Intermittent Chaos Research Articles

Four-scroll attractor on the dynamics of a novel Hopfield neural network based on bi-neurons without bias current

The dynamics of a neural network under several factors (bias current and electromagnetic induction effect) are recently used to simulate activities of the brain under different excitation. In this paper, we introduce a novel Hopfield neural network (HNN) based on two neurons with a memristive synaptic weight connected between neuron one and two based of flux controlled memristor recently proposed by Hua M. et al., in 2022. Using analysis tools, we proved that this model can develop rich dynamical characteristics such as various number of equilibrium points when the parameters are varied, four-scroll attractors, transient chaos, multistability of more than three different attractors and intermittency chaos phenomenon are reported. Moreover, when increasing a synaptic weight, the model shows bursting oscillations phenomenon. To obtain the normal state of the brain, the control of multistability to a strange monostable state is carry out. Finally, microcontroller implementation of the model is considered to verify the numerical analysis.

Dynamic Effects of a MCNN-CS Under Electromagnetic Induction and Its Application

Cellular neural network (CNN) is one of the best artificial neural network models due to its hardware implementability and many applications for image processing. The memristor-based cellular neural network (MCNN) combines the distinct benefits of memristors and neural networks, and it excels in mimicking chaotic systems under the effect of electromagnetic induction. In this work, a novel CNN chaotic system based on a flux-controlled memristor (MCNN-CS) with electromagnetic induction effects is constructed. Dynamical behaviors are examined by controlling the magnetic flux leakage strength and the intensity of the electromagnetic induction related to the memristor. Based on stability analyses and numerical simulation, forward and reverse period-doubling bifurcations, various coexisting attractors, multiperiods, chaos, several periodic windows, and chaotic crisis are observed. Besides, a number of complicated phenomena, including transient chaos, intermittent chaos, sustained chaos and bistability are also observed, demonstrating that this chaotic system has a wide range of dynamic properties. Finally, we generate pseudo-random number sequences based on this new system and apply them to image encryption. The obtained results show that this chaotic system offers superior randomness and is hence appropriate for image encryption applications.

IoT botnet detection with feature reconstruction and interval optimization

The existing botnet detection methods have the problems of uneven sampling, poor feature selection, and weak generalization ability, resulting in low detection and classification results and poor adaptability to the internet of things (IoT) environment with limited computing and storage resources. This paper proposes an IoT botnet detection method using feature reconstruction and interval optimization to solve the above problems. Through the designed address triple and time window-based IP aggregation and feature reconstruction method (ATTW-IP-FR), the network traffic samples obtained from the IoT gateway are integrated, and the flow features are reconstructed to attain the reconstructed sample set. The proposed self-corrected hybrid weighted sampling algorithm balances the normal and botnet flow samples in the reconstructed sample set to get the resampling sample set. The introduced multiattribute decision-making and adjacency relation chain-based sequential forward selection algorithm is applied to eliminate the redundant features in the resampling sample set, and the optimal feature subset is obtained. The resampling sample set filtered by the optimal feature subset is detected and classified through the designed two-stage hybrid heterogeneous model optimized by the intermittent chaos and bald eagle search algorithm-based interval optimization algorithm. The experimental results show that the proposed method effectively detects the botnet in two real IoT scenarios. The detection accuracy is 99.17 % $ \% $ , the Matthews correlation coefficient is 98.35 % $ \% $ , the false positive rate is 0.25 % $ \% $ , and the false negative rate is 1.27 % $ \% $ , which are better than the existing methods. This method can effectively reduce sampling and feature selection time and space overhead and better adapt to the resource-constrained IoT environment.

