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This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field [Formula: see text] (the set of integers modulo [Formula: see text]) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.

This paper is concerned with a reaction–diffusion–advection chemostat model with two species growing and competing for a single-limited resource. By taking the growth rates of the two species as variable parameters, we study the effect of growth rates on the dynamics of this system. It is found that there exist several critical curves, which may classify the dynamics of this system into three scenarios: (1) extinction of both species; (2) competitive exclusion; (3) coexistence. Moreover, we take numerical approaches to further understand the potential behaviors of the above critical curves and observe that the bistable phenomenon can occur, besides competitive exclusion and coexistence. To further study the effect of advection and diffusion on the dynamics of this system, we present the bifurcation diagrams of positive equilibrium solutions of the single species model and the two-species model with the advection rates and the diffusion rates increasing, respectively. These numerical results indicate that advection and diffusion play a key role in determining the dynamics of two species competing in a flow reactor.

Deoxyribonucleic Acid (DNA) coding technology is a new research field developed by the combination of computer science and molecular biology, that has been gradually applied in the field of image encryption in recent years. Furthermore, sensitivity to initial conditions, pseudo-random properties, and state ergodicity of coupled chaotic maps can help produce good pseudo-random number generators and meet the requirements of an image encryption system well. In this paper, an image encryption algorithm based on high-dimensional coupled chaotic maps and DNA coding is proposed. A pseudo-random sequence is generated by a long short-term memory (LSTM) architecture using the proposed maps and evaluated through a set of statistical tests to show the high performance of the proposed generator. All intensity values of an input image are converted to a binary sequence, which is scrambled globally by the high-dimensional coupled chaotic maps. The DNA operations are performed on the scrambled binary sequences instead of binary operations to increase the algorithm efficiency. Simulation results and performance analyses demonstrate that the proposed encryption scheme is extremely sensitive to small changes in secret keys, provides high security and can resist differential attack.

In this paper, by the example of 3D model of enzyme reaction, we study mechanisms of noise-induced generation of complex multimodal chaotic oscillations in the monostability zone where only simple deterministic cycles are observed. In such a generation, a constructive role of deterministic toroidal transients is revealed. We perform a statistical analysis of these phenomena and localize the intensity range of the noise that causes stochastic [Formula: see text]- and [Formula: see text]-bifurcations connected with transitions to chaos and qualitative changes in the probability density. Constructive possibilities of the stochastic sensitivity function technique in the analytical study of these phenomena are demonstrated.

The stochastic nature of magnetization dynamics of dipole–dipole interactions described by the Landau–Lifshitz–Gilbert equation without considering the Gilbert damping parameter is investigated. It is shown that the occurrence of the complex dynamic states depends on the spatial anisotropy of interactions on one hand and the lattice geometry on the other. It is observed from the higher-order moments of the magnetization fluctuations that two significant dynamical regimes, regular and chaos, may be obtained depending on the perturbation strength. Relying on the Hurst exponent obtained by the standard deviation principle, the correlation and persistence of the magnetization fluctuations are analyzed. The results also exhibit a transition from an anti-correlated to a positively correlated system as the relevant parameters of the system vary.

In this paper, we study a family of planar piecewise linear systems with saddles separated by two parallel lines, and mainly investigate the existence of four-intersection-point limit cycles. We provide complete conclusions on the existence of a special four-intersection-point limit cycle and a heteroclinic loop. And, based on these results, we give some sufficient conditions for the existence of general four-intersection-point limit cycles. Some examples are given to illustrate the main results.

During the last twenty years there has been increasing interest in studying the piecewise differential systems, mainly due to their many applications in natural science and technology. Up to now the most studied differential systems are in dimension two, here we study them in dimension three. One of the main difficulties for studying these differential systems consists in controlling the existence and nonexistence of limit cycles, and the numbers when they exist. In this paper, we study the nonsymmetric limit cycles for a family of three-dimensional piecewise linear differential systems with three zones separated by two parallel planes. For this class of differential systems we study the nonexistence, existence and uniqueness of their limit cycles.

Hidden chaotic attractors is a fascinating subject of study in the field of nonlinear dynamics. Jerk systems with a stable equilibrium may produce hidden chaotic attractors. This paper seeks to enhance our understanding of hidden chaotic dynamics in jerk systems of three variables [Formula: see text] with nonlinear terms from a predefined set: [Formula: see text], where [Formula: see text] is a real parameter. The behavior of the systems is analyzed using rigorous Hopf bifurcation analysis and numerical simulations, including phase portraits, bifurcation diagrams, Lyapunov spectra, and basins of attraction. For certain jerk systems with a subcritical Hopf bifurcation, adjusting the coefficient of a linear term can lead to hidden chaotic behavior. The adjustment modifies the subcritical Hopf equilibrium, transforming it from an unstable state to a stable one. One such jerk system, while maintaining its equilibrium stability, experiences a sudden transition from a point attractor to a stable limit cycle. The latter undergoes a period-doubling route to chaos, which may be followed by a reverse route. Therefore, by perturbing certain jerk systems with a subcritical Hopf equilibrium, we can gain insights into the formation of hidden chaotic attractors. Furthermore, adjusting the coefficient of the nonlinear term [Formula: see text] in certain systems with a stable equilibrium can also lead to period-doubling routes or reverse period-doubling routes to hidden chaotic dynamics. Both findings are significant for our understanding of the hidden chaotic dynamics that can emerge from nonlinear systems with a stable equilibrium.

