Intermittent Chaos Research Articles

Stability, bifurcation, and chaos in a class of scalar quartic polynomial delay systems.

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos.

The design, circuit realization and applications of chaotic system with offset-boosting and multistability

Based on the foundation of a three-dimensional Jerk chaotic system, a novel four-dimensional chaotic system based on a memristor with cosine magneto-memristor is proposed. The system’s dynamical behavior is analyzed using bifurcation diagrams, Lyapunov exponents, multiple bifurcation diagrams, two-parameter Lyapunov exponents, complexity, and attractor basins. It is found that this new chaotic system exhibits characteristics such as multistability, transient chaos, intermittent chaos, spike discharge, controllable multi-vortex, and offset-boosting. The chaotic system is designed and simulated in analog circuit and realized in a digital circuit using a Field-Programmable Gate Array (FPGA). The consistency between simulation results and implementation results demonstrates the existence and feasibility of the system. Finally, the significance and superiority of the control scheme are verified based on the adaptive anti-synchronization control strategy, and the synchronization strategy is applied to image encryption, and the results show that the encryption effect is excellent and the anti-jamming ability is strong.

A novel underwater weak signal detection method based on parameter optimized VMD and 3D chaotic system

Based on chaos theory, combined with improved variational mode decomposition, a novel method for underwater weak signal detection and frequency extraction is proposed. The aim of this research is to address the challenge of detecting underwater signals and extracting their frequencies under ultra-low Signal-to-Noise Ratio (SNR). A novel three-dimensional (3D) chaotic system with infinitely shrinking attractors is proposed, which incorporates a cosine function to achieve chaos control and enhancement. The nonlinear characteristics of the system, such as Lyapunov exponent and bifurcation, are analyzed in detail. Then, we design a frequency array and perform scale transformation on the chaotic system to construct a detection model that covering all frequencies. The model can transition from critical chaos to large period or intermittent chaos in the presence of external periodic signals by adjusting the parameters of the cosine function. Considering the relationship between the generation of intermittent chaos and the frequency of target signal, a frequency extraction algorithm based on chaos sparrow search algorithm optimization variational mode decomposition (CSSA-VMD) and Hilbert transform is designed. The algorithm can estimate the real frequency of underwater signals by performing noise reduction and envelope analysis on intermittent chaotic signals. The experimental results demonstrate that the proposed system can detect weak signals in complex marine backgrounds and extract their frequency features, with a detection SNR of -48 dB and a frequency error of less than 0.01%. Importantly, the circuit of the detection system has been implemented, verifying its feasibility for detecting and tracking underwater signals.

Dynamic analysis based on a memristive hyperchaotic system with stable unfixed point and its synchronization application

A novel two-memristor hyperchaotic system is obtained by introducing a cubic magnetic-controlled memristor and a hyperbolic sine function memristor. The dynamics of the new system are analyzed by various techniques such as Lyapunov exponents, complexity, 0–1 test, bifurcation diagram and phase diagram. The results demonstrate that the new system exhibits complex dynamic behaviors, including transient chaos, transient transition, intermittent chaos, and offset-boosting. Notably, a rare phenomenon with stable unfixed point has been discovered in this newly proposed system. The largest Lyapunov exponent of the stable unfixed point fluctuates around 0 and remains predominantly less than or equal to 0. Despite this, the new system still partially exhibits chaotic characteristics, indicating that the stable unfixed point can be regarded as a local chaotic attractor. Furthermore, there are four types of coexisting attractors with period-period, chaos-chaos, chaos-stable unfixed point and stable unfixed point-stable unfixed point in the new system. The circuit design is implemented to validate the accuracy of the memristive chaotic system, and the consistency between numerical calculations and simulation results is confirmed. Finally, the coupling synchronization and tracking synchronization methods are designed, which hold practical applications in the field of secure communication, control systems and signal processing.