A Novel four - Wing chaotic system with multiple attractors based on hyperbolic sine: Application to image encryption*

A novel symmetrical four-wing fourth-order chaotic system is constructed. The basic dynamic characteristics of the system were analyzed by phase diagram, bifurcation diagram, Lyapunov index spectrum, Poincare cross section diagram and 0–1 test. In addition, the chaotic attractors under different parameters in the system are analyzed. In the dynamic analysis of the new system, it is found that the new system has some characteristics, such as multi-stability, offset boosting, multi-state transition phenomenon, transient chaos and intermittent chaos, and coexistence of multiple attractors. These features have the value of in-depth analysis compared to previous systems and can make it promising for more applications. Moreover, after calculating the complexity of the system by C0 algorithm at different initial values, it is found that the complexity of the system has been stable. Due to the existence of many characteristics, the new chaotic system has attracted great attention. At the same time, the circuit design of the system is realized by using Multisim simulation software and FPGA digital hardware circuit. Finally, a novel and efficient image encryption algorithm is designed by combining multi-direction pixel scrambling and DNA dynamic encryption. The NIST test, key space, encryption histogram, adjacent pixel correlation, robustness and information entropy are analyzed by encrypting images using chaotic sequences of the new system. In a word, its plentiful characteristics and intricate phenomena have significant reference value in the field of chaotic image encryption.

A practical underwater information sensing system based on intermittent chaos under the background of L\xe9vy noise

Nowadays, the increasingly complex and changeable marine environment makes the signals received by the underwater sensing equipment not only contain the weak signals radiated by underwater targets but also accompanied by marine solid background noise, which leads to the degradation and distortion of underwater acoustic signals and the decline of underwater communication quality. Under the severe influence of ocean noise, the underwater acoustic sensing and acquisition system will have the problems of high SNR ratio threshold, minimal sensing bandwidth, and unable to sense the signal with unknown frequency effectively. The Lévy noise model has been selected to describe the marine noise environment and explain its scientificity in this paper. A parameter estimation method for Lévy noise is proposed. Under the condition of characteristic index alpha =1.5 and noise intensity D=0.1 of the Lévy noise model, the estimated mean values of parameters are 1.5026 and 1.1664. The estimated variances are 0.0034 and 0.0046, which proves the effectiveness and applicability of the estimation method. Then, an improved dual-coupled Duffing oscillator sensing system is proposed to sense the weak signals with unknown frequency under Lévy noise. Under the background of Lévy with characteristic index alpha =1.5, deflection parameter beta =0 and noise intensity D=0.1, the sensing error rate of our system with unknown frequency is 0.054%, the lowest sensing signal amplitude is A=0.010, the lowest sensing SNR ratio is − 23.9254 dB, and the frequency of multi-frequency weak signals to be measured can be obtained. The estimation error of frequency sensing is 0.33%.

A New Generalized Hamiltonian Chaotic System with Transient Quasi-Periodic Flows and Intermittent Chaos

The paper reports a new generalized Hamiltonian chaotic system with transient quasi-periodic flows and intermittent chaos by investigating the Kolmogorov-type transformation for a 3D chaotic system. The new system meets Hamiltonian energy conservation when [Formula: see text], and it is always conservative in volume because of a zero divergence. In addition, its Hamiltonian energy is found to be only related to the initial points. Subsequently, some dynamics analyses are presented to investigate its conservative characteristics and coexisting flows including periodic flows, quasi-periodic flows, and chaotic flows. Besides, both transient quasi-periodic flows and intermittent chaos are also found to occur, which further shows complex dynamics of the new generalized Hamiltonian chaotic system. Finally, an FPGA circuit is designed to implement the new generalized Hamiltonian chaotic system, and the results from the circuit experiments are consistent with those from numerical analyses. Complex dynamic characteristics existing in the new generalized Hamiltonian chaotic system seem more appropriate to be used in image encryption and secure communication. And the FPGA circuit not only shows the new generalized Hamiltonian chaotic system from a physical viewpoint, but also provides a new pseudo-random signal generator for engineering applications.