Neuroendocrine system mainly consists of hypothalamus, anterior pituitary, and target organ. In this paper, a three-state-variable delayed Goodwin model with two Hill functions is considered, where the Hill functions with delays denote the hormonal feedback suppressions from target organ to hypothalamus and to anterior in the reproductive hormonal axis. The existence of Hopf bifurcation shows the circadian rhythms of neuroendocrine system. The direction and stability of Hopf bifurcation are also analyzed using the normal form theory and the center manifold theorem for functional differential equations. Furthermore, based on the sparse identification algorithm, it is verified that the transient time series generated from the delayed Goodwin model cannot be equivalently presented by ordinary differential equations from the viewpoint of data when considering that a library of candidates are at most cubic terms. The reason is because the solution space of delayed differential equations is of infinite dimensions. Finally, we report that reservoir computing can predict the periodic behaviors of the delayed Goodwin model accurately if the size of reservoir and the length of data used for training are large enough. The predicting performances are evaluated by the mean squared errors between the trajectories generated from the numerical simulations and the reservoir computing.

Reservoir computing (RC) has attracted much attention as a brain-like neuromorphic computing algorithm for time series processing. In addition, the hardware implementation of the RC system can significantly reduce the computing time and effectively apply it to edge computing, showing a wide range of applications. However, many hardware implementations of RC use different hardware to implement standard RC without further expanding the RC architecture, which makes it challenging to deal with relatively complex time series tasks. Therefore, we propose a bidirectional hierarchical light reservoir computing method using optoelectronic memristors as the basis for the hardware implementation. The approach improves the performance of hardware-implemented RC by allowing the memristor to capture multilevel temporal information and generate a variety of reservoir states. Ag[Formula: see text]GQDs[Formula: see text]TiOx[Formula: see text]FTO memristors with negative photoconductivity effects can map temporal inputs nonlinearly to reservoir states and are used to build physical reservoirs to accomplish higher-speed operations. The method’s effectiveness is demonstrated in multivariate time series classification tasks: a predicted accuracy of 98.44[Formula: see text] is achieved in voiceprint recognition and 99.70[Formula: see text] in the mobile state recognition task. Our study offers a strategy for dealing with multivariate time series classification issues and paves the way to developing efficient neuromorphic computing.

The construction of multidimensional discrete hyperchaotic maps with ergodicity and larger Lyapunov exponents is desired in cryptography. Here, we have designed a general [Formula: see text]D ([Formula: see text]) discrete hyperchaotic map ([Formula: see text]D-DHCM) model that can generate any nondegenerate [Formula: see text]D chaotic map with Lyapunov exponents of desired size through setting the control matrix. To verify the effectiveness of the [Formula: see text]D-DHCM, we have provided two illustrative examples and analyzed their dynamic behavior, and the results demonstrated that their state points have ergodicity within a sufficiently large interval. Furthermore, to address the finite precision effect of the simulation platform, we analyzed the relationship between the size of Lyapunov exponent and the randomness of the corresponding state time sequence of the [Formula: see text]D-DHCM. Finally, we designed a keyed parallel hash function based on a 6D-DHCM to evaluate the practicability of the [Formula: see text]D-DHCM. Experimental results have demonstrated that [Formula: see text]D discrete chaotic maps constructed using [Formula: see text]D-DHCM have desirable Lyapunov exponents, and can be applied to practical applications.

The analysis of time series is essential in many fields, and reservoir computing (RC) can provide effective temporal processing that makes it well-suited for time series analysis and prediction tasks. In this study, we introduce a new discrete memristor model and a corresponding two-dimensional hyperchaotic map with complex dynamic properties that are well-suited for reservoir computing. By applying this map to the RC, we enhance the state richness of the reservoir, resulting in improved performance. The paper evaluates the performance of the proposed RC approach using time series data for sunspot, exchange rate, and solar-E forecasting tasks. Our experimental results demonstrate that this approach is highly effective in handling temporal data with both accuracy and efficiency. And comparing with other discrete memristive chaotic maps, the proposed map is the best for improving the RC performance. Furthermore, the proposed RC model is characterized by a simple structure that enables it to fully exploit the time-dependence of the state values of the hyperchaotic map.