A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption

By coupling memristors to nonlinear circuits, more complex dynamical behaviors can be induced. However, to date, there has been insufficient attention given to high-dimensional chaotic systems based on memristors. In this paper, a magnetic-controlled memristor is combined with a three-dimensional chaotic system, resulting in a five-dimensional memristive chaotic system. Through dynamic analysis and numerical simulations, the chaotic nature of the system is elucidated based on fundamental system behaviors, including Lyapunov dimension, dissipativity, stability of equilibrium points, 0–1 test, and Poincaré mapping. During the complex dynamical analysis of this system, unique dynamical behaviors are discovered, including intermittent chaos, transient chaos, extreme multistability, and offset-boosting. Moreover, the consistency between numerical calculations and the physical implementation of the actual system is verified through equivalent circuit design. Finally, this system is applied to image encryption, leading to the design of an efficient and secure hyper-chaotic image encryption algorithm, whose effectiveness is confirmed through several security tests.

Stabilization of Laminars in Chaos Intermittency

Chaos intermittency is composed of a laminar regime, which exhibits almost periodic motion, and a burst regime, which exhibits chaotic motion; it is known that in chaos intermittency, switching between these regimes occurs irregularly. In the laminar regime of chaos intermittency, the periodic solution before the saddle node bifurcation is closely related to its generation, and its behavior becomes periodic in a short time; the laminar is not, however, a periodic solution, and there are no unstable periodic solutions nearby. Most chaos control methods cannot be applied to the problem of stabilizing a laminar response to a periodic solution since they refer to information about unstable periodic orbits. In this paper, we demonstrate a control method that can be applied to the control target with laminar phase of a dynamical system exhibiting chaos intermittency. This method records the time series of a periodic solution prior to the saddle node bifurcation as a pseudo-periodic orbit and feeds it back to the control target. We report that when this control method is applied to a circuit model, laminar motion can be stabilized to a periodic solution via control inputs of very small magnitude, for which robust control can be obtained.

Observation of Intermittent Chaos Caused by Delayed Feedback in a Laboratory Plasma

Observation of Intermittent Chaos Caused by Delayed Feedback in a Laboratory Plasma

Routes to chaos and generation of cnoidal wave envelopes in exciton-polariton microcavities

We show the presence of intermittent chaos as a function of the variation of the chemical potential during the condensation of exciton-polaritons using the bifurcation diagram and the Lyapunov Exponent. We then show the possibility of controlling the chaos in the homogeneous solution by the variation of the chemical potential when the pump increases. The influence of the chemical potential is demonstrated in stationary state by studying stability using the Routh-Hurwitz criterion. We also prove the existence of cnoidal waves and through phase portraits, their stabilities in propagation are established using the notion of symmetry. For this, we use a dissipative Gross-Pitaevskii equation for the polariton field, coupled to a reservoir dynamics equation, commonly used to describe polariton condensates under non-resonant pumping.

Design of memristor hyperchaotic circuit with burst oscillation and infinite attractor coexistence and its application

A 4D hyperchaotic system with symmetry based on memristors is proposed and the chaotic properties of the circuit are verified by phase diagrams, Poincaré cross sections, bifurcation, Lyapunov exponents, and 0–1 tests. In addition, the system is found to have rich dynamical behaviors such as transient chaos, intermittent chaos, offset enhancement, burst oscillation, and infinite attractor coexistence. The complexity comparison analysis reveals that this hyperchaotic system has good symmetry as well as the initial values of the system that are more suitable for image encryption. Circuit simulation of the system using simulation software proved the usability of the system. A field programmable gate array (FPGA) is used to implement the actual digital circuit of the system. The final FPGA results agree with the numerical simulation and prove the practical value of the constructed memristor hyperchaotic system. Finally, chaotic circuits are applied to DNA image encryption. It is discovered to have a very large key space, strong key sensitivity, outstanding anti-cropping, noise immunity and high security performance. Thus, the proposed hyperchaotic system has good advantages for application in the field of image encryption.