Memory Storage Systems Utilizing Chaotic Attractor-Merging Bifurcation

In nonlinear dynamical systems with barriers/thresholds, the signal response against a weak external input signal is enhanced by an appropriate additive noise (stochastic resonance). In recent years, progress in the application of stochastic resonance shows that the existence of additive noise heightens the memory storage functions in memory elements using bistable oscillations even with extremely low power consumption. By not restricting the additive noise, the deterministic chaos (an internal fluctuation) induces a similar phenomenon known as chaotic resonance. Chaotic resonance appears in nonlinear dynamical systems and is accompanied by chaos–chaos intermittency, where the chaotic orbit intermittently transitions among separated attractor regions through attractor-merging bifurcation. Previously, a higher chaotic resonance sensitivity than that of stochastic resonance was reported in various types of systems. In this study, we hypothesize that chaotic-resonance-based memory devices can store information with lower power consumption than that of stochastic-resonance-based devices. To prove this hypothesis, we induced attractor-merging bifurcation in a cubic map system, which is the simplest model for the emergence of chaotic resonance. Thereafter, we adjusted the internal system parameter under a noise-free system as the chaotic resonance and applied stochastic noise similar to the condition for inducing stochastic resonance. The results of this study reveal that, even with weaker memory storage input signals, the former exhibits a higher memory storage capability than the latter. The approach using chaotic resonance could facilitate the development of memory devices that were hitherto restricted to the application of stochastic resonance.

Analysis, circuit implementation and synchronization control of a hyperchaotic system

A four-dimensional autonomous hyperchaotic system is constructed, and the basic characteristics of the system are studied by the power spectrum, Poincaré maps, 0–1 test and Lyapunov exponents. The system has rich dynamical behaviors, such as bursting oscillations, offset boosting, transient chaos, intermittent chaos and coexistence of attractors. In addition, by studying the coexisting phenomenon and spectral entropy (SE) complexity of different initial values, an initial value that is more suitable for chaotic secure communication is selected. The circuit simulation of the system using Multisim and then the actual hardware implementation of the system by Field Programmable Gate Array (FPGA), these prove the practical existence of the system. Finally, combining the methods of backstepping control, multi-switching synchronization and synchronization of different systems, a set of controllers are proposed which can realize the backstepping multi-switching synchronization of this system with a memristive chaotic system.

Complex dynamical behaviors in a Bertrand game with service factor and differentiated products

In this paper, taking the factor of product service provided by the manufacturers into consideration, a static Bertrand duopoly game with service factor is studied first, in which these two oligarchs produce differentiated products. A dynamic Bertrand duopoly game with bounded rationality is established by using the gradient mechanism. Keeping the adjustment speeds in a relatively small range may help the long-term stable operation of the market. It is found that there is another 2-cycle, different from the one flip bifurcated from the fixed point, which may appear through a saddle-node bifurcation. The unstable set of the saddle cycle, connecting the saddle to the node, gives a closed invariant curve. In addition, the emergence of intermittent chaos implies that the established economic system has the capability of self-regulating, where PM-I intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated.

Dynamics analysis, hardware implementation and engineering applications of novel multi-style attractors in a neural network under electromagnetic radiation

This article studies the interesting dynamics of a small neural network with three neurons under electromagnetic radiation. The electromagnetic radiation strength can change the number of the equilibrium points in the neural network, which leads to the diversification of the attractor’s trajectory. Thus, the novel multi-style attractors like one-to-two-spiral attractors and one-to-four-scroll attractors can be generated from the neural network stimulated by electromagnetic radiation. Besides, the plentiful dynamical behaviors are observed in the neural network, such as transitional coexisting attractors, hypogenetic attractors, periodic patterns, firing patterns, transient chaos and intermittent chaos. In terms of hardware implementation, we utilize FPGA to digitally implement the constructed neural network model. The experimental verification results are highly consistent with the numerical simulation results. In the aspects of engineering application, we apply it to pseudo-random number generator and image encryption respectively, test the random performance of different chaotic attractors by NIST test suite, describe the image encryption scheme based on neural network and estimate its security performances. The ultimate outcomes demonstrate that the neural network model with chaotic behavior has superior randomness and wonderful security, which is very suitable for engineering applications based on chaos.

