by coupled forced negative conductance circuits is presented. The nonlinear system exhibiting two types of chaotic attractors is studied for complete synchronization of the identical chaotic attractors through phase portraits under one type of coupling. The stability of the synchronized states is observed for different coupling schemes of the circuit variables through {\emph{Master Stability Function}}. The Conditional lyapunov exponents explaining the dynamical behaviour of the driven system is presented.

Nonlinear dynamics of a generalized higher-order nonlinear Schrödinger equation with a periodic external perturbation

The nonlinear dynamics of a generalized higher-order nonlinear Schrodinger (HNLS) equation with a periodic external perturbation is investigated numerically. Via the phase plane analysis, we find that both the homoclinic orbits and heteroclinic orbits can exist for the unperturbed HNLS equation under certain conditions, which respectively corresponds to the bell-shaped and kink-shaped soliton solutions. Moreover, under the effect of the periodic external perturbation, the quasi-periodic bifurcations arise and can evolve into the chaos. The dynamical responses of the perturbed system varying with the perturbation strength and two types of chaotic attractors are discussed to show the existence of the chaotic motions. Via the feedback control methods, such chaotic motions are found to be controlled effectively and finally evolve into the stable quasi-periodic orbits. All the results are helpful to understand the dynamical properties of the nonlinear system.

Is a Unifying Chaotic Dynamical System Possible ?

Strange attractors are classified into three principal classes: hyperbolic, Lorenz-type, and quasi-attractors. This paper discusses the possibility of finding a rigorous mathematical model describing the three types of chaotic attractors and the transitions among them.

Interactions between two neural populations: A mechanism of chaos and oscillation in neural mass model

Neural mass model developed by Lopes da Silva et al. is able to describe limit cycle behavior in Electroencephalography (EEG) of alpha rhythm and exhibit complex dynamics between cortical areas. In this work, we extend Grimbert and Faugeras's work to study the dynamical behavior caused by interaction of cortical areas. The model is developed with the coupling of two neural populations. We show that various attractors, including equilibrium points, periodic solutions and chaotic strange attractors, could coexist in different ways with different value of the connectivity parameters. The main findings are that: (1) The stable equilibrium points only appear with a small value of the parameter. (2) While the alpha activities always exist for both two populations with proper initial conditions. Interestingly, the coexistence of the multiple alpha-to-epileptic activities implies the multiple coupling ways for these activities in phase. Two neuronal populations with epileptic activities could interact with multiple rhythms depending on their connectivity. (3) For particular interest, chaotic behaviors are identified in four regions divided by the connectivity parameter with the positive maximal Lyapunov exponent. The four types of chaotic attractors have their own structures, but all of them are related to the epileptic activities.

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.

Generate -Scroll Attractor via Composite Switching Controls

We propose a method for designing chaos generators. We introduce a switched system with three-dimensional space functions for generating a new type of chaotic attractor, and then we introduce saturated function series for generating -scroll chaotic attractor. Moreover, we present some examples with numerical simulations that illustrate the efficiency of our method. The statistic behavior is also discussed, which reveals the regularities in the complex dynamics.

The chaotic dynamics of vibrational mechanisms with energy sources of limited power

The dynamics of a vibrational mechanism with an energy source of limited power is considered. A system of two degrees of freedom is reduced to a system of the Lorenz type by the method of averaging. The existence of one of the types of chaotic attractors in a dynamical system which is a vibrational mechanism, that is, a Lorenz attractor, is established by this. The existence of a Feigenbaum attractor and intermittence is also established. Chaotic limit sets determine the chaotic behaviour of the instantaneous frequency of rotation of an asynchronous motor. The qualitative patterns of the rotational characteristic are constructed for different values of the parameters of the system and a physical interpretation of the results is given.

Symmetry of attractors and the Perron-Frobenius operator

We study the stochastic aspects of dynamics for equivariant dynamical systems, in particular those aspects that relate to symmetric chaos. By now, symmetric chaos is a rich field with both numerical evidence and theoretical classification of admissible symmetry types of chaotic attractors and their bifurcations. In this paper, we present a novel framework based upon the Perron-Frobenius (P-F) formalism to study these questions. We define appropriate notions of symmetry for basic objects relevant to the study of stochastic aspects of equivariant dynamics. We use these definitions to show that the resulting P-F operator commutes with the action of the symmetry group. We apply group representation theory to analyse both the algebraic and the stochastic implications of this commutativity. We use the P-F formalism to re-derive the necessary conditions for admissible symmetry types of attractors and for admissible symmetry types for the bifurcation of attractors in the presence of reflections. In particular, we show that these conditions are an easy consequence of the commutativity property of the P-F operator.

The multi‐mode chaotic behaviors: N+N and 2D N‐scroll chaotic attractors

PurposeThe purpose of this paper is to present a simulation and realization of the different types of chaotic attractors using the generalized Chua's circuit equations.Design/methodology/approachThis paper presents n+n‐scroll and 2D n‐scroll chaotic attractors by introducing multiple breakpoints to the nonlinearity. Two piece‐wise linear elements controlled with x‐ and y‐state space variables are used. Chaotic scrolls are originated through both x‐ and y‐axes. The current feedback operational amplifiers are preferred in the experimental circuits because of their wide bandwidth.FindingsIt is possible to increase the number of the scrolls in two directions by varying the number of breakpoints in the piece‐wise linear characteristics or changing the location of equilibrium points of the system on 2D‐plane. Theoretically developed behaviors are also experimentally tested.Originality/valueThe excellent adaptation is observed between theoretical and experimental results. This paper also provides useful information about how to build the multiple breakpoints nonlinear elements using the simulation results.

Chaotic Dynamics of Delayed Feedback Klystron Oscillator and its Control by External Signal

Complex nonlinear dynamics and applications of a microwave klystron oscillator with delayed feedback are studied in this paper on the basis of a system of delayed differential equations. Results of numerical simulation of self-modulation and chaotic regimes are presented, unveiling a complex bifurcation pattern in a wide range of parameters. Various types of chaotic attractors and transition to chaos scenarios are described. Experimental results for the five-cavity klystron oscillator are presented as well, showing good qualitative agreement with numerical simulations. Special attention is paid to application of the oscillator in direct chaotic communications. Information transmission using chaos shift keying (CSK) modulation and coherent reception is demonstrated via numerical simulation. The driving by an external signal is used to control the chaotic states of the transmitter oscillator for the purpose of generation of CSK basis functions.

PIECEWISE-LINEAR CHAOTIC SYSTEMS WITH A SINGLE EQUILIBRIUM POINT

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented.

A New Type of Chaotic Attractor

Abstract An experimentally discovered inverted spiral-type chaotic attractor is reproduced by a model equation. Does there exist a simple equation for the attractor re-cently found both in an electronic and a hydrodynamical system [1]? After a first, unpublished attempt by Sven Sahle to mirror a classical spiral-type attractor using a tube put into the middle, which yielded "messy" equa-tions, a fairly simple ordinary differential equation (ODE) was found and will be presented in the following. An experimental result is reproduced in Figure 1. It Deviation Fig. 1. Experimental attractor obtained with a hydrodynami-cal system, cf. [1]