We apply the [Formula: see text]-equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of [Formula: see text]-equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described.

Let [Formula: see text] be a piecewise increasing mapping satisfying some generalized convexity condition, so that it possesses an invariant density that is a decreasing function. We show that this invariant density can be computed by a family of Markov finite approximations that preserve the monotonicity of integrable functions. We also construct a quadratic spline Markov method and demonstrate its merits numerically.

Feed-forward effect strongly modulates collective behavior of a multiple-layer neuron network and usually facilitates synchronization as signals are propagated to deep layers. However, a full synchronization of neuron system corresponds to functional disorder. In this work, we focus on a network containing two layers as the simplest model for multiple layers to investigate pattern selection during interaction between two layers. We first confirm that the chimera state emerges in layer 1 and it also induces chimera in layer 2 when the feed-forward effect is strong enough. A cluster is discovered as a transient state which separates full synchronization and chimera state and occupy a narrow region. Second, both feed-forward and back-forward effects are considered and we discover chimera states in both layers 1 and 2 under the same parameter for a large range of parameters selection. Finally, we introduce adaptive dynamics into inter-layer rather than intra-layer couplings. Under this circumstance, chimera state can still be induced and coupling matrix will be self-organized under suitable phase parameter to guarantee chimera formation. Indeed, chimera, cluster and synchronization can propagate from one layer to another in a regular multiple network for a corresponding parameter selection. More importantly, adaptive coupling is proved to control pattern selection of neuron firing in a network and this plays a key role in encoding scheme.

This paper aims to study the bifurcation structures of the homographic [Formula: see text]-Ricker maps in a four-dimensional parameter space. The generalized Lambert [Formula: see text] functions are used to establish upper bounds for the number of fixed points of these population growth models. The variation of the number of fixed points and the cusp points organization is stipulated. This study also observes a vital characteristic on the Allee effect phenomenon in a class of bimodal Allee’s maps. Some numerical studies are included to illustrate the Allee effect and big bang local bifurcations.

In this paper, a perturbed quintic BBM equation with weak backward diffusion and dissipation effects is investigated. By applying geometric singular perturbation theory and analyzing the perturbations of a Hamiltonian system with a hyper-elliptic Hamiltonian of degree six, we prove the existence of isolated periodic wave solutions with certain wave speed in an open interval. It is also shown that isolated periodic wave solutions persist for any energy parameter [Formula: see text] in an open interval under small perturbation. Furthermore, we prove that the wave speed [Formula: see text] of periodic wave is strictly monotonically increasing with respect to [Formula: see text] by analyzing Abelian integral having three generating elements. Moreover, the upper and lower bounds of the limiting wave speed are obtained. Our analysis is mainly based on Melnikov theory, Chebyshev criteria, and symbolic computation, which may be useful for other problems.

In this paper, we propose a reaction–diffusion mathematical model augmented with self/cross-diffusion in 2D domain which describes the oncolytic virotherapy treatment of a tumor with its growth following the logistic law. The tumor cells are divided into uninfected and infected cells and the virus transmission is supposed to be in a direct mode (from cell to cell). In the absence of cross-diffusion, we establish well posedness of the problem, non-negativity and boundedness of solutions, nonexistence of positive solutions, local and global stability of the nontrivial steady-state and the nonoccurrence of Turing instability. In the presence of cross-diffusion, we prove the occurrence of Turing instability by using the cross-diffusion coefficient of infected cells as a parameter. To have an idea about different patterns, we derive the corresponding amplitude equation by using the nonlinear analysis theory. In the end, we perform some numerical simulations to illustrate the obtained theoretical results.

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

Assessing the complexity of signals or dynamical systems is important in disease diagnosis, mechanical system defect, astronomy analysis, and many other fields. Although entropy measures as complexity estimators have greatly improved, the majority of these measures are quite sensitive to specified parameters and are impacted by short data lengths. This paper proposes a novel entropy algorithm to enhance the existing complexity assessment methods based on classical dispersion entropy (DE) and Rényi entropy (RE) by introducing refined composite multiscale coarse-grained treatment and phase transformation. The proposed refined composite multiscale phase Rényi dispersion entropy (PRRCMDE) addresses the flaws of various existing entropy approaches while still incorporating their merits. Several simulated signals from logistic mapping, AR model, MIX process, and additive WGN periodic signals are adopted to examine the performance of PRRCMDE from multiple perspectives. It demonstrates that the efficacy of the suggested algorithm can be increased by modifying the DE and RE parameters to a reasonable range. As a real-world application, the bearings’ varied fault types and levels can also be recognized clearly.