Synchronization transitions in a hyperchaotic SQUID trimer.

The phenomena of intermittent and complete synchronization between two out of three identical, magnetically coupled Superconducting QUantum Interference Devices (SQUIDs) are investigated numerically. SQUIDs are highly nonlinear superconducting oscillators/devices that exhibit strong resonant and tunable response to applied magnetic field(s). Single SQUIDs and SQUID arrays are technologically important solid-state devices, and they also serve as a testbed for exploring numerous complex dynamical phenomena. In SQUID oligomers, the dynamic complexity increases considerably with the number of SQUIDs. The SQUID trimer, considered here in a linear geometrical configuration using a realistic model with experimentally accessible control parameters, exhibits chaotic and hyperchaotic behavior in wide parameter regions. Complete chaos synchronization as well as intermittent chaos synchronization between two SQUIDs of the trimer is identified and characterized using the complete Lyapunov spectrum of the system and appropriate measures. The passage from complete to intermittent synchronization seems to be related to chaos-hyperchaos transitions as has been conjectured in the early days of chaos synchronization.

An Approach for Stabilizing Abnormal Neural Activity in ADHD Using Chaotic Resonance.

Reduced integrity of neural pathways from frontal to sensory cortices has been suggested as a potential neurobiological basis of attention-deficit hyperactivity disorder. Neurofeedback has been widely applied to enhance reduced neural pathways in attention-deficit hyperactivity disorder by repeated training on a daily temporal scale. Clinical and model-based studies have demonstrated that fluctuations in neural activity underpin sustained attention deficits in attention-deficit hyperactivity disorder. These aberrant neural fluctuations may be caused by the chaos–chaos intermittency state in frontal-sensory neural systems. Therefore, shifting the neural state from an aberrant chaos–chaos intermittency state to a normal stable state with an optimal external sensory stimulus, termed chaotic resonance, may be applied in neurofeedback for attention-deficit hyperactivity disorder. In this study, we applied a neurofeedback method based on chaotic resonance induced by “reduced region of orbit” feedback signals in the Baghdadi model for attention-deficit hyperactivity disorder. We evaluated the stabilizing effect of reduced region of orbit feedback and its robustness against noise from errors in estimation of neural activity. The effect of chaotic resonance successfully shifted the abnormal chaos-chaos intermittency of neural activity to the intended stable activity. Additionally, evaluation of the influence of noise due to measurement errors revealed that the efficiency of chaotic resonance induced by reduced region of orbit feedback signals was maintained over a range of certain noise strengths. In conclusion, applying chaotic resonance induced by reduced region of orbit feedback signals to neurofeedback methods may provide a promising treatment option for attention-deficit hyperactivity disorder.

Designing Hyperchaos and Intermittency in Semiconductor Superlattices.

Weakly coupled semiconductor superlattices under dc voltage bias are excitable systems with many degrees of freedom that may exhibit spontaneous chaos at room temperature and act as fast physical random number generator devices. Superlattices with identical periods exhibit current self-oscillations dueto the dynamics of charge dipole waves but chaotic oscillations exist on narrow voltage intervals. They disappear easily due to variation in structural growth parameters. Based on numerical simulations, we predict that inserting two identical sufficiently separated wider wells increases superlattice excitability by allowing wave nucleation at the modified wells and more complex dynamics. This system exhibits hyperchaos and varieties of intermittent chaos in extended dc voltage ranges. Unlike in ideal superlattices, our chaotic attractors are robust and resilient against noises and against controlled random disorder due to growth fluctuations.

