Transient Chaos Research Articles

Transient chaos analysis of string scattering

It has long been thought that a highly excited string can be regarded as a black hole: the correspondence principle between strings and a black hole, while recent studies found that black holes are characterized by chaos. This suggests that highly excited strings are the source of the black hole chaoticity. We study the chaoticity of a string amplitude where a tachyon is scattered by a highly excited string. Our strategy to extract the chaos in the amplitude is a generalization of the transient chaos analysis for classical scattering. We look for the fractal structure in the plots of incoming/outgoing scattering angles, where the outgoing angle is defined as the maximum pole of the amplitude. Within our strategy, we could not identify any fractal structure in the scattering data. We also discuss other possible setups and strategies to extract the chaos, hoping that our present work serves as a step toward the formulation of chaos in string scattering amplitudes.

A New Four-Dimensional Chaotic System with Multistability and Its Predefined-Time Synchronization

A new chaotic system is obtained by modifying the Sprott-C system. Then the phase diagrams, power spectra, 0–1 tests, Poincaré maps, Lyapunov exponential spectra, time sequences, and complexity are studied. Research indicates that the system is sensitive to parameters and initial conditions, and bursting oscillation, transient chaos and multistability are investigated. The complexity of the new system is calculated using the Sample Entropy (SE) complexity algorithm, including selecting more suitable initial values and parameters for the application. In addition, the system circuit designed by Multisim and the actual digital circuit realized by Field Programmable Gate Array (FPGA) verify the feasibility of the system. Finally, to obtain a more appropriate synchronization result for practical applications, a synchronous controller is designed to successfully implement the predefined-time synchronization of the system in different dimensions. The simulation results demonstrate that the predefined-time synchronization can control the synchronous time and is unaffected by the initial conditions. The results demonstrate that this synchronization method is well accommodated to practical applications.

Dynamics of stimuli-based fractional-order memristor-coupled tabu learning two-neuron model and its engineering applications

External stimulus has an impact on the functional behavior of the biological nervous system, and appropriate stimulus helps the organism to maintain neural function. Inspired by this, the effects of different external stimuli on the dynamical behaviors of neuron model are studied in this paper. Firstly, a fractional-order (FO) memristor-coupled tabu learning two-neuron model whose equilibrium points are symmetric about the origin and unstable is constructed. The dynamical behaviors of this neuron model under different stimuli are further discussed, which are no external stimulus, external forced current stimulus, and electromagnetic radiation (EMR), respectively. The neuron model without external stimulus has periodic attractors and transient chaos, but applying external stimulus to one of the neurons can produce abundant two-scroll chaotic attractors and multistability; especially, when the neuron model is stimulated by EMR, it can generate hyperchaotic attractors that have not been observed in the tabu learning neuron model before. Besides, the transient transition behaviors of the model under different stimuli are also studied. Then, a pseudo-random number generator is designed and its random performance is tested with NIST suite. Finally, it is applied to voice encryption, and the result shows that it has good encryption effect. Therefore, it can be said that the FO memristor-coupled tabu learning two-neuron model has superior randomness and is suitable for chaotic-based engineering applications.

Prediction and diagnostics of crises and critical states in an unusual vibro-impact system with soft impact

In nonlinear dynamical systems, whenchanging control parameters, critical states may occur. To predict these undesirable events, it is necessary to study the system dynamic behavior with changing in different control parameters. Two-body 2-DOF vibro-impact system is under consideration. The huge mass of the upper body, which can break away from the lower one and be in “free flight” for a short time, may be attributed to its specific and is a cause to consider it unusual. Some changes in the parameters of both external loading and the system itself can increase the efficiency of the machine, the mathematical model of which is the considered vibro-impact system. But the same changes lead to appearance of unwanted nonlinear phenomena, such as permanent sustained chaos, transient chaos, crisis-induced intermittency, and coexisting regimes with larger oscillatory amplitudes. The article shows their occurrence, traces the precursors, carries out their diagnostics by both traditional and less common methods. The description of many numerical experiments is accompanied by numerous figures and tables that clearly and convincingly demonstrate the results obtained.

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.