In this paper, we consider a delayed single population model with patch structure, and investigate Hopf bifurcations when the dispersal rate or the scaling parameter of the growth rate is small. The effect of dispersal patterns is analyzed. We show that dispersal patterns affect the occurrence of Hopf bifurcations when the scaling parameter of the growth rate is small, and affect the values of Hopf bifurcations when the dispersal rate is small.

In this paper, we study a class of piecewise smooth near-Hamiltonian systems with piecewise polynomial perturbations. We first give the expression of the first order Melnikov function, and then estimate the number of limit cycles bifurcated from periodic annuluses by Melnikov function method. In addition, we discuss the number of limit cycles that can appear simultaneously near both sides of a generalized homoclinic or generalized double homoclinic loop.

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected(I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) into an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes., Comment: 21 pages, 9 figures

This work studies a three-dimensional predator–prey model with gestation delay and stage structure between aphidophagous coccinellids and aphid pests, where the interaction between mature coccinellids and aphids is inscribed by Crowley–Martin functional response function, and immature coccinellids and aphids act in the form of Holling-I type. We prove the positivity and boundedness of the solution of the nondelayed system and analyze its equilibrium point, local asymptotic stability, and global stability. In addition to the delays, the critical values of Hopf bifurcation occurring for different parameters are also found from the numerical simulation. The stability of the delayed system and Hopf bifurcation with different delays as parameters are also discussed. Our model analysis shows that the time delay essentially governs the system’s dynamics, and the stability of the system switches as delays increase. We also investigate the direction and stability of the Hopf bifurcation using the normal form theory and center manifold theorem. Finally, we perform computer simulations and depict diagrams to support our theoretical results.

This paper proposes a new nonlinear transverse-torsional coupled model for single-stage gear transmission system, by taking transmission error, time-varying meshing stiffness, backlash, bearing clearances and the self-adaptive double-parameter control module into account. The nonlinear differential governing equation of system motion is derived and solved by applying variable step-size Runge–Kutta numerical integration method. The system’s nonlinear dynamic characteristics and stability are investigated systematically by a bifurcation diagram of the Poincaré map and parameter stability region. Firstly, the velocity bifurcation diagrams have shown that, under the same damping ratio and backlash and with the increase of control parameter [Formula: see text], the route to chaos in the subcritical velocity region is first experienced from crisis to periodic doubling, and to crisis again, but the route that reverts to periodic motion in the super-critical velocity region is not affected. Additionally, the backlash is found to be the key parameter to affect the route to chaos as well. With the increase of the backlash, the crisis becomes the unique route to chaos in sub-critical region no matter what the [Formula: see text] is, but the increase of [Formula: see text] could change the route that reverts to periodic motion from 3T-periodic attractor to 2T-periodic attractor. Secondly, with the increase of the control parameter [Formula: see text], the system starts to enter the chaotic motion and exit the chaos state at different critical points and through different routes. Besides, the unstable region could shrink dramatically and the route to crisis is suppressed as well with the increase of damping ratio. Thirdly, the motion stability region analysis established in full range of double-parameter and velocity provides a mathematical reference model and is stored in control module, which could be utilized to make the control module seek a nearest parameter set automatically that could make the motion stable again in the quickest way under unstable working condition. Finally, according to global motion stability diagram, the forbidden zones that cannot make the system motion stable by adjusting single control parameter are revealed, which has remarkable guiding value during the practical operation especially under the manual adjusting working condition.

Periodicity analysis of sequences generated by a deterministic system is a long-standing challenge in both theoretical research and engineering applications. To overcome the inevitable degradation of the logistic map on a finite-precision circuit, its numerical domain is commonly converted from a real number field to a ring or a finite field. This paper studies the period of sequences generated by iterating the logistic map over ring [Formula: see text] from the perspective of its associated functional network, where every number in the ring is considered as a node, and the existing mapping relation between any two nodes is regarded as a directed edge. The complete explicit form of the period of the sequences starting from any initial value is given theoretically and verified experimentally. Moreover, conditions on the control parameter and initial value are derived, ensuring the corresponding sequences to achieve the maximum period over the ring. The results can be used as ground truth for dynamical analysis and cryptographical applications of the logistic map over various domains.

This paper considers the experimental and numerical study of an electromechanical arm powered by a DC motor and subjected to the action of permanent magnets. The magnetic torques arise from permanent magnets mounted at the free end of the arm and along a circle. The electrical subsystem is powered by two forms of input signal (DC and AC voltage sources). For each case, we determine the condition for complete rotation of the mechanical arm versus the parameters of the system such as the arm length, the number of magnets, and the frequency of the external signal. The nonlinear dynamics of the system is examined by means of time-histories, bifurcation diagrams, Lyapunov exponents and phase portraits. Chaotic and periodic dynamics are detected numerically and confirmed experimentally.