A new five-dimensional non-Hamiltonian conservative hyperchaos system with multistability and transient properties

Recently, the majority of research in chaos field has focused on dissipative chaos systems, while conservative chaos systems, particularly conservative hyperchaos systems, have been largely overlooked. However, these systems possess more intricate dynamic properties and greater potential for practical applications than general chaos systems. This paper aims to address this gap by introducing two linear terms to the system equation of an existing five-dimensional Hamiltonian conservative hyperchaos system, based on the construction techniques of the existing conservative chaos system. As a result, a five-dimensional non-Hamiltonian conservative hyperchaos system without time-reversal symmetry is obtained, which exhibits complex dynamic properties. Specifically, the system displays multi-stable characteristics with consistent parameters but varying initial conditions. Furthermore, during the study of the system's transient characteristics, a unique phenomenon was observed, namely intermittent chaos and intermittent quasi-periodic characteristics. Finally, the system's sequences were evaluated using the NIST test, and the results indicated their good level of pseudo-randomness.

Multi-vortex hyperchaotic systems based on memristors and their application to image encryption

In this paper, a new memristor-based fourth-order hyperchaotic circuit that can generate multiple vortex volumes is constructed with the aid of a cosine memristor proposed by scholars. Thanks to the properties of cosine memristors, we find new hyperchaotic systems with full even symmetry based on parameters. The new system can produce multi-vortex properties with infinite equilibrium points and exhibits extreme multi-stability phenomena with the coexistence of an infinite number of attractors. After further study of phase trajectory diagrams, Lyapunov exponential spectrum, bifurcation diagrams, intermittent chaos and transient phenomena, we find that there are complex state transfer phenomena such as period 1 windows, chaos, multiperiodic bifurcations, multi-periodic windows and hyperchaotics, which are constantly alternating. In terms of circuit implementation, Multisim was used to build the circuit and to realise the characteristics of the memristor and the simulation of the hyperchaotic circuit. Finally, a DNA encryption algorithm is designed based on memristor hyperchaotic system, with emphasis on adjacent pixel correlation, encryption histogram, key sensitivity, anti-cropping and anti-noise performance. The results show that the new system has a very large key space, strong key sensitivity, outstanding anti-cropping, and anti-noise performance. The proposed hyperchaotic system thus has high security performance when applied to image encryption.

Design and DSP implementation of a hyperchaotic map with infinite coexisting attractors and intermittent chaos based on a novel locally active memcapacitor

This paper proposes a novel discrete locally active memcapacitor model and uses it to construct a novel memcapacitive hyperchaotic map. The fixed points and stability are analyzed, and the dynamical behaviors of the map with different parameters are studied in detail. In addition, special phenomena such as infinite coexisting attractors and intermittent chaos are discovered. Finally, the DSP implementation of the map is completed and the realizability of the theoretical analysis is demonstrated. The results of the theoretical analysis and digital circuit implementation show that the memcapacitive hyperchaotic map has rich dynamical behaviors and practical application potential.

The mechanism of periodic and chaotic bursting patterns in an externally excited memcapacitive system

In this paper, a multi-time-scale-coupled nonlinear dynamic system is constructed by introducing an external excitation into a charge-controlled memcapacitive system. We report the complex bursting oscillation mechanism of this system under different parameter conditions by taking advantage of the fast-slow analysis method. Firstly, we discuss several different types of periodic bursting behaviors consisting of “asymmetric and symmetric ‘delayed pitchfork/Hopf/delayed Homoclinic/Homoclinic/Hopf’ bursting type”, “asymmetric and symmetric ‘delayed Hopf/delayed Homoclinic/Homoclinic/Hopf’ mixed-mode bursting oscillation”, “symmetric ‘delayed Hopf/delayed Homoclinic/delayed pitchfork’ mixed-mode bursting oscillation” and “asymmetric ‘cycle/point’ mixed-mode bursting type via ‘delayed Hopf/delayed pitchfork’ hysteresis loop”. The numerical simulation results express that different delay-bifurcation intervals under the slow passage effect will induce completely various bursting types. Furthermore, we explore a novel route with complex structure from periodic bursting to chaotic bursting. By gradually increasing the value of control parameter, we can observe a variety of dynamic behaviors including complex structure such as symmetric breaking bifurcations, intermittent period-doubling bursting type, period-doubling bifurcations and inverse period-doubling bifurcations. With the further increase of the parameter, the system generates crisis effect, and we can obtain the chaotic attractors in the periodic window. We discover that different parameter conditions can produce chaotic attractors which can induce “‘point/chaos/period/chaos/point’ intermittent chaos type”, “‘point/chaos/point’ intermittent chaos type” and “‘cycle/chaos/cycle’ intermittent chaos type” until complete chaos. Chaotic bursting types are corroborated by the maximum Lyapunov exponents. Our research results enrich the route to bursting oscillation and deepen the understanding of the mechanism of complex bursting oscillation types.