Complex Dynamics and Sliding Bifurcations of the Filippov Lorenz–Chen System

In this paper, we propose a Filippov switching model which is composed of the Lorenz and Chen systems. By employing the qualitative analysis techniques of nonsmooth dynamical systems, we show that the new Filippov system not only inherits the properties of the Lorenz and Chen systems but also presents new dynamics including new chaotic attractors such as four-wing butterfly attractor, Lorenz attractor with sliding segments, etc. In particular, we find that different new attractors can coexist such as the coexistence of two-point attractors and chaotic attractor, the coexistence of two-point attractors and quasi-periodic solution, the coexistence of transient transition chaos and quasi-periodic solution. Furthermore, nonsmooth bifurcations and numerical analyses reveal that the proposed Filippov system has a series of new sliding bifurcations including a symmetric pair of sliding mode bifurcations, a symmetric pair of sliding Hopf bifurcations, and a symmetric pair of Hopf-like boundary equilibrium bifurcations.

Two-Neuron Based Memristive Hopfield Neural Network with Synaptic Crosstalk

Synaptic crosstalk is an important biological phenomenon that widely exists in neural networks. The crosstalk can influence the ability of neurons to control the synaptic weights, thereby causing rich dynamics of neural networks. Based on the crosstalk between synapses, this paper presents a novel two-neuron based memristive Hopfield neural network with a hyperbolic memristor emulating synaptic crosstalk. The dynamics of the neural networks with varying memristive parameters and crosstalk weights are analyzed via the phase portraits, time-domain waveforms, bifurcation diagrams, and basin of attraction. Complex phenomena, especially coexisting dynamics, chaos and transient chaos emerge in the neural network. Finally, the circuit simulation results verify the effectiveness of theoretical analyses and mathematical simulation and further illustrate the feasibility of the two-neuron based memristive Hopfield neural network hardware.

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.

Chaotic instability in the BFSS matrix model

Chaotic scattering is a manifestation of transient chaos realized by the scattering with non-integrable potential. When the initial position is taken in the potential, a particle initially exhibits chaotic motion, but escapes outside after a certain period of time. The time to stay inside the potential can be seen as lifetime and this escape process may be regarded as a kind of instability. The process of this type exists in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model in which the potential has flat directions. We discuss this chaotic instability by reducing the system with an ansatz to a simple dynamical system and present the associated fractal structure. We also show the singular behavior of the time delay function and compute the fractal dimension. This chaotic instability is the basic mechanism by which membranes are unstable, which is also common to supermembranes at quantum level.

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.

Noise-induced stabilization of the FitzHugh-Nagumo neuron dynamics: Multistability and transient chaos.

The nonlinear dynamics of a FitzHugh-Nagumo (FHN) neuron driven by an oscillating current and perturbed by a Gaussian noise signal with different intensities D is investigated. In the noiseless case, stable periodic structures [Arnold tongues (ATS), cuspidal and shrimp-shaped] are identified in the parameter space. The periods of the ATSs obey specific generating and recurrence rules and are organized according to linear Diophantine equations responsible for bifurcation cascades. While for small values of D, noise starts to destroy elongations ("antennas") of the cuspidals, for larger values of D, the periodic motion expands into chaotic regimes in the parameter space, stabilizing the chaotic motion, and a transient chaotic motion is observed at the periodic-chaotic borderline. Besides giving a detailed description of the neuronal dynamics, the intriguing novel effect observed for larger D values is the generation of a regular dynamics for the driven FHN neuron. This result has a fundamental importance if the complex local dynamics is considered to study the global behavior of the neural networks when parameters are simultaneously varied, and there is the necessity to deal the intrinsic stochastic signal merged into the time series obtained from real experiments. As the FHN model has crucial properties presented by usual neuron models, our results should be helpful in large-scale simulations using complex neuron networks and for applications.

Design of heterogeneous time-lags system with multi-stability and its analog circuit

A novel heterogeneous time-lags chaotic system is constructed and explored in this work. By choosing appropriate control parameters, the state of this system can be switched between conservative and dissipative. When different initial conditions are selected, this system can produce multiple coexisting attractors or flows, that is, the system has the property of multi-stability. More interestingly, the attractors generated by this system can be distributed in 1-D lines, 2-D lattices and 3-D grids which means the circumstance of multi-directional attractors occurs. And the rare phenomenon of transient chaos is analyzed in this system as well. It is worth noting that the implementation of hardware circuits for time-lags chaotic systems has been a major obstacle in engineering applications. To remove this bottleneck and further explore the physical presence of this proposed system, the hardware analog circuit of the proposed system is achieved. The experimental observations further verify the feasibility of this system. Additionally, the experiments of spectral entropy complexity and image encryption performance verify the good performance of the proposed system.