In this paper, we show that any switching hypersurface of n-dimensional continuous piecewise linear systems is an (n − 1)-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e., equilibria at the switching line) and singular continuum (i.e., continuum of non-isolated equilibria) between two parallel switching lines. The definition of the index of singular continuum is introduced. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines. 2010 Mathematics Subject Classification. Primary 34C05; 34A26; 49J52.

Detection of Stretching And Folding (SAF) traits in a time series is still controversial and of great interest. Also, visuo-manual tracking studies did not pay attention to SAF in hand motion trajectories. This research aims to find out the relevance of SAF to the discontinuities in chaotic dynamics of hand motion through target tracking tasks. Specifically, a new method is constructed based on this relation in which SAF can extract accurately trajectories in both time domain and phase space. Consequently, we designed experiments to track sinusoidal and trapezoidal target movements shown on a monitor. In these experiments, fourteen participants were instructed to move the joystick handle by wrist flexion-extension movements. Results confirm intermittency in significant human motor control behavior which results in discontinuities in hand motion trajectories. The relation between SAF and these discontinuities is realized by chaotic and intermittent behaviors of tracking dynamics. Verification of the method’s accuracy is also carried out by taking advantage of the Poincaré section. Our method can provide insight into the dynamical behaviors of chaotic and intermittent systems involving mechanisms in human motor control. It can be applied to general systems with intermittent behavior and other systems with modification.

Based on China’s green energy development strategy, this paper constructs a basic model of recycling and a channel expansion model for the circular economy foundation of Zaozhuang, Shandong Province, China. Through numerical simulation, it is found that each member of the supply chain should control the rate of price adjustment, otherwise it will cause market disruption. The model is controlled based on a chaos control method. Then, based on the fuzzy comprehensive evaluation method, an early warning system for the circular economy of Zaozhuang City is constructed. It is found that the economic development of Zaozhuang is a serious warning, resources are moderate warning, and the environment is not in an alarm state. In addition to paying attention to energy conservation and emission reduction of enterprises, the government should pay attention to creating awareness of energy conservation and emission reduction in society, and strengthen the technological investment in reducing pollutant emissions. This paper provides a strategic reference for the circular economy model in Zaozhuang, Shandong, China.

In this paper, a novel coupled code shifted M-ary differential chaos shift keying modulation scheme is proposed as a simple and low-cost solution to reference removal. In this scheme, only two information-bearing signals are transmitted. Both signals are modulated by selected Walsh code sequences with specific indices determined by the index bits, and one of them also carries an additional M-ary information symbol. Since these two information-bearing signals are coupled by the same chaotic message bearer, data bits can be easily recovered according to the correlation between these two signals so that the transmission of reference signals can be completely avoided. Theoretical Bit Error Rate (BER) expressions are derived over the additive white Gaussian noise and multipath Rayleigh fading channels. Furthermore, simulations and comparisons indicate that the proposed system can achieve better BER performance with lower complexity.

Image encryption is an effective way to protect images in secure transmission or storage. In this paper, we propose a novel double color image visual encryption algorithm based on the improved Chebyshev map (ICM) and compressed sensing. Firstly, a new nonlinear term is introduced into the classical one-dimensional Chebyshev map, and then the ICM is used to generate the secret code stream for the encryption algorithm. Next, the key-controlled sensing measurement matrices are constructed through the ICM, and they are used to compress the integer wavelet coefficients of two plain images. Subsequently, the compressed images are dislocated by dislocation matrices and diffused by an ICM-generated diffusion matrix, respectively. Finally, the encrypted images are embedded into the carrier image using the least significant bit embedding algorithm. Experimental results demonstrate that the proposed method has good visual safety, large key space, and high key sensitivity.

Since the invention of Chua’s circuit, numerous generalizations based on the substitution of the nonlinear function have been reported. One of the generalizations is obtained by substituting cubic nonlinearity for piece-wise linear (PWL) nonlinearity. Although hidden chaotic attractors with a PWL nonlinearity have been discovered in the classical Chua’s circuit, chaotic attractors with a smooth cubic nonlinearity have long been known as self-excited attractors. Through a systematically exhaustive computer search, this paper identifies coexisting hidden attractors in the smooth cubic Chua’s circuit. Either self-excited or coexisting hidden attractors are now possible in the smooth cubic Chua’s circuit with algebraically elegant values of both initial points and system parameters. The newly found coexisting attractors exhibit an inversion symmetry. Both initial points and system parameters are equally required to localize hidden attractors. Basins of attraction of individual equilibria are illustrated and clearly show critical areas of multistability where a tiny drift in an initial point potentially induces jumps among different basins of attraction and coexisting states. Such multistability poses potential threats to engineering applications. The dynamical regions of hidden and self-excited attractors, and areas of stability of equilibria, are illustrated against two parameter spaces. Both illustrations reveal that two nonzero equilibrium points of the smooth cubic Chua’s circuit have a transition from unstable to stable equilibrium points, leading to generations of self-excited and hidden attractors simultaneously.