Intermittency, fluctuations and maximal chaos in an emergent universal state of active turbulence

The phenomenon of active turbulence, a complex organization of matter driven at the scale of its constituent agents, is puzzling. Specifically, the lack of scale-separation in low-Reynolds-number active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. Here, using a generalized hydrodynamic model developed for bacterial turbulence, we provide compelling analytical and numerical evidence that, beyond a critical drive, active turbulence indeed attains universality akin to inertial turbulence. In this asymptotic state, the energy spectrum scales as k−3/2, reminiscent of some classes of inertial turbulence. The flow also exhibits spatio-temporal intermittency beyond the transition, as seen from non-Gaussian fluctuations in velocity differences. With these tell-tale fingerprints, active turbulence is placed closer in phenomenology to inertial turbulence than previously held. We show, however, that as a consequence of a finite range of scales, the degree of chaoticity and hence mixing efficiency saturates to a maximum in the asymptotic regime, unlike unbounded chaos in inertial turbulence. We conclude that active turbulence, depending on the level of drive, can switch between fundamentally distinct non-universal and universal states. Active fluids exhibit regimes with a complex spatio-temporal structure reminiscent of inertial turbulence. Now, inertial and active turbulence are theoretically shown to be closely related indeed.

Analysis and circuit implementation of a non-equilibrium fractional-order chaotic system with hidden multistability and special offset-boosting.

In order to obtain a system of higher complexity, a new fractional-order chaotic system is constructed based on the Sprott system. It is noteworthy that the system has no equilibrium point yet exhibits chaotic properties and has rich dynamical behavior. Its basic properties are analyzed by Lyapunov exponents, phase diagrams, and smaller alignment index tests. The change of its state is observed by changing parameters and order, during which the new system is found to have intermittent chaos phenomena. Surprisingly, the new proposed system has a special offset-boosting phenomenon, where only a boosting-controller makes the system undergo a multi-directional offset, and the shape of the generated hidden attractor changes. In addition, changing the initial value brings kinds of coexisting attractors in the system, which proves the existence of multistability. Because the new system is very sensitive to the initial value, the complexity of the new system is calculated based on the complexity algorithm, and the initial value with higher complexity is gained by contrast. Finally, the field programmable gate array is used to implement the actual circuit of the new system to verify its feasibility. This system provides an example for the study of fractional-order chaotic systems and a complex system for fractional-order chaotic applications.

Chaotic precession of antiferromagnetic domain walls

In contrast with extensive investigations on the translation of domain walls (DWs) in antiferromagnetic (AFM) materials, the precessional motion of an AFM DW is seldom concerned partly because of the lack of an effective driving method. In this Letter, we show that under an alternating spin-polarized current with polarization along the AFM anisotropy axis, an AFM DW can be excited to precess. Especially, chaotic precession occurs at a moderate interfacial Dzyaloshinskii-Moriya interaction (DMI), which contributes to a nonlinear term in the dynamic equation of DW precession. Crisis-induced intermittent chaos appears when the current density is higher than a critical value. This Letter not only paves a way to unravel the rich dynamics of a current-induced AFM texture but also provides guidelines for developing ultrafast spintronic devices to exploit the nonlinear chaotic magnetization dynamics with promising applications.