Compound Chaotic Systems with Composite Attractors

Chaotic systems with either a self-excited attractor or a hidden attractor are prevalent in the research paradigm. But a composite attractor of a chaotic system that exhibits both self-excited attractors and hidden attractors with time is rare. Compounding of a chaotic system is devised to enhance the richness of the system as it exhibits different behaviors with time. This paper presents a two-stage technique for generating a compound-chaotic system by combining eight subsystems from Chameleon chaotic system. The first stage is the design of a nonautonomous piecewise function. The second stage is to generate a compound system using the designed function. The compound system exhibits a composite attractor that is a mix of self-excited and hidden attractors with time. The composite attractor is generated with the variation of the equilibrium points in the eight subsystems due to the designed function. The resultant attractor of the compound-chaotic system is termed as a composite attractor since it has both the self-excited and hidden attractors. Transient chaos and coexistence of attractors are also observed in the system. The proposed compound-chaotic system is analyzed using the available tools and the paper’s claims are validated.

Analysis and finite-time synchronization of a novel double-wing chaotic system with transient chaos

A chaotic system that can generate double-wing attractor is proposed. Through the analysis of Lyapunov exponents, bifurcation diagram, phase diagram and complexity, it is found that the system has rich dynamical phenomenon. The topological structure of the system will change with the change of parameters, which is analyzed by using two-parameter space. In addition, the novel system has phenomenon such as offset-boosting, transient chaos and coexisting attractors. The analog simulation circuit of the system is designed based on Multisim, and the actual digital circuit is realized with field programmable gate array (FPGA). The experimental phase diagram, numerical simulation results and simulation circuit results agree well, which prove the feasibility of the chaotic system. Finally, a controller is designed according to finite-time theory to achieve finite-time synchronization of systems with different structures. The simulation results are consistent with the theoretical analysis, which proves the feasibility of the controller and lays the foundation for the next application research.

Multistable dynamics in a Hopfield neural network under electromagnetic radiation and dual bias currents

This paper investigates a Hopfield neural network under the simulation of external electromagnetic radiation and dual bias currents, in which the fluctuation of magnetic flux across the neuron membrane is used to emulate the influence of electromagnetic radiation. Utilizing conventional analytical methods, the basic properties of the proposed Hopfield neural network are discussed. Due to the addition of electromagnetic radiation and dual bias currents, the Hopfield neural network shows high sensitivity to system parameters and initial conditions. The proposed Hopfield neural network possesses multistability with periodic attractor, quasi-periodic attractor, chaotic attractor and transient chaotic attractor, and all of the attractors are hidden attractors because there is no equilibrium point in the system. In particular, when the neuron membrane magnetic flux is different, the system can present transient chaos with different chaotic times. More interestingly, with the change of system parameters, the proposed Hopfield neural network can exhibit parallel bifurcation behaviors. Finally, the Multisim simulation and hardware experiment results based on discrete electronic components are conducted to support the numerical ones. These results could give useful information to the study of nonlinear dynamic characteristics of the Hopfield neural network.

Transient behaviors and equilibria-analysis-based boundary crisis analysis in a smooth 4D dynamical system

This paper addresses the transient behavior mechanism analysis of a new chaotic system. Interesting transient behaviors of the new system are analyzed under the attractor merging crisis. The transient chaos and transient quasi-periodic behaviors are observed, under the boundary crisis, when the system has two stable equilibria. Furthermore, the reason of the two transient behaviors is revealed, with the conditions and relevant threshold for the boundary crisis are found, by introducing a simplified method of Lorenz time series. The lifetime of transient quasi-periodicity and parameters are found to meet the power-law distribution and have long memory, which indicates that the lifetime of transient quasi-periodicity is predictable. Therefore, the transient quasi-periodicity has potential values in the field of weak signal amplitude prediction.

Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes

It is well known in quantum mechanics that the theory of quantum transitions is based on the convergence of the series of time-dependent perturbation theory. This series converges in atomic and nuclear physics. However, in molecular and chemical physics, this series converges only in the Born–Oppenheimer adiabatic approximation and due to the application of the Franck–Condon principle, and it diverges as a result of going beyond the adiabatic approximation and the Franck–Condon principle. This divergence (singularity) is associated with the incommensurability of the masses of light electrons and heavy nuclei which jointly participate in the full-fledged movement in the transient state of molecular “quantum” transitions. In a new physical theory—quantum–classical mechanics (Egorov, V.V. Heliyon Physics 2019, 5, e02579)—this singularity is damped by introducing chaos into the transient state. This transient chaos is introduced by replacing the infinitely small imaginary additive in the energy denominator of the spectral representation of the total Green’s function of the system with a finite value and is called dozy chaos. In this article, resonance at the nanoscale (nano-resonance) between electron and nuclear reorganization motions in the quantum–classical (dozy-chaos) mechanics of elementary electron transfers in condensed media and their applications to polymethine dyes and J-aggregates in solutions are reviewed. Nano-resonance explains the resonant character of the transformation of the shape of the optical absorption band in a series of polymethine dyes in which the length of the polymethine chain changes, as well as the nature of the red-shifted absorption band of the J-aggregates of polymethine dyes (J-band), which is narrow and intense. The process of dye aggregation in an aqueous solution with an increase in its concentration by the formation of J-aggregates is considered a structural tuning of the “polymethine dye + environment” system into resonance with light absorption. For J-aggregates in Langmuir films, the asymmetry of the luminescence and absorption bands, as well as the small value of their Stokes shifts, are explained. The parasitic transformation of the resonant shape of the optical absorption band of a polymethine dye in solution during the transition from one-photon to two-photon absorption is also explained, and the conditions for the restoration of this nano-resonance shape are predicted.

Noise-induced behavioral change driven by transient chaos

We study behavioral change in the context of a stochastic, non-linear consumption model with preference adjusting, interdependent agents. Changes in long-run consumption behavior are modelled as noise induced transitions between coexisting attractors. A particular case of multistability is considered: two fixed points, whose immediate basins have smooth boundaries, coexist with a periodic attractor, with a fractal immediate basin boundary. If a trajectory leaves an immediate basin, it enters a set of complexly intertwined basins for which final state uncertainty prevails. The standard approach to predicting transition events rooted in the stochastic sensitivity function technique due to Mil'shtein and Ryashko (1995) does not apply since the required exponentially stable attractor, for which a confidence region could be constructed, does not exist. To solve the prediction problem we propose a heuristic based on the idea that a vague manifestation of a non-attracting chaotic set (chaotic repellor) - could serve as a surrogate for an attractor. A representation of the surrogate is generated via an algorithm for generating the boundary of an absorbing area due to Mira et al. (1996). Then a confidence domain for the surrogate is generated using the approach due to Bashkirtseva and Ryashko (2019). The intersections between this confidence region and the immediate basins of the coexisting attractors can then be used to make predictions about transition events. Preliminary assessments show that the heuristic indeed explains the transition probabilities observed in numerical experiments.

Controlling chaotic transients in the Hénon and the Lozi map with the safety function

In this work, we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work, we will apply a new approach of the partial control method that has been recently developed. This new approach is based on finding a special function called the safety function which allows to automatically find the minimum control necessary to avoid the escape of the orbits. Furthermore, we will show the strong connection between the safety function and the classical safe set. To illustrate that, we will compute for the first time, safety functions for the two-dimensional Hénon and Lozi maps, where we also show the strong dependence of this function with the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits.

Multistability, transient chaos and hyperchaos, synchronization, and chimera states in wireless magnetically coupled VDPCL oscillators

Systems in nature are for the most part rich in connection, interaction, and communication in various ways that are generally complex. Synchronization is one of the main phenomenons associated with couple systems. Synchronization may be understood as a process where several systems adjust time scales of their oscillations or a given property of their motion over time due to their interactions. In compact designs, unconnected circuits may interact through the magnetic field. This interaction can be avoided or exploited in some cases. This work investigated the dynamics and the synchronization study of a set of magnetically coupled van der Pol oscillators coupled to a linear circuit (the VDPCL oscillator) in wireless interaction. The coupling coefficient between the coupled but unconnected oscillators is used as the only control parameter allowing investigating the dynamics and the synchronization of the oscillators. We first derive a mathematical model of the ring of the magnetically coupled oscillators. After this, some basic properties of the model are derived. The dynamical study of the model is carried out in the case of two and three sub-circuits including the phase synchronization, by using the well-known classical tools, that is, the bifurcation diagrams, the Lyapunov exponents, time series, and phase portraits. It is found that the system displays a rich catalog of behaviors such as regular, chaotic, and hyperchaotic (for a weak coupling) behaviors in terms of the control parameter. Moreover, it is found that the systems exhibit the coexistence of many attractors. The system is shown to exhibit global and partial synchronization (chimera state) in the case of three sub-circuits. Transient chaos/hyperchaos are also highlighted in the two sub-circuits cases. PSPICE based simulations and real laboratory measurements are included and are consistent with the theoretical/numerical results.