A dynamic modeling method for Multistage Planetary Gear Transmission (MPGT) is proposed based on the concept of integral planetary gearbox modeling. The integrated interaction of multiple nonlinear parameters is considered in the dynamic model. The time-varying mesh stiffness of each gear pair is calculated by the energy method. The effects of input torque, gear backlash, and meshing damping on the chaos and impact characteristics of the system are analyzed in detail. The results show that the dynamic behavior of the system is closely related to the Dynamic Meshing Force (DMF). When the system is in the states of chaos, bifurcation, and jumping, the DMF fluctuates violently, and the stability and reliability of the system are seriously affected. With the increase of input torque and meshing damping, the system exits chaos through the inverse period-doubling bifurcation path, which indicates that increasing the input torque and meshing damping can suppress the chaotic motion and enhance the stability of the system. The backlash has a significant effect on the nonlinear behavior and meshing impact characteristics of the system. When the backlash is small, the system is in bilateral impact, and the meshing impact tends to be stable as the backlash increases. In order to improve the vibration characteristics of the system, a slightly larger backlash is necessary. The results can be used to guide the dynamic characteristics design and vibration control of the MPGT.

It is important for functional neurons of animals or human beings to adapt to external stimuli, such as sound, pressure, and light. Regarding this aspect, autaptic neuron enables itself to utilize historical information to modulate its instant dynamics, such that it may be able to behave adaptively. In this paper, a FitzHugh–Nagumo based autaptic neuron is employed to investigate the capability of a sound-sensitive neural circuit’s adaptation and filtering to analog acoustic signals. Extensive simulations are performed for excitatory and inhibitory types of autaptic neurons. The results show that the time-delayed feedback of the excitatory chemical autapse can be tuned to play the role of a narrow-band filter in response to a broadband acoustic signal. While the excitatory chemical autaptic neuron cannot saturate its response amplitude due to its positive feedback gain, the inhibitory chemical autapse can drive the neuron’s amplitude to converge as the intensity of external drive increases, which reveals the capability of adaptation. What’s more, the inhibitory chemical autaptic neuron can also exhibit a novel bursting adaptation, in which the number of spikings contained in one bursting changes as the electrical activity evolves. For electrical autaptic neurons, it is also found that both time-delay feedback gains can effectively modulate the response of neuron to acoustic signal. While the variation of time-lags mainly changes the spiking rates of the excitatory electrical autaptic neuron, the feedback gain alters its response amplitude. Lastly, by carefully tuning the time-lags, the expected subthreshold dynamics for larger inhibitory feedback gains can be switched to nearby quasi-periodic firings, which implies a competing relation between the time-delays and the feedback gains in the spiking dynamics of the inhibitory electrical autaptic neurons. The diverse emerging phenomena are expected to facilitate the design of online or interactive learning artificial neural networks with these functional autaptic neurons.

In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.

This paper considers the discontinuous characteristics of a real aero-engine rotor system, that is, the existence of bolted connection characteristics, and establishes a new bolted connection rotor system model. Taking into account the bending stiffness and the nonlinear Hertzian contact force of the rolling bearing, the Newmark-[Formula: see text] numerical method is used to solve the system response, and the influence of the bending stiffness on the system is studied. Moreover, the effects of bending stiffness and eccentricity on the system dynamics are analyzed. The results show that the nonlinear phenomena of the system are more abundant and the critical speed of the system is higher when the bending stiffness is involved. With the increase of bending stiffness, the critical speed of the system increases, and the frequency component of the system becomes more complex. Then, the influence of eccentricity on the system is studied based on the bending stiffness. It is found that the greater the eccentricity, the greater the critical speed of the rotor and the greater the amplitude of the main frequency. In the case of the same eccentricity, the main frequency increases as the rotational speed increases, and the frequency doubling component appears in the 2-period motion. This paper provides a basis for predicting the nonlinear response of bolted rotor-bearing system.

The generation and evolution of chaotic motions in a hybrid Rayleigh–Van der Pol–Duffing oscillator driven by parametric and amplitude-modulated excitation forces are investigated analytically and numerically. By using the Melnikov method, the conditions for the appearance of horseshoe chaos in our system are derived in the case where the modulation frequency [Formula: see text] and the forcing frequency [Formula: see text] are the same [Formula: see text]. The obtained results show that the chaotic region decreases and increases in certain ranges of frequency. The numerical simulations based on the basin of attraction of initial conditions validate the obtained analytical predictions. It is also found that in the case where [Formula: see text] is irrational, the increase of amplitude-modulated force accentuates the fractality of the basin of attraction. The global dynamical changes of our model are numerically examined. It is found that our model displays a rich variety of bifurcation phenomena and remarkable routes to chaos. In addition, the presence of the hybrid Rayleigh–Van der Pol damping force reduces the chaotic domain in the absence of amplitude-modulated force. But when the amplitude-modulated force acts on the system, the chaotic oscillations decrease and disappear. Further, the geometric shape of the chaotic attractors considerably decreases in the presence of the amplitude-modulated excitation force. On the other hand, the system presents transient chaos, torus-chaos and torus of different topologies when [Formula: see text] is irrational.