A new Duffing detection method for underwater weak target signal

On the basis of the double coupled Duffing oscillator, a differential double coupled Duffing oscillator method, called the proposed method 1, is proposed. The large scale state is judged by the differential timing sequence diagram, and its detection result is more intuitive. To overcome the frequency limitation of differential double coupled Duffing oscillator in detecting unknown frequency periodic signal, according to the proposed method 1, based on the intermittent chaos phenomenon, the differential double coupled Duffing oscillator variable step size detection method, called the proposed method 2, is proposed. This method can be used to detect Gaussian noise background periodic signal and K-distributed noise background periodic signal. Compared with the double Duffing method and the double coupled Duffing method, its average signal-to-noise ratio (SNR) threshold is reduced by 4.94 dB and 1.68 dB respectively, and the detection result is more intuitive to judge through the differential timing sequence diagram. The accuracy and universality of the proposed method 2 are proved by the detection of three measured underwater weak target signals. Experiments show that it has some characteristics such as intuitive detection result, high sensitivity, low SNR threshold, few oscillators, low system complexity and so on.

Analysis of the Dynamic Stiffness, Hysteresis Resonances and Complex Responses for Nonlinear Spring Systems in Power-Form Order

Power-form nonlinear contact force models are widely adopted in relatively moving parts of macro (e.g., rolling bearings considering Hertzian contact restoring force between rolling elements and bearing raceways) or micro (e.g., the micro cantilever probe system of atomic force microscopy) scale mechanical systems, and contact resonance could cause serious problems of wear, contact fatigue, vibration, and noise, which has attracted widespread attention. In the present paper, the softening/hardening stiffness characteristics of continuous and one-sided contact power-form nonlinear spring models are addressed, respectively, by the analysis of the monotone features of resonant frequency-response skeleton lines. Herein, the period-n solution branch and its stability characteristics are obtained by the harmonic balance and alternating frequency/time domain (HB–AFT) method and Floquet theory. Compared with previous studies, this paper will furtherly clarify the influences of externally normal load, the power form exponent term, and excitation amplitude on the softening/hardening stiffness characteristics of general power-form spring systems. In addition, for a power-form system with a one-sided contact, the phenomena of primary and super/sub-harmonic hysteretic resonances inducing period-doubling, folding bifurcation, the coexistence of multiple solutions are demonstrated. Besides, it gives the evolution mechanism of two types of intermittency chaos in a one-sided contact system. The overall results may have certain basic theoretical significance and engineering values for the control of vibration and noise in contact mechanical systems.

A new multi-scroll Chua’s circuit with composite hyperbolic tangent-cubic nonlinearity: Complex dynamics, Hardware implementation and Image encryption application

In this paper, a new method for generating multi-scroll chaotic attractors via constructing a compound hyperbolic tangent-cubic nonlinear function in canonical Chua’s circuit is presented. The basic dynamic characteristics of the system are analyzed, including equilibrium points, bifurcation diagrams, Lyapunov exponents, phase portraits, time-domain diagrams and attractive basins. What is surprising is that the proposed multi-scroll Chua’s circuit also exhibits rich dynamic behaviors like coexisting multiple attractors, transient period, intermittent chaos and offset boosting. In addition, we put forward the application of the system in chaotic image encryption, and analyzed some security performance evaluation indexes to show that the new Chua’s chaotic cipher system has high security and reliable encryption performance. Finally, the hardware design and experiments of the chaotic digital circuits and image encryption are carried out. Both numerical simulation and FPGA experimental results verify the feasibility and usability of the proposed new multi-scroll Chua’s system.

Coexistence of infinite attractors in a fractional-order chaotic system with two nonlinear functions and its DSP implementation

In this paper, a new five-dimensional fractional-order chaotic system based on two nonlinear functions is constructed. The rich dynamical behaviors of the system are analyzed by phase diagram, bifurcation diagram and Lyapunov exponents spectrum. In addition, the complexity of the fractional-order system is analyzed through Spectral Entropy (SE) and Permutation Entropy (PE) algorithms. Meanwhile the phenomenon of coexisting infinite attractors is analyzed. Of particular concern is that the phenomenon of multi-state transition and intermittent oscillation chaos is found in this new chaotic system. Furthermore, the system is implemented on the DSP platform. To the best of the knowledge, these rich dynamical characteristics and complicated phenomena are of great reference value in chaotic image encryption and other fields.