Numerical simulation of the bifurcation-remerging process and intermittency in an undriven direct current glow discharge.

As a complex nonlinear medium, gas discharge plasma can exhibit various nonlinear discharge behaviors. In this study, in order to investigate the chaos phenomenon in the subnormal glow region of an undriven direct current glow discharge, a two-dimensional plasma fluid model is established coupled with a circuit model as a boundary condition. Using the applied voltage as control parameter in the simulation, the complete period-doubling bifurcation and inverse period-doubling bifurcation processes in the oscillation region are found, and the influence of the applied voltage on the spatiotemporal distribution of plasma parameters during the bifurcation-remerging process is examined. In addition, the spatial distribution of the plasma parameters of the bifurcation-remerging process is also examined. Also, a series of periodic windows are present in the chaotic region, where the positions and relative order are generally consistent with the universal sequence. Additionally, this study showed that the intermittent chaos appears near the period-3 window, and the bursts appearing in the approximate periodic motion becomes more and more frequent as the control parameters move away from the saddle-node bifurcation point, which shows the typical type-I intermittent chaos characteristics.

Forecasting and diagnostics of critical states in platform-vibrator with shock.

A platform-vibrator with shock is a low-frequency machine used in the construction industry for compaction and molding of large concrete products. Its mathematical model is a two-degree-of-freedom two-body vibro-impact system with a soft impact. Some changes in its parameters can increase the machine performance and improve the product quality, but these same changes may lead to the emergence of critical states, such as coexisting regimes in hysteresis zone, chaotic motion, intermittency and crisis-induced intermittency, crises, and transient chaos. Some of them can be undesirable and dangerous. This article shows their diagnostics and recognition, the possibility of their prediction, as well as the criterion determining the set of parameter ranges where critical states can occur. Diagnostics is carried out both by traditional tools and by the less common ones, such as the construction of fractal structures and wavelet characteristics.

Hyperchaos, Intermittency, Noise and Disorder in Modified Semiconductor Superlattices.

Weakly coupled semiconductor superlattices under DC voltage bias are nonlinear systems with many degrees of freedom whose nonlinearity is due to sequential tunneling of electrons. They may exhibit spontaneous chaos at room temperature and act as fast physical random number generator devices. Here we present a general sequential transport model with different voltage drops at quantum wells and barriers that includes noise and fluctuations due to the superlattice epitaxial growth. Excitability and oscillations of the current in superlattices with identical periods are due to nucleation and motion of charge dipole waves that form at the emitter contact when the current drops below a critical value. Insertion of wider wells increases superlattice excitability by allowing wave nucleation at the modified wells and more complex dynamics. Then hyperchaos and different types of intermittent chaos are possible on extended DC voltage ranges. Intrinsic shot and thermal noises and external noises produce minor effects on chaotic attractors. However, random disorder due to growth fluctuations may suppress any regular or chaotic current oscillations. Numerical simulations show that more than 70% of samples remain chaotic when the standard deviation of their fluctuations due to epitaxial growth is below 0.024 nm (10% of a single monolayer) whereas for 0.015 nm disorder suppresses chaos.

Numerical study of type-I intermittency in an atmospheric-pressure dielectric barrier discharge

Atmospheric-pressure dielectric barrier discharge (AP-DBD) as a complex dissipative dynamical system exhibits rich temporal nonlinear and chaotic behavior. The intermittent chaos in AP-DBD, where the abrupt intensive currents break the regular periodic current oscillations during a discharge process, is studied by numerical simulation. By increasing the control parameter of applied voltage amplitude, random intensive currents occur more frequently, reducing the average laminar length (time interval of regular oscillations). It is found that the average laminar length depending on the control parameter follows the scaling law with an exponent of −0.5, implying a type-I intermittent discharge. This study further supplements and strengthens the nonlinear theory in plasma discharge.