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Vector-borne disease malaria is transmitted to humans by arthropod vectors (mosquitoes) and contributes significantly to the global disease burden. TV and social media play a key role to disseminate awareness among people by broadcasting awareness programs. In this paper, a nonlinear model is formulated and analyzed in which cumulative number of advertisements through TV and social media is taken as dynamical variable that propagates awareness among people to control the prevalence of vector-borne disease. The human population is partitioned into susceptible, infected and aware classes, while the vector population is divided into susceptible and infected classes. Humans become infected and new cases arise when bitten by infected vectors (mosquitoes) and susceptible vectors get infected as they bite infected humans. The feasibility of equilibria is justified and their stability conditions are discussed. A crucial parameter, basic reproduction number, which measures the disease transmission potentiality is obtained. Bifurcation analysis is performed by varying the sensitive parameters, and it is found that the proposed system shows different kinds of bifurcations, such as transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, etc. The analysis of the model shows that reduction in vector population due to intervention of people of aware class would not efficiently reduce the infective cases, rather we have to minimize the transmission rates anyhow, to control the disease outbreak.

The Recurrence Quantification Analysis (RQA), a pattern recognition-based time series analysis method, can be successfully utilized for short, nonstationary, nonlinear, and chaotic time series. These RQA measures quantify several properties of time series, including predictability, regularity, stability, randomness, and complexity. In this regard, first, we analyzed the intraday seasonality with RQA and demonstrated how RQA measures change among the intraday periods by using 160 million row matched orders of 100 shares from Borsa Istanbul Equity Market between 2019M10 and 2020M02. We selected 50 stocks from the BIST50 Index group and 50 stocks from outside of the BIST100 Index group. Since these two share groups exhibit similar intraday RQA seasonality, our results are robust. Second, we explained intraday volatility with RQA measures and found a relationship between RQA measures and intraday volatility using a regression model.

The paper presents a very compact dual memelement function simulator using only one active building block (ABB) namely modified Voltage Differencing Current Conveyor (mVDCC), two MOSFETs, and two grounded passive elements. The proposed emulator can realize the function of memristor, meminductor, and memcapacitor-dual, which can be achieved via the proper selection of only one grounded passive element as R, L, and C. The proposed multimemelement emulator (MME) is fully electronically tunable and exhibits nonvolatile storage property. Also, the emulator can exhibit memristor response up to MHz range of frequency. The PSPICE-generated simulation results verify the working of the given floating MME for the realization of all three elements using 0.18 [Formula: see text]m CMOS technology node. The presented CMOS layout shows that the proposed emulator implementation occupies an area of [Formula: see text]. Along with the CMOS-based structure, the presented MME is verified through commercial ICs-based implementation. The given application example of the chaotic circuit also proves the working of the presented MME.

The Investment Savings-Liquidity preference Money supply (IS-LM) model is represented as a graph depicting the intersection of products and the money market. It elaborates how an equilibrium of money supply versus interest rates may keep the economy in control. In this paper, we combine the basic business cycle IS-LM model with Kaldor’s growth model in order to create an augmented model. The IS-LM model, when coupled with a certain economics expansion (in our instance, the Kaldor–Kalecki Business Cycle Model), provides a comprehensive description of a developing but robust economy. Right after the introduction of capital stock into the system, it cannot be employed and also, while making some investment choices, this requires some time in execution, which ultimately alters resources, i.e. capital. Thus, in the capital accumulation, we will be incorporating double time delays in Gross product and Capital Stock. These time delays represent the time periods during which investment decisions were made and executed and the time spent in order for the capital to be put to productive use. After formulating a mathematical model using delayed differential equations, dynamic functioning of the system around equilibrium point is examined where three instances appeared based on time delays. These cases are: when both delays are not in action, when only one delay is in action and when both delays are in action. It is shown that time delay affects the stability of the equilibrium point and, as the delay crosses a critical point, Hopf bifurcation exists. It is observed that by using Kaldor type investment function, the delay residing in capital stock only will destabilize in less time as compared to when both the delays are present in the system. The system is sensitive to certain parameters which is also analyzed in this work.