Bifurcation and chaos of axially moving nanobeams considering two scale effects based on non-local strain gradient theory

The nonlinear vibration of axially moving nanobeams at the microscale exhibits remarkable scale effects. A model of an axially moving nanobeam is established based on non-local strain gradient theory and considering two scale effects. The discrete equation of a non-autonomous planar system is obtained using the Galerkin method. The response characteristics of the system are determined using phase diagrams and Poincaré sections, and the effects of the scale parameters on the form of the motion are analyzed. The results show that as the non-local parameter and the material characteristic length parameter vary, the system undergoes multiple forms of motion, including periodic, period-doubling and chaotic motions. Two routes to chaos — period-doubling bifurcation and intermittent chaos — are identified in the variation ranges of the two scale parameters.

Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of large deployable space antenna subjected to thermal load and radial pre-stretched membranes with 1:3 internal resonance

Abstract Andronov-Hopf bifurcations, Pomeau-Manneville intermittent chaos and nonlinear vibrations of the large deployable space antenna (LDSA) subjected to the thermal load with the case of 1:3 internal resonance are studied for the first time. The frequencies and vibration modes of the LDSA are analyzed by using the finite element method. It is found that there may exist an approximate threefold relationship between the fourth-order and first-order frequencies of the LDSA. The LDSA is simplified to a composite laminated equivalent cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at two ends subjected to the thermal load. Considering the case of 1:3 internal resonance, four-dimensional nonlinear averaged equations are obtained by using the method of multiple scales. The amplitude-frequency response curves and amplitude-force response curves are obtained for the LDSA subjected to the thermal load through the prediction-correction continuation algorithm. The fold bifurcation and Andronov-Hopf bifurcation points are located on these resonant response curves. The equivalent model of the LDSA has the hardening spring characteristics. The thermal load has very significant influences on the stability of the LDSA. The nonlinear dynamics of the equivalent model for the LDSA are investigated by using the fourth-order Runge-Kutta algorithm. These nonlinear dynamic behaviors are described by the bifurcation diagrams, waveforms, phase plots, and Poincare maps. The periodic doubling bifurcation and Pomeau-Manneville type intermittent chaos appear in this nonlinear dynamical system. According to the topological evolution of the phase trajectories, the phase-locking phenomenon and a special evolution path of the chaotic attractors are found.

Spatiotemporal spread of perturbations in a driven dissipative Duffing chain: An out-of-time-ordered correlator approach.

Out-of-time-ordered correlators (OTOCs) have been extensively used as a major tool for exploring quantum chaos, and recently there has been a classical analog. Studies have been limited to closed systems. In this work, we probe an open classical many-body system, more specifically, a spatially extended driven dissipative chain of coupled Duffing oscillators using the classical OTOC to investigate the spread and growth (decay) of an initially localized perturbation in the chain. Correspondingly, we find three distinct types of dynamical behavior: the sustained chaos, transient chaos, and nonchaotic region, as clearly exhibited by different geometrical shapes in the OTOC heat map. To quantify such differences, we look at instantaneous speed (IS), finite-time Lyapunov exponents (FTLEs), and velocity-dependent Lyapunov exponents (VDLEs) extracted from OTOCs. Introduction of these quantities turns out to be instrumental in diagnosing and demarcating different regimes of dynamical behavior. To gain control over open nonlinear systems, it is important to look at the variation of these quantities with respect to parameters. As we tune drive, dissipation, and coupling, FTLEs and IS exhibit transition between sustained chaos and nonchaotic regimes with intermediate transient chaos regimes and highly intermittent sustained chaos points. In the limit of zero nonlinearity, we present exact analytical results for the driven dissipative harmonic system, and we find that our analytical results can very well describe the nonchaotic regime as well as the late-time behavior in the transient regime of the Duffing chain. We believe that this analysis is an important step forward towards understanding nonlinear dynamics, chaos, and spatiotemporal spread of perturbations in many-particle open systems.