Vibration impact is often used in the piezoelectric energy harvesting (PEH) system to increase the effective bandwidth of the harvester. Viscoelastic materials have been used successfully to mitigate vibration problems in various types of mechanical systems such as buildings, cars, aircraft and industrial equipment. However, less research has been done on the energy harvesting system with impact and viscoelastic force driven by random excitation. Stochastic response of an impact PEH system with viscoelastic force under Gaussian white noise excitation is investigated in this paper. Firstly, by transforming the variables, viscoelastic force can be substituted with the stiffness and damping terms to get an approximately equivalent system without viscoelastic term. Secondly, the approximate analytical solutions are acquired by the stochastic averaging method and nonsmooth coordinate transformation. The validity of this theoretical approach is confirmed by comparing the analytical solutions with the numerical solutions derived from the Monte Carlo method. Then, the effect of noise intensity and nonlinear damping coefficient on the stochastic response of the system is discussed. It is concluded that the restitution coefficient, viscoelastic component, relaxation time and linear damping coefficient can induce the occurrence of stochastic P-bifurcation. Finally, the roles of system parameters on the mean square voltage and average output power of the energy harvester are investigated respectively.

This paper designs a locally active memristor with two variable parameters based on Chua’s unfolding theorem. The dynamical behavior of the memristor is analyzed by employing pinched hysteresis loop, power-off plot (POP), DC V–I curve, small-signal analysis, and edge-of-chaos theory. It is found that the proposed memristor exhibits nonvolatile and bistable behaviors because of coexisting pinched hysteresis loops. And the variable parameters can realize the rotation of the coexisting pinched hysteresis loops, regulate the range of the locally active region and even transform the shape of the DC V–I curve into S-type or N-type. Furthermore, a simple oscillation circuit is constructed by connecting this locally active memristor with an inductor, a capacitor, a resistance, and a bias voltage. It is shown by analysis that the memristive circuit can generate complex nonlinear dynamics such as multiscroll attractor, initial condition-based dynamics switching, transient phenomenon with the same dynamical state but different offsets and amplitudes, and symmetric coexisting attractors. The measurement observed from the implementation circuit further verifies the numerical results of the oscillation circuit.

Recently, a chaotic secure video communication system based on FPGA is proposed, which essentially encrypts the original image data of each frame in the video in multiple rounds by using the 6D self-synchronizing chaotic stream cipher. In this paper, the security performance of the multiround encryption algorithm is analyzed based on divide-and-conquer attack. In the case of one round encryption, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 30 appropriate initial conditions according to known-plaintext attack. After that, according to chosen-ciphertext attack, the chaotic iterative equation of the decryption end is degenerated into a linear one if ciphertexts are all set to zero. Under this condition, the equations between the unknown keys and the plaintext-ciphertext pairs can be obtained by setting 24 appropriate initial conditions, and the keys [Formula: see text] [Formula: see text] can be deciphered by solving the equations in Mathematica. Secondly, in order to reduce the multiplication times of the unknown keys in the case of two-round encryption, the number of pixels in an image can be selected as little as possible. According to chosen-ciphertext attack, when the number of the pixels in an image is only one, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 30 appropriate initial conditions. Then, when the number of pixels in an image is two, the keys [Formula: see text] [Formula: see text] can be deciphered by setting 24 appropriate initial conditions. However, in the case that the encryption round is three or more, the keys cannot be deciphered by the attack method proposed in this paper. The analysis results show that the multiround image encryption algorithm is not secure when the number of encryption rounds is one or two.

This paper introduces a new 3D conservative chaotic system. The oscillator preserves the energy over time, according to the Kaplan–Yorke dimension computation. It has a line of unstable equilibrium points that are investigated with the help of eigenvalues and also numerical analysis. The bifurcation diagrams and the corresponding Lyapunov exponents show various behaviors, for example, chaos, limit cycle, and torus with different parameters. Other dynamical properties, such as Poincaré section and basin of attraction, are investigated. Additionally, an oscillator’s electrical circuit is designed and implemented to demonstrate its potentiality.

The literature posits that an introduced predator population is able to drive its target pest population to extinction, if supplemented with high quality additional food of sufficient quantity. We show this approach actually leads to infinite time blow-up of the predator population, so is unpragmatic as a pest management strategy. We propose an alternate model in which the additional food induces predator competition. Analysis of this model indicates that depending on the competition parameter [Formula: see text], one can have global stability of the pest-free state, bistability dynamics, or up to three interior equilibria. As [Formula: see text] and the additional food quantity [Formula: see text] are varied standard codimension one and codimension two bifurcations are observed. We also use structural symmetries to construct several nonstandard bifurcations such as saddle-node-transcritical bifurcation (SNTC) in codimension two and a cusp-transcritical bifurcation (CPTC), also in codimension two. We further use symmetry to construct a novel pitchfork-transcritical bifurcation (PTC) in codimension two, thus explicitly characterizing a new organizing center of the model. Our findings indicate that increasing additional food in predator–pest models can hinder bio-control, contrarily to some of the literature. However, additional food that also induces predator competition, leads to rich dynamics and enhances bio-control